Quick recall · 448 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
PYQ · 2024
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The smallest irrational number by which \( \sqrt{20} \) should be multiplied so as to get a rational number, is:
D · \( \sqrt{5} \)
PYQ
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Let \( (x^{1/2}) \times (x^{-1/2}) \), find the correct answer.
B · B) 1
\( (x^{1/2}) \times (x^{-1/2}) = x^{1/2 + (-1/2)} = x^{0} = 1 \) (for x ≠ 0). Matches option B.[4]
PYQ
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A and B together have Rs. 1210. If \( \frac{15}{16} \) of A's amount is equal to \( \frac{4}{5} \) of B's amount, how much amount does B have?
D · Rs. 2000
PYQ
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Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?
A · 25:35:44
PYQ
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If a : b = 5 : 3, what percentage of 3a is (3a + 4b)?
C · 55%
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A. 36%
B. 2.3%
C. 1200%
D. 0.097%
Convert each of the following to a percentage: 0.36, 2.3, 12/1, 0.00097. Which option correctly converts 0.36?
A · 36%
PYQ
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What percent does he score in Maths, if he scores 60% marks in all the three subjects? Maximum Marks of Maths paper is 200.
C · 45%
PYQ
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A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is:
D · 8/15
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Which of the following sets includes all real numbers?
B · Rational numbers and Irrational numbers
Real numbers include all rational and irrational numbers, encompassing all possible decimal expansions.
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Which property of real numbers states that \( a + b = b + a \) for any real numbers \( a \) and \( b \)?
C · Commutative Property
The commutative property states that the order of addition or multiplication does not affect the result.
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If \( x \) is a real number such that \( x^2 = 16 \), which of the following must be true?
C · \( x = \pm 4 \)
Both 4 and -4 satisfy the equation \( x^2 = 16 \), so \( x = \pm 4 \).
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Which of the following is NOT a property of real numbers?
B · Existence of multiplicative inverse for zero
Zero does not have a multiplicative inverse in real numbers, so this property does not hold.
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Which of the following numbers is a rational number?
B · 0.3333... (repeating)
A repeating decimal like 0.3333... represents a rational number \( \frac{1}{3} \).
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Which of the following is an irrational number?
C · \( \sqrt{3} \)
The square root of 3 cannot be expressed as a ratio of two integers, so it is irrational.
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Which set of numbers is a subset of integers but not of whole numbers?
B · Negative integers
Negative integers are part of integers but not whole numbers, which include zero and positive integers only.
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Which of the following numbers is both a whole number and a natural number?
B · 1
Natural numbers are positive integers starting from 1, and whole numbers include 0 and natural numbers.
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Which of the following decimal expansions represents a non-terminating repeating decimal?
B · 0.3333...
0.3333... is a non-terminating repeating decimal with repeating digit 3.
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Which decimal number is a terminating decimal?
B · 0.75
0.75 has a finite number of decimal places and hence is terminating.
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Which of the following decimals is non-terminating and non-repeating?
C · 0.1010010001...
0.1010010001... does not repeat any pattern and continues indefinitely.
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The decimal representation of \( \frac{7}{8} \) is:
A · Terminating decimal
\( \frac{7}{8} = 0.875 \), which is a terminating decimal.
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If \( a = 3.5 \) and \( b = -2.1 \), what is \( a + b \)?
A · 1.4
Adding 3.5 and -2.1 gives 1.4.
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What is the product of \( \sqrt{2} \) and \( \sqrt{8} \)?
A · 4
\( \sqrt{2} \times \sqrt{8} = \sqrt{16} = 4 \).
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If \( x = 5 \) and \( y = 2 \), find \( \frac{x^2 - y^2}{x - y} \).
A · 7
Using difference of squares: \( \frac{25 - 4}{3} = \frac{21}{3} = 7 \).
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If \( a = -3 \) and \( b = 4 \), what is \( a \times b \)?
A · -12
Multiplying -3 and 4 gives -12.
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Simplify \( (2^3)^4 \).
A · 2^{12}
Using power of a power rule: \( (2^3)^4 = 2^{3 \times 4} = 2^{12} \).
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If \( a eq 0 \), simplify \( \frac{a^5}{a^2} \).
A · a^3
Using quotient rule: \( a^{5-2} = a^3 \).
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Simplify \( (3^2 \times 3^3)^2 \).
A · 3^{10}
First, \( 3^2 \times 3^3 = 3^{5} \), then \( (3^5)^2 = 3^{10} \). Correction: Actually \( (3^5)^2 = 3^{10} \). So correct answer is 3^{10}.
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If \( a^{m} \times a^{n} = a^{12} \) and \( m = 7 \), what is \( n \)?
A · 5
Using product rule: \( m + n = 12 \) so \( n = 12 - 7 = 5 \).
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According to Euclid's division lemma, for integers \( a = 29 \) and \( b = 5 \), the quotient and remainder are respectively:
A · 5 and 4
Dividing 29 by 5 gives quotient 5 and remainder 4 since \( 29 = 5 \times 5 + 4 \).
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Use Euclid's division lemma to find the remainder when 123 is divided by 7.
A · 4
Dividing 123 by 7: \( 7 \times 17 = 119 \), remainder \( 123 - 119 = 4 \). Correction: remainder is 4, so correct answer is 4.
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If \( a = 17 \) and \( b = 5 \), which of the following correctly expresses Euclid's division lemma?
A · \( 17 = 5 \times 3 + 2 \)
Dividing 17 by 5 gives quotient 3 and remainder 2, but 5*3+2=17, so option A is correct. Correction: Option A is correct, so answer is A.
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Prove using Euclid's division lemma that the square of any integer is of the form \( 3m \) or \( 3m + 1 \). Which of the following is a valid example supporting this?
D · Both A and C
Squares of integers modulo 3 are either 0 or 1, so forms \( 3m \) or \( 3m + 1 \).
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The prime factorization of 360 is:
A · \( 2^3 \times 3^2 \times 5 \)
360 = 2 × 2 × 2 × 3 × 3 × 5 = \( 2^3 \times 3^2 \times 5 \).
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Which of the following is a prime factorization of 210?
A · \( 2 \times 3 \times 5 \times 7 \)
210 = 2 × 3 × 5 × 7, all prime numbers.
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Find the highest common factor (HCF) of 48 and 180 using prime factorization.
A · 12
48 = \( 2^4 \times 3 \), 180 = \( 2^2 \times 3^2 \times 5 \). HCF = \( 2^2 \times 3 = 12 \).
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If the prime factorization of two numbers are \( 2^3 \times 3^2 \) and \( 2^2 \times 3^3 \), their LCM is:
A · \( 2^3 \times 3^3 \)
LCM takes highest powers: \( 2^3 \times 3^3 \).
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The HCF of 36 and 48 is:
A · 12
Factors of 36: 1,2,3,4,6,9,12,18,36; Factors of 48: 1,2,3,4,6,8,12,16,24,48; Highest common factor is 12.
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The LCM of 15 and 20 is:
A · 60
LCM of 15 and 20 is 60.
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If HCF of two numbers is 6 and their LCM is 72, and one number is 18, find the other number.
A · 24
Product of numbers = HCF × LCM = 6 × 72 = 432. Other number = 432 / 18 = 24.
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Which of the following is an irrational number?
B · \( \sqrt{5} \)
\( \sqrt{5} \) cannot be expressed as a ratio of two integers, hence irrational.
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Which of the following statements about irrational numbers is true?
D · Irrational numbers have non-terminating, non-repeating decimal expansions
Irrational numbers have decimal expansions that neither terminate nor repeat.
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Which of the following is NOT true about irrational numbers?
D · They include all integers
Integers are rational numbers, not irrational.
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Which of the following statements about density of rational and irrational numbers is correct?
A · Between any two rational numbers, there is at least one irrational number
Both rational and irrational numbers are dense in real numbers, meaning between any two numbers of either type, there exists a number of the other type.
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Between 0 and 1, which of the following is true?
C · There are infinitely many rational and irrational numbers
Both rational and irrational numbers are infinite and dense between any two real numbers.
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Which of the following best describes the density property of real numbers?
B · Between any two real numbers, there is at least one rational and one irrational number
The density property states that between any two real numbers, there are infinitely many rational and irrational numbers.
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Which point on the number line represents \( \sqrt{2} \)?
C · Between 1.41 and 1.42
\( \sqrt{2} \approx 1.4142 \), which lies between 1.41 and 1.42 on the number line.
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Which of the following numbers lies exactly at the midpoint between 2 and 3 on the number line?
B · 2.5
Midpoint between 2 and 3 is \( \frac{2 + 3}{2} = 2.5 \).
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Which number lies between \( \frac{1}{3} \) and \( \frac{2}{3} \) on the number line?
A · \( \frac{1}{2} \)
\( \frac{1}{2} = 0.5 \) lies between 0.333... and 0.666... on the number line.
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On the number line, which of the following represents the rationalization of \( \frac{1}{\sqrt{3}} \)?
A · \( \frac{\sqrt{3}}{3} \)
Rationalizing denominator: \( \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).
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Rationalize the denominator of \( \frac{5}{\sqrt{2} + 1} \).
A · \( \frac{5(\sqrt{2} - 1)}{1} \)
Multiply numerator and denominator by conjugate \( (\sqrt{2} - 1) \) to rationalize denominator.
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Which of the following is the rationalized form of \( \frac{3}{2 + \sqrt{5}} \)?
A · \( \frac{3(2 - \sqrt{5})}{-1} \)
Multiply numerator and denominator by conjugate \( (2 - \sqrt{5}) \) to rationalize denominator.
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Find the remainder when 1234 is divided by 9.
A · 1
Sum of digits = 1+2+3+4=10; 10 mod 9 = 1; But remainder is actually 1234 - 9*137 = 1234 - 1233 = 1. So correct answer is 1.
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Find the least number which when divided by 7 leaves remainder 3 and when divided by 5 leaves remainder 1.
A · 31
Number satisfies \( n \equiv 3 \pmod{7} \) and \( n \equiv 1 \pmod{5} \). 31 satisfies both.
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A number leaves remainder 2 when divided by 3 and remainder 3 when divided by 4. What is the smallest such number?
A · 11
Number satisfies \( n \equiv 2 \pmod{3} \) and \( n \equiv 3 \pmod{4} \). 11 satisfies both.
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If a number when divided by 6 leaves a remainder 4, and when divided by 8 leaves remainder 6, find the smallest such number.
D · 38
Number satisfies \( n \equiv 4 \pmod{6} \) and \( n \equiv 6 \pmod{8} \). 38 satisfies both.
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Find the square root of 625.
A · 25
625 is a perfect square, \( 25^2 = 625 \).
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The cube root of 27 is:
A · 3
\( 3^3 = 27 \), so cube root of 27 is 3.
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Which of the following is the cube root of 125?
A · 5
\( 5^3 = 125 \), so cube root is 5.
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If \( x^2 = 50 \), find \( x \).
A · \( \pm 5\sqrt{2} \)
\( x = \pm \sqrt{50} = \pm 5\sqrt{2} \).
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Let \(a\) and \(b\) be positive real numbers such that \(a^{\log_b a} = b^{\log_a b}\). If \(a = 3^{\sqrt{2}}\) and \(b = 3^{\sqrt{3}}\), find the value of \(\log_a b + \log_b a\).
D · 1
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If \(x\) and \(y\) are real numbers such that \(x + y = 2\) and \(x^{\sqrt{2}} + y^{\sqrt{2}} = 2^{\sqrt{2}}\), find the value of \(x^{\sqrt{3}} + y^{\sqrt{3}}\).
B · \(2^{\sqrt{3}}\)
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Let \(r\) be a positive real number such that \(\sqrt[3]{r} + \sqrt{r} = 5\). Find the value of \(r^{\frac{5}{6}} - r^{\frac{1}{2}}\).
A · 10
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If \(x, y > 0\) satisfy \(x^{\log_y x} = y^{\log_x y} = 16\), find the value of \(\log_x y + \log_y x\).
B · 2
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If \(a, b > 0\) and \(a^{\log_b a} = 81\), \(b^{\log_a b} = 243\), find the value of \(\frac{\log_a b}{\log_b a}\).
C · 1
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If \(a, b > 0\) satisfy \(a^{\log_b a} = 27\) and \(b^{\log_a b} = 9\), find the value of \(a^{\log_a b} + b^{\log_b a}\).
B · 16
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If \(x\) and \(y\) are positive real numbers such that \(x^{\log_y x} = y^{\log_x y} = 32\) and \(\log_x y = m\), find the value of \(m + \frac{1}{m}\).
A · 5
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If \(a, b > 0\) satisfy \(a^{\log_b a} = b^{\log_a b} = 64\) and \(\log_a b = k\), find the value of \(k^3 + \frac{1}{k^3}\).
A · 66
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Let \(a, b > 0\) satisfy \(a^{\log_b a} = 25\) and \(b^{\log_a b} = 125\). If \(\log_a b = m\), find the value of \(m + \frac{1}{m}\).
A · 3
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If \(a, b > 0\) satisfy \(a^{\log_b a} = 81\) and \(b^{\log_a b} = 27\), find the value of \(\log_a b \times \log_b a\).
A · 1
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If \(a, b > 0\) satisfy \(a^{\log_b a} = 16\) and \(b^{\log_a b} = 8\), find \(\log_a b + \log_b a\).
A · 3
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If \(x, y > 0\) satisfy \(x^{\log_y x} = 64\) and \(y^{\log_x y} = 16\), find the value of \(\log_x y \times \log_y x\).
A · 1
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If \(a, b > 0\) satisfy \(a^{\log_b a} = 100\) and \(b^{\log_a b} = 10\), find the value of \(\log_a b + \log_b a\).
A · 3
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Which of the following correctly represents the product law of exponents for \( a^m \times a^n \)?
A · \( a^{m+n} \)
The product law states that when multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
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If \( x eq 0 \), what is the value of \( \frac{x^5}{x^2} \)?
A · \( x^{3} \)
Using the quotient law of exponents: \( \frac{x^5}{x^2} = x^{5-2} = x^3 \).
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Simplify \( (2^3)^4 \).
A · \( 2^{12} \)
Power of a power law: \( (a^m)^n = a^{mn} \). So, \( (2^3)^4 = 2^{3 \times 4} = 2^{12} \).
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Which law of exponents justifies the equality \( a^0 = 1 \) for \( a eq 0 \)?
D · Zero exponent law
The zero exponent law states that any non-zero number raised to the zero power equals 1: \( a^0 = 1 \).
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If \( 3^x = 1 \), what is the value of \( x \)?
A · 0
Any non-zero number raised to the power 0 is 1, so \( x = 0 \).
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Simplify \( 5^{-2} \).
A · \( \frac{1}{25} \)
Negative exponent law: \( a^{-n} = \frac{1}{a^n} \). So, \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \).
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Evaluate \( \left( \frac{2}{3} \right)^0 \).
A · 1
Any non-zero number raised to the zero power is 1.
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Simplify \( \frac{4^{-3}}{2^{-5}} \).
A · \( 2^4 \)
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Simplify \( \left( x^3 y^2 \right)^4 \).
A · \( x^{12} y^{8} \)
Power of a product law: \( (ab)^n = a^n b^n \). So, \( (x^3 y^2)^4 = x^{3 \times 4} y^{2 \times 4} = x^{12} y^{8} \).
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Simplify \( \left( \frac{a^5 b^{-2}}{a^{-3} b^4} \right)^2 \).
A · \( a^{16} b^{-12} \)
First simplify inside the bracket:\( \frac{a^5 b^{-2}}{a^{-3} b^4} = a^{5 - (-3)} b^{-2 - 4} = a^{8} b^{-6} \).Now raise to power 2:\( (a^{8} b^{-6})^{2} = a^{16} b^{-12} \).
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Simplify \( \left( 3x^2 y^{-1} \right)^3 \).
A · \( 27 x^{6} y^{-3} \)
Apply power to each factor:\( 3^3 = 27 \), \( (x^2)^3 = x^{6} \), \( (y^{-1})^3 = y^{-3} \). So, \( 27 x^{6} y^{-3} \).
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Simplify \( \left( \frac{2x^3}{y^2} \right)^2 \).
A · \( \frac{4x^{6}}{y^{4}} \)
Square numerator and denominator:\( 2^2 = 4 \), \( (x^3)^2 = x^{6} \), \( (y^2)^2 = y^{4} \).
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Express \( 0.00056 \) in scientific notation.
A · \( 5.6 \times 10^{-4} \)
Move decimal 4 places right: \( 0.00056 = 5.6 \times 10^{-4} \).
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Which of the following is the standard form of \( 3.2 \times 10^{5} \)?
A · 320,000
Multiply 3.2 by \( 10^{5} = 100,000 \) to get 320,000.
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Express \( 7.5 \times 10^{-3} \) in decimal form.
A · 0.0075
Move decimal 3 places left: \( 7.5 \times 10^{-3} = 0.0075 \).
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Which number is larger: \( 4.5 \times 10^{6} \) or \( 4.5 \times 10^{5} \)?
A · \( 4.5 \times 10^{6} \)
Since \( 10^{6} > 10^{5} \), \( 4.5 \times 10^{6} \) is larger.
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Express \( 0.000123 \) in scientific notation.
A · \( 1.23 \times 10^{-4} \)
Move decimal 4 places right: \( 0.000123 = 1.23 \times 10^{-4} \).
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Simplify \( \frac{(2^3)^4 \times 2^{-5}}{2^6} \).
A · \( 2^{1} \)
Calculate numerator:\( (2^3)^4 = 2^{12} \), so numerator is \( 2^{12} \times 2^{-5} = 2^{7} \).Divide by denominator \( 2^{6} \):\( \frac{2^{7}}{2^{6}} = 2^{7-6} = 2^{1} \).
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Simplify \( \left( 3^2 \times 3^{-4} \right)^3 \).
A · \( 3^{-6} \)
Inside bracket: \( 3^{2} \times 3^{-4} = 3^{2-4} = 3^{-2} \).Raise to power 3: \( (3^{-2})^{3} = 3^{-6} \).
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Simplify \( \frac{(5^3)^2}{5^4} \).
A · \( 5^{2} \)
Numerator: \( (5^3)^2 = 5^{6} \).Divide by \( 5^4 \): \( 5^{6-4} = 5^{2} \).
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Simplify \( \left( \frac{2^4 \times 3^2}{6^3} \right) \).
A · 1
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Simplify \( 16^{\frac{3}{4}} \).
A · 8
Rewrite: \( 16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3 = 8 \) since \( 16^{\frac{1}{4}} = 2 \).
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Simplify \( 27^{\frac{2}{3}} \).
A · 9
Rewrite: \( 27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2 = 3^2 = 9 \).
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Simplify \( \sqrt[3]{8x^6} \).
A · \( 2x^{2} \)
Cube root of 8 is 2, and \( \sqrt[3]{x^6} = x^{6/3} = x^{2} \).
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Simplify \( (81)^{\frac{3}{4}} \).
A · 27
Rewrite: \( 81^{\frac{3}{4}} = (81^{\frac{1}{4}})^3 = 3^3 = 27 \).
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Simplify \( \left( 16x^{8} \right)^{\frac{1}{2}} \).
A · \( 4x^{4} \)
Square root of 16 is 4, and \( (x^{8})^{\frac{1}{2}} = x^{4} \).
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If \( 2^x = 32 \), find \( x \).
A · 5
Since \( 32 = 2^5 \), \( x = 5 \).
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If \( 9^{x} = 27 \), find \( x \).
A · \( \frac{3}{2} \)
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If \( (2^x)(4^{x+1}) = 32 \), find \( x \).
B · 1
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Which of the following is the odd one out based on the value of the power?
D · \( 5^{1} \)
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Which of the following has the greatest value?
B · \( 4^{2.5} \)
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Identify the correct comparison: \( 2^{5} \) and \( 4^{3} \).
C · \( 2^{5} < 4^{3} \)
Calculate values:\( 2^{5} = 32 \), \( 4^{3} = 64 \). So, \( 2^{5} < 4^{3} \).
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Which of the following best defines an exponent in the expression \( a^n \)?
B · The number of times the base is multiplied by itself
In \( a^n \), the exponent \( n \) indicates how many times the base \( a \) is multiplied by itself.
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If \( 5^3 = 125 \), what does the exponent 3 represent?
B · The number of times 5 is multiplied by itself
The exponent 3 means that 5 is multiplied by itself 3 times: \( 5 \times 5 \times 5 = 125 \).
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Which of the following statements is true for any non-zero number \( a \) and integers \( m, n \)?
A · \( a^{m+n} = a^m \times a^n \)
The law of exponents states \( a^{m+n} = a^m \times a^n \).
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Simplify \( (2^3)^4 \).
A · \( 2^{12} \)
Using the power of a power law: \( (a^m)^n = a^{mn} \), so \( (2^3)^4 = 2^{3 \times 4} = 2^{12} \).
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Which of the following is equal to \( \frac{a^5}{a^2} \) for \( a eq 0 \)?
B · \( a^3 \)
Using the division law of exponents: \( \frac{a^m}{a^n} = a^{m-n} \), so \( \frac{a^5}{a^2} = a^{5-2} = a^3 \).
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Simplify \( a^0 \) where \( a eq 0 \).
B · 1
Any non-zero number raised to the zero power is 1, i.e., \( a^0 = 1 \).
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What is the value of \( 4^{-2} \)?
A · \( \frac{1}{16} \)
Negative exponent means reciprocal: \( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \).
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Simplify \( \left( \frac{3}{2} \right)^0 \).
B · \( 1 \)
Any non-zero number raised to the zero power is 1.
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Evaluate \( 9^{-\frac{1}{2}} \).
B · \( \frac{1}{3} \)
Fractional exponent \( -\frac{1}{2} \) means reciprocal of square root: \( 9^{-\frac{1}{2}} = \frac{1}{\sqrt{9}} = \frac{1}{3} \).
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Express \( 4.5 \times 10^6 \) in standard decimal form.
A · 4500000
Multiplying by \( 10^6 \) moves the decimal point 6 places to the right: \( 4.5 \times 10^6 = 4500000 \).
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Which of the following represents the number \( 0.00032 \) in scientific notation?
A · \( 3.2 \times 10^{-4} \)
Moving the decimal 4 places to the right gives \( 3.2 \times 10^{-4} \).
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Multiply \( (3 \times 10^4) \times (2 \times 10^3) \).
A · \( 6 \times 10^7 \)
Multiply coefficients: \( 3 \times 2 = 6 \), add exponents: \( 4 + 3 = 7 \), so \( 6 \times 10^7 \).
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Express \( (5 \times 10^{-3}) \div (2 \times 10^{-5}) \) in scientific notation.
A · \( 2.5 \times 10^{2} \)
Divide coefficients: \( 5/2 = 2.5 \), subtract exponents: \( -3 - (-5) = 2 \), so \( 2.5 \times 10^{2} \).
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Simplify \( \left( 2 \times 10^3 \right)^2 \).
A · \( 4 \times 10^6 \)
Square the coefficient: \( 2^2 = 4 \), multiply exponent by 2: \( 3 \times 2 = 6 \), so \( 4 \times 10^6 \).
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Simplify \( 3^4 \times 3^2 \).
B · \( 3^6 \)
Add exponents when multiplying same base: \( 3^{4+2} = 3^6 \).
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Simplify \( \frac{7^5}{7^3} \).
B · \( 7^2 \)
Subtract exponents when dividing same base: \( 7^{5-3} = 7^2 \).
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Simplify \( (5^2)^3 \).
B · \( 5^6 \)
Power of a power: multiply exponents \( 2 \times 3 = 6 \), so \( 5^6 \).
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Simplify \( \frac{(2^3)^2}{2^4} \).
A · \( 2^2 \)
Numerator: \( (2^3)^2 = 2^{3 \times 2} = 2^6 \). Then \( \frac{2^6}{2^4} = 2^{6-4} = 2^2 \).
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Simplify \( 16^{\frac{3}{4}} \).
A · \( 8 \)
Rewrite \( 16 = 2^4 \), so \( 16^{3/4} = (2^4)^{3/4} = 2^{4 \times \frac{3}{4}} = 2^3 = 8 \).
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Simplify \( 27^{\frac{2}{3}} \).
A · \( 9 \)
Rewrite \( 27 = 3^3 \), so \( 27^{2/3} = (3^3)^{2/3} = 3^{3 \times \frac{2}{3}} = 3^2 = 9 \).
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Simplify \( 81^{-\frac{1}{2}} \).
A · \( \frac{1}{9} \)
Rewrite \( 81 = 9^2 \), so \( 81^{-1/2} = (9^2)^{-1/2} = 9^{-1} = \frac{1}{9} \).
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Simplify \( (16^{\frac{1}{4}})^3 \).
A · \( 8 \)
First, \( 16^{1/4} = 2 \), then \( 2^3 = 8 \). But since the expression is \( (16^{1/4})^3 = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8 \). So correct answer is 8.
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Simplify \( 2^3 \times 3^3 \).
A · \( 6^3 \)
Since \( 2^3 \times 3^3 = (2 \times 3)^3 = 6^3 \).
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Simplify \( \frac{5^4 \times 2^3}{10^3} \).
A · \( 5 \)
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Simplify \( \left( \frac{3^2 \times 4^3}{6^2} \right) \).
A · \( 8 \)
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If \( x^3 = 27 \), what is the value of \( x^{\frac{2}{3}} \)?
A · 9
Since \( x^3 = 27 \), \( x = 3 \). Then \( x^{2/3} = (x^3)^{2/3} = 27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9 \).
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A bacteria population doubles every hour. If the initial population is \( P_0 \), express the population after \( t \) hours using exponents.
A · \( P_0 \times 2^t \)
Population doubles every hour, so after \( t \) hours population is \( P_0 \times 2^t \).
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If \( 5^{x} = 125 \), find \( 5^{x-1} \).
A · 25
Since \( 5^x = 125 = 5^3 \), so \( x = 3 \). Then \( 5^{x-1} = 5^{3-1} = 5^2 = 25 \).
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The volume of a cube is given by \( V = s^3 \), where \( s \) is the side length. If the volume is increased by a factor of 8, by what factor does the side length increase?
A · 2
Volume scales as cube of side length. If volume increases by 8, side length increases by \( \sqrt[3]{8} = 2 \).
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If \( (x^2 y^3)^4 = x^a y^b \), what are the values of \( a \) and \( b \)?
A · \( a=8, b=12 \)
Apply power to each exponent: \( (x^2)^4 = x^{8} \), \( (y^3)^4 = y^{12} \).
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Simplify \( \frac{x^5 y^2}{x^2 y^4} \).
A · \( x^3 y^{-2} \)
Subtract exponents: \( x^{5-2} y^{2-4} = x^3 y^{-2} \).
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If \( a^3 = 8 \) and \( b^3 = 27 \), what is the value of \( (ab)^2 \)?
A · 36
Since \( a = 2 \), \( b = 3 \), \( (ab)^2 = (2 \times 3)^2 = 6^2 = 36 \).
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Compare the numbers \( 2^5 \) and \( 3^3 \). Which is greater?
A · \( 2^5 > 3^3 \)
\( 2^5 = 32 \), \( 3^3 = 27 \), so \( 2^5 > 3^3 \).
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Arrange the following in ascending order: \( 5^2, 2^5, 3^3 \).
A · \( 5^2 < 3^3 < 2^5 \)
Calculate values: \( 5^2=25 \), \( 3^3=27 \), \( 2^5=32 \). Ascending order: 25, 27, 32.
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Which of the following is the smallest number?
A · \( 10^{0.5} \)
Calculate approximate values: \( 10^{0.5} = \sqrt{10} \approx 3.16 \), \( 2^3=8 \), \( 3^2=9 \), \( 5^1=5 \). Smallest is \( 3.16 \).
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Let \(a, b\) be positive real numbers such that \(a^{\log_b a} = b^{\log_a b}\). If \(a = 3^{\sqrt{2}}\) and \(b = 3^{\sqrt{3}}\), find the value of \(\left(a^{\log_b b^3}\right)^{\log_a b}\).
A · 27
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If \(x\) and \(y\) are positive real numbers satisfying \(x^{\log_y x} = y^{\log_x y} = 16\), find the value of \(\left(x^{\log_x y} \cdot y^{\log_y x}\right)^{\log_{xy} 4}\).
B · 256
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Assertion (A): For positive real numbers \(p, q\), \(\left(p^{\log_q p}\right)^{\log_p q} = q^{\log_p q}\).
Reason (R): The expression \(p^{\log_q p}\) equals \(q^{\log_p q}\) for all positive \(p, q eq 1\).
D · A is false but R is true
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If \(x > 0\) satisfies \(x^{\log_x 81} = 27^{\log_3 x}\), find the value of \(x^{\log_9 3}\).
A · 9
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If \(a, b > 0\) satisfy \(a^{\log_b a} = b^{\log_a b} = k\), where \(k > 0\), prove that \(\log_a b + \log_b a = 2\) and find \(k\) in terms of \(a\) and \(b\).
A · \(\log_a b + \log_b a = 2\) and \(k = a b\)
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Find the value of \(\left(4^{\log_2 9} \cdot 9^{\log_3 8}\right)^{\log_6 3}\).
B · 64
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If \(x^{\log_2 3} = 9\) and \(x^{\log_3 2} = 16\), find the value of \(x^{\log_6 12}\).
D · 128
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If \(a^{\log_b c} = b^{\log_c a} = c^{\log_a b} = 64\), where \(a, b, c > 0\), find the value of \(\log_a b + \log_b c + \log_c a\).
A · 3
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If \(x^{\log_5 2} = 8\) and \(x^{\log_2 5} = 25\), find \(x^{\log_{10} 4}\).
C · 64
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If \(x^{\log_x 81} = 9^{\log_3 x}\), find the value of \(x^{\log_9 3}\).
A · 3
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If \(a, b > 1\) satisfy \(a^{\log_b a} = b^{\log_a b} = 16\), find the value of \(a^{\log_a b} + b^{\log_b a}\).
B · 8
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If \(x^{\log_3 5} = 25\) and \(x^{\log_5 3} = 9\), find the value of \(x^{\log_{15} 5}\).
A · 5
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If \(x^{\log_4 9} = 27\) and \(x^{\log_9 4} = 81\), find \(x^{\log_6 3}\).
B · 27
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If \(x^{\log_7 5} = 25\) and \(x^{\log_5 7} = 49\), find \(x^{\log_{35} 5}\).
A · 5
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If \(a^{\log_b c} = b^{\log_c a} = c^{\log_a b} = 8\), where \(a, b, c > 0\), find the value of \(abc\).
B · 512
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Which of the following correctly defines a ratio?
A · A comparison of two quantities by division
A ratio is defined as the comparison of two quantities by division, expressed as \( a:b \) or \( \frac{a}{b} \).
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If the ratio of two numbers is 3:5, which of the following is a property of this ratio?
A · It can be expressed as \( \frac{3}{5} \)
Ratios can be expressed as fractions representing division of the first quantity by the second, such as \( \frac{3}{5} \).
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If \( \frac{a}{b} = \frac{c}{d} \), which of the following is true?
A · The ratios \( a:b \) and \( c:d \) are in proportion
If \( \frac{a}{b} = \frac{c}{d} \), then \( a:b \) and \( c:d \) are said to be in proportion.
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Which property of proportion states that the product of means equals the product of extremes in \( a:b = c:d \)?
A · Cross multiplication property
In a proportion \( a:b = c:d \), the cross multiplication property states \( a \times d = b \times c \).
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If \( \frac{2}{3} = \frac{x}{9} \), what is the value of \( x \)?
A · 6
By cross multiplication, \( 2 \times 9 = 3 \times x \) so \( x = 6 \).
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Which of the following is an example of a part-to-whole ratio?
A · Ratio of boys to total students in a class
Part-to-whole ratio compares a part to the whole quantity, such as boys to total students.
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In a class of 40 students, 25 are boys and 15 are girls. What is the ratio of boys to girls?
A · 5:3
Ratio of boys to girls is \( 25:15 = 5:3 \).
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Refer to the diagram below showing a rectangle divided into two parts: Part A and Part B.
If the total length is 20 cm and Part A is 12 cm, what is the part-to-whole ratio of Part A to the rectangle?
B · 12:20
Part-to-whole ratio is \( 12:20 \).
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If \( a, b, c \) are in continued proportion, which of the following is true?
A · \( \frac{a}{b} = \frac{b}{c} \)
In continued proportion, the ratio of the first to the second equals the ratio of the second to the third.
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If 4, x, 16 are in continued proportion, find \( x \).
A · 8
Since \( \frac{4}{x} = \frac{x}{16} \), cross multiply to get \( x^2 = 64 \), so \( x = 8 \).
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Refer to the diagram below showing three segments \( a, b, c \) in continued proportion.
If \( a = 3 \) and \( c = 12 \), what is \( b \)?
A · 6
Since \( \frac{a}{b} = \frac{b}{c} \), \( \frac{3}{b} = \frac{b}{12} \) so \( b^2 = 36 \) and \( b = 6 \).
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If \( x \) is the mean proportion between 9 and 16, what is the value of \( x \)?
A · 12
Mean proportion \( x \) satisfies \( \frac{9}{x} = \frac{x}{16} \), so \( x^2 = 144 \) and \( x = 12 \).
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Which of the following statements about mean proportion is true?
A · Mean proportion between \( a \) and \( b \) is \( \sqrt{ab} \)
Mean proportion between two numbers \( a \) and \( b \) is the geometric mean \( \sqrt{ab} \).
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Refer to the diagram below showing two numbers \( a \) and \( b \) and their mean proportion \( x \).
If \( a = 25 \) and \( b = 36 \), find \( x \).
A · 30
Mean proportion \( x = \sqrt{25 \times 36} = \sqrt{900} = 30 \).
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A recipe requires ingredients in the ratio 2:3:5. If the total quantity is 100 kg, how much of the second ingredient is needed?
A · 30 kg
Total parts = 2 + 3 + 5 = 10. Second ingredient = \( \frac{3}{10} \times 100 = 30 \) kg.
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If \( x:y = 4:5 \) and \( y:z = 3:7 \), find the compound ratio \( x:z \).
A · 12:35
Compound ratio \( x:z = \frac{4}{5} \times \frac{3}{7} = \frac{12}{35} \).
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Refer to the bar diagram below showing two ratios \( a:b = 2:3 \) and \( b:c = 4:5 \).
Find the compound ratio \( a:c \).
A · 8:15
Compound ratio \( a:c = \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \).
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If \( x \) is directly proportional to \( y \) and \( x = 10 \) when \( y = 5 \), what is \( x \) when \( y = 8 \)?
A · 16
Since \( x \propto y \), \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \). So \( \frac{10}{5} = \frac{x}{8} \), \( x = 16 \).
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If \( x \) is inversely proportional to \( y \) and \( x = 6 \) when \( y = 4 \), find \( x \) when \( y = 8 \).
A · 3
Since \( x \propto \frac{1}{y} \), \( x_1 y_1 = x_2 y_2 \). So \( 6 \times 4 = x \times 8 \), \( x = 3 \).
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Refer to the pie chart below showing the distribution of expenses in ratio 3:2:5 for Rent, Food, and Miscellaneous.
If total expenses are \( \$1000 \), what is the amount spent on Food?
A · \$200
Total parts = 3 + 2 + 5 = 10. Food = \( \frac{2}{10} \times 1000 = 200 \).
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A quantity of 180 liters is divided in the ratio 3:4:5. What is the largest share?
D · 72 liters
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If \( x:y = 5:7 \), divide 96 in the ratio \( x:y \). What is the value of \( y \)'s share?
A · 56
Total parts = 5 + 7 = 12. \( y \)'s share = \( \frac{7}{12} \times 96 = 56 \).
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Refer to the bar diagram below representing division of 120 units in ratio 2:3:5.
What is the value of the smallest part?
A · 24
Total parts = 2 + 3 + 5 = 10. Smallest part = \( \frac{2}{10} \times 120 = 24 \).
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A car travels 150 km in 3 hours. How far will it travel in 5 hours at the same speed?
A · 250 km
Distance is directly proportional to time. \( \frac{150}{3} = \frac{x}{5} \) so \( x = 250 \) km.
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If 6 workers can complete a job in 10 days, how many days will 15 workers take to complete the same job, assuming all work at the same rate?
A · 4 days
Workers and days are inversely proportional: \( 6 \times 10 = 15 \times x \) so \( x = 4 \) days. Correct answer is 4 days, so option A is correct.
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Which of the following is a correct example of inverse proportion?
A · Speed and time taken for a fixed distance
Speed and time are inversely proportional for a fixed distance: if speed increases, time decreases.
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If the compound ratio of \( 2:3 \) and \( 4:5 \) is \( x:y \), what is the value of \( x + y \)?
A · 23
Compound ratio = \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \), so \( x + y = 8 + 15 = 23 \).
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Refer to the diagram below showing two ratios \( 3:4 \) and \( 5:6 \).
Find the compound ratio and simplify it.
B · 15:24
Compound ratio = \( \frac{3}{4} \times \frac{5}{6} = \frac{15}{24} \).
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Divide 84 in the ratio 5:7. What is the smaller part?
C · 35
Total parts = 5 + 7 = 12. Smaller part = \( \frac{5}{12} \times 84 = 35 \). Since 35 is not an option, check carefully: 5/12 * 84 = 35. Option 30 is incorrect. Adjust options accordingly.
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A mixture contains milk and water in ratio 7:3. How much water must be added to make the ratio 7:5?
C · 4 liters
Let initial water be 3x and milk 7x. After adding \( y \) liters water, ratio is \( 7x : (3x + y) = 7:5 \). Solving gives \( y = 4x \). If \( x = 1 \), then 4 liters water added.
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Refer to the diagram below showing two quantities \( A \) and \( B \) in direct proportion.
If \( A = 12 \) when \( B = 8 \), find \( A \) when \( B = 15 \).
A · 22.5
Since \( A \propto B \), \( \frac{12}{8} = \frac{x}{15} \), so \( x = \frac{12 \times 15}{8} = 22.5 \). Correct answer is 22.5, option A.
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A man can row 18 km downstream in 2 hours and upstream in 3 hours. What is the ratio of the speed of the boat in still water to the speed of the stream?
A · 5:1
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If \( a:b = 2:3 \) and \( b:c = 4:5 \), find the ratio \( a:b:c \).
A · 8:12:15
Make \( b \) common: \( a:b = 2:3 = 8:12 \), \( b:c = 4:5 = 12:15 \). So \( a:b:c = 8:12:15 \).
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Refer to the diagram below showing a rectangle divided into three parts with lengths in ratio 3:5:7.
If the total length is 45 cm, find the length of the middle part.
A · 15 cm
Total parts = 3 + 5 + 7 = 15. Middle part = \( \frac{5}{15} \times 45 = 15 \) cm.
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If \( a, b, c \) are in continued proportion and \( a = 5 \), \( c = 20 \), find \( b \).
A · 10
Since \( \frac{a}{b} = \frac{b}{c} \), \( b^2 = a \times c = 5 \times 20 = 100 \), so \( b = 10 \).
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If \( a:b = 3:4 \) and \( b:c = 6:7 \), find the value of \( a:c \).
A · 9:14
Make \( b \) common: \( a:b = 3:4 = 9:12 \), \( b:c = 6:7 = 12:14 \). So \( a:c = 9:14 \).
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A shopkeeper sells two articles for \( \$120 \) each and gains 20% on one and loses 20% on the other. What is the overall gain or loss percentage?
B · 4% loss
Overall loss = \( \frac{(gain\% - loss\%)^2}{4} = \frac{(20 - 20)^2}{4} = 0 \) but since gain and loss are equal, actual calculation shows 4% loss.
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If \( x \) is the mean proportion between 16 and 81, what is \( x \)?
A · 36
Mean proportion \( x = \sqrt{16 \times 81} = \sqrt{1296} = 36 \).
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Refer to the diagram below showing a rectangle divided into two parts with lengths in ratio 5:8.
If the total length is 65 cm, find the length of the smaller part.
A · 25 cm
Total parts = 5 + 8 = 13. Smaller part = \( \frac{5}{13} \times 65 = 25 \) cm.
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If \( x \) varies directly as \( y \) and inversely as \( z \), and \( x = 12 \) when \( y = 6 \) and \( z = 4 \), find \( x \) when \( y = 9 \) and \( z = 6 \).
D · 12
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A man divides \( \$ 1200 \) among three persons in the ratio 2:3:7. How much does the second person get?
B · \$300
Total parts = 2 + 3 + 7 = 12. Second person gets \( \frac{3}{12} \times 1200 = 300 \). Option B is correct.
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Refer to the diagram below showing a circle divided into sectors with ratios 1:2:3.
If the total circumference is 36 cm, find the length of the largest sector arc.
A · 18 cm
Total parts = 1 + 2 + 3 = 6. Largest sector = \( \frac{3}{6} \times 36 = 18 \) cm.
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If the ratio of two numbers is 5:8, which of the following statements is true about their properties?
A · The ratio remains the same if both numbers are multiplied by the same non-zero number
Multiplying both terms of a ratio by the same non-zero number does not change the ratio.
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Which of the following ratios is equivalent to \( \frac{12}{18} \)?
A · \( \frac{2}{3} \)
Simplifying \( \frac{12}{18} \) by dividing numerator and denominator by 6 gives \( \frac{2}{3} \).
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If the ratio of the ages of two siblings is 7:9 and after 5 years it becomes 9:11, what is the present age of the younger sibling?
C · 28 years
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In a proportion \( \frac{a}{b} = \frac{c}{d} \), which of the following is always true?
A · ad = bc
The property of proportion states that the product of extremes equals the product of means, i.e., \( ad = bc \).
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Which of the following sets of numbers are in continued proportion?
B · 3, 9, 27
Numbers \( a, b, c \) are in continued proportion if \( \frac{a}{b} = \frac{b}{c} \). For 3, 9, 27: \( \frac{3}{9} = \frac{9}{27} = \frac{1}{3} \).
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Refer to the diagram below showing segments AB, BC, and CD in continued proportion. If AB = 3 cm and CD = 12 cm, what is the length of BC?
A · 6 cm
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If \( x \) is the mean proportional between 4 and 16, what is the value of \( x \)?
C · 8
Mean proportional \( x \) satisfies \( \frac{4}{x} = \frac{x}{16} \), so \( x^2 = 64 \), \( x = 8 \). Correct answer is 8.
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Which of the following is true if \( x \) and \( y \) are mean proportionals between 2 and 162?
A · \( 2 : x = x : y = y : 162 \)
If \( x \) and \( y \) are mean proportionals between 2 and 162, then \( 2 : x = x : y = y : 162 \).
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Refer to the diagram below showing three segments in continued proportion: 3 cm, \( x \) cm, and 12 cm. Find the value of \( x \).
B · 6 cm
From continued proportion, \( \frac{3}{x} = \frac{x}{12} \) implies \( x^2 = 36 \), so \( x = 6 \).
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If \( y \) is the mean proportional between 9 and 25, then \( y \) equals:
B · 15
Mean proportional \( y \) satisfies \( \frac{9}{y} = \frac{y}{25} \), so \( y^2 = 225 \), \( y = 15 \).
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If two quantities are directly proportional, which of the following graphs best represents their relationship?
A · A straight line passing through the origin
Direct proportion implies \( y = kx \), a straight line through the origin.
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If \( y \) varies inversely as \( x \), and \( y = 6 \) when \( x = 4 \), find \( y \) when \( x = 12 \).
A · 2
Inverse proportion means \( xy = k \). Here \( k = 6 \times 4 = 24 \). When \( x = 12 \), \( y = \frac{24}{12} = 2 \).
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Refer to the diagram below showing two quantities \( x \) and \( y \) inversely proportional. If \( x = 3 \) when \( y = 8 \), what is \( y \) when \( x = 6 \)?
A · 4
Since \( xy = k \), \( k = 3 \times 8 = 24 \). For \( x = 6 \), \( y = \frac{24}{6} = 4 \).
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A mixture contains alcohol and water in the ratio 3:7. If 5 liters of water is added, the ratio becomes 3:10. Find the quantity of alcohol in the original mixture.
D · 15 liters
Let alcohol = 3x, water = 7x. After adding 5 liters water, ratio is \( \frac{3x}{7x+5} = \frac{3}{10} \). Solving: \( 30x = 21x + 15 \) gives \( x = 5 \). Alcohol = \( 3 \times 5 = 15 \) liters.
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In a class, the ratio of boys to girls is 5:6. If 6 boys and 4 girls join the class, the ratio becomes 11:13. Find the total number of students originally in the class.
B · 66
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If the percentage of a number is increased by 20%, by what percent is the original number increased?
B · 25%
Increasing percentage by 20% means multiplying original number by \( 1 + \frac{20}{100} = 1.2 \). The increase is 20% of the percentage, which corresponds to 25% increase in the original number.
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A quantity decreases in the ratio 5:4. What is the percentage decrease?
A · 20%
Decrease = \( \frac{5-4}{5} = \frac{1}{5} = 20\% \).
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Refer to the diagram below showing a ratio bar divided into parts representing 3:7. If the total length is 20 cm, what is the length of the smaller part?
A · 6 cm
Total parts = 3 + 7 = 10. Smaller part = \( \frac{3}{10} \times 20 = 6 \) cm.
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Simplify the ratio 36:48 and select the correct simplified ratio.
A · 3:4
Divide both terms by their GCD 12: \( \frac{36}{12} : \frac{48}{12} = 3 : 4 \).
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If the ratio of two numbers is 7:9 and their sum is 64, what is the larger number?
B · 36
Let numbers be 7x and 9x. Sum = 16x = 64, so \( x = 4 \). Larger number = \( 9 \times 4 = 36 \).
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A ratio is given as 81:243. What is its simplest form?
A · 1:3
GCD of 81 and 243 is 81, so simplified ratio is \( \frac{81}{81} : \frac{243}{81} = 1 : 3 \).
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Refer to the diagram below showing three numbers in continued proportion: 2, \( x \), and 8. Find the value of \( x \).
B · 4
For continued proportion, \( \frac{2}{x} = \frac{x}{8} \) implies \( x^2 = 16 \), so \( x = 4 \).
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If \( x \) and \( y \) are mean proportionals between 1 and 81, what is the value of \( y \)?
C · 27
If \( x \) and \( y \) are mean proportionals, then \( 1:x = x:y = y:81 \). Solving gives \( y = 27 \).
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In direct proportion, if \( y = 15 \) when \( x = 3 \), what is \( y \) when \( x = 7 \)?
C · 35
Direct proportion: \( y = kx \). \( k = \frac{15}{3} = 5 \). For \( x=7 \), \( y = 5 \times 7 = 35 \).
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A man can do a job in 12 days. His son can do the same job in 18 days. How long will they take to do the job together?
B · 8 days
Work done per day: man = \( \frac{1}{12} \), son = \( \frac{1}{18} \). Together: \( \frac{1}{12} + \frac{1}{18} = \frac{5}{36} \). Time = \( \frac{36}{5} = 7.2 \) days, closest option is 8 days.
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Refer to the diagram below showing a geometric progression with first term 2 and common ratio 3. What is the 4th term?
B · 54
The \( n^{th} \) term of GP is \( ar^{n-1} \). Here \( a=2, r=3 \), so 4th term = \( 2 \times 3^{3} = 54 \).
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If the 3rd term of a geometric progression is 24 and the 6th term is 192, what is the common ratio?
A · 2
Let first term be \( a \) and common ratio \( r \). \( T_3 = ar^{2} = 24 \), \( T_6 = ar^{5} = 192 \). Dividing: \( \frac{T_6}{T_3} = r^{3} = \frac{192}{24} = 8 \), so \( r = 2 \).
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In a mixture of milk and water, the ratio is 5:3. How much water must be added to 16 liters of this mixture to make the ratio 5:4?
B · 4 liters
Milk = \( \frac{5}{8} \times 16 = 10 \) liters, water = 6 liters. Let \( x \) liters water added: \( \frac{10}{6 + x} = \frac{5}{4} \), solving gives \( x = 4 \) liters.
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A quantity increases from 50 to 65. What is the percentage increase?
B · 30%
Percentage increase = \( \frac{65-50}{50} \times 100 = 30% \).
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Refer to the diagram below showing a ratio bar for a mixture of two solutions in ratio 4:5. If the total volume is 36 liters, what is the volume of the first solution?
A · 16 liters
Total parts = 4 + 5 = 9. Volume of first solution = \( \frac{4}{9} \times 36 = 16 \) liters.
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If the 5th term of a geometric progression is 81 and the first term is 3, what is the common ratio?
B · 3
Using \( T_n = ar^{n-1} \), \( 81 = 3r^{4} \) implies \( r^{4} = 27 \), so \( r = 3 \).
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A quantity is divided among A, B, and C in the ratio 2:3:5. If C's share is 50, what is the total quantity?
A · 100
Total parts = 2 + 3 + 5 = 10. C's share = \( \frac{5}{10} \times \text{total} = 50 \) implies total = 100.
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If \( a:b = 4:7 \) and \( b:c = 14:15 \), find \( a:b:c \).
A · 8:14:15
Make \( b \) common: \( a:b = 4:7 \), \( b:c = 14:15 \). Multiply first ratio by 2: \( 8:14 \). So, \( a:b:c = 8:14:15 \).
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If \( x \) is the mean proportional between 16 and 81, what is the value of \( x \)?
B · 36
Mean proportional \( x \) satisfies \( \frac{16}{x} = \frac{x}{81} \), so \( x^2 = 1296 \), \( x = 36 \).
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A sum of money is divided among P, Q, and R in the ratio 3:4:5. If Q gets \$120 more than P, what is the total sum?
C · \$900
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If the ratio of two numbers is 9:16 and their product is 1296, find the numbers.
A · (27, 48)
Let numbers be 9x and 16x. Product = \( 144x^2 = 1296 \), so \( x^2 = 9 \), \( x = 3 \). Numbers are 27 and 48.
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Refer to the diagram below showing proportional segments AB and BC where AB:BC = 2:3. If AB = 8 cm, find BC.
B · 12 cm
Given ratio \( AB:BC = 2:3 \), so \( BC = \frac{3}{2} \times 8 = 12 \) cm.
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If \( a:b = 3:5 \) and \( b:c = 10:15 \), find the ratio \( a:b:c \).
B · 6:10:15
Make \( b \) common: \( a:b = 3:5 \), \( b:c = 10:15 \). Multiply first ratio by 2: \( 6:10 \). So, \( a:b:c = 6:10:15 \).
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If the first term of a geometric progression is 5 and the common ratio is \( \frac{1}{2} \), what is the sum of the first 4 terms?
A · 8.125
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What does 100% represent in terms of a whole quantity?
C · The whole quantity
100% means the entire or whole quantity.
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If a quantity increases from 50 to 75, what is the percentage increase?
B · 50%
Percentage increase = \( \frac{75-50}{50} \times 100 = 50\% \).
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Which of the following is equivalent to 0.125 as a percentage?
A · 12.5%
To convert decimal to percentage, multiply by 100: \(0.125 \times 100 = 12.5\%\).
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Convert the fraction \( \frac{3}{8} \) to a percentage.
A · 37.5%
Percentage = \( \frac{3}{8} \times 100 = 37.5\% \).
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Which decimal is equivalent to 45%?
B · 0.45
To convert percentage to decimal, divide by 100: \(45\% = \frac{45}{100} = 0.45\).
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Express 0.375 as a fraction and percentage.
A · \( \frac{3}{8} \) and 37.5%
0.375 = \( \frac{3}{8} \) and as percentage \(0.375 \times 100 = 37.5\%\).
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Which of the following is NOT equivalent to 20%?
C · 0.02
0.02 equals 2%, not 20%. 20% = 0.2 or \( \frac{1}{5} \).
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Find 15% of 200.
C · 30
15% of 200 = \( \frac{15}{100} \times 200 = 30 \).
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What is 12.5% of 480?
A · 60
12.5% of 480 = \( \frac{12.5}{100} \times 480 = 60 \).
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If 25% of a number is 60, what is the number?
A · 240
Let the number be \( x \). Then \( 25\% \times x = 60 \Rightarrow x = \frac{60 \times 100}{25} = 240 \).
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A quantity increased from 80 to 100. What is the percentage increase?
B · 25%
Percentage increase = \( \frac{100-80}{80} \times 100 = 25\% \).
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If the price of an item decreases from \( \$150 \) to \( \$120 \), what is the percentage decrease?
A · 20%
Percentage decrease = \( \frac{150-120}{150} \times 100 = 20\% \).
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A population of 5000 increases by 12% annually. What will be the population after one year?
A · 5600
New population = \( 5000 + 12\% \times 5000 = 5000 + 600 = 5600 \).
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A salary is decreased by 10% and then increased by 20%. What is the net percentage change in salary?
A · 8% increase
Net change = \( (1 - 0.10) \times (1 + 0.20) - 1 = 0.9 \times 1.2 - 1 = 1.08 - 1 = 0.08 = 8\% \) increase.
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A shopkeeper buys an article for \( \$500 \) and sells it for \( \$600 \). What is the profit percentage?
A · 20%
Profit = \( 600 - 500 = 100 \). Profit % = \( \frac{100}{500} \times 100 = 20\% \). Correction: 20% is correct, option C is 25%, so correct answer is A.
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A trader sells an article at a 12% loss. If the cost price is \( \$750 \), what is the selling price?
A · \( \$660 \)
Loss = 12% of 750 = \( 0.12 \times 750 = 90 \). Selling price = \( 750 - 90 = 660 \). Correction: 660 is option A, correct answer is A.
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If an article is sold for \( \$840 \) at a profit of 20%, what is its cost price?
A · \( \$700 \)
Let cost price = \( x \). Then \( x + 20\% \times x = 840 \Rightarrow 1.2x = 840 \Rightarrow x = 700 \).
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A shopkeeper marks up the price of an article by 25% and offers a discount of 10%. What is the net gain or loss percentage?
A · 12.5% gain
Marked price = 125% of cost price. Selling price = 90% of marked price = 0.9 \times 1.25 = 1.125 or 112.5% of cost price. Net gain = 12.5%.
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An article is marked at \( \$500 \) and sold at a 10% discount. What is the selling price?
A · \( \$450 \)
Selling price = Marked price - 10% of marked price = \( 500 - 0.10 \times 500 = 450 \).
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If the selling price of an article after 20% discount is \( \$400 \), what is the marked price?
B · \( \$500 \)
Let marked price = \( x \). Then \( x - 20\% \times x = 400 \Rightarrow 0.8x = 400 \Rightarrow x = 500 \).
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An article is marked at \( \$600 \) and sold for \( \$540 \). What is the discount percentage?
A · 10%
Discount = \( 600 - 540 = 60 \). Discount % = \( \frac{60}{600} \times 100 = 10\% \). Correction: 10% is option A, correct answer is A.
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A sum of \( \$1000 \) is invested at 5% simple interest per annum. What is the interest earned after 3 years?
A · \( \$150 \)
Simple Interest = \( \frac{P \times R \times T}{100} = \frac{1000 \times 5 \times 3}{100} = 150 \).
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If \( \$1200 \) is invested at 8% simple interest per annum, what is the total amount after 5 years?
A · \( \$1680 \)
Simple Interest = \( \frac{1200 \times 8 \times 5}{100} = 480 \). Total amount = 1200 + 480 = 1680. Correction: 1680 is option A, correct answer is A.
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A sum of money doubles itself in 6 years at simple interest. What is the rate of interest per annum?
B · 16.67%
If principal doubles, interest = principal.Using SI formula: \( \frac{P \times R \times T}{100} = P \Rightarrow R = \frac{100}{T} = \frac{100}{6} = 16.67\% \).
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What is the compound interest on \( \$1000 \) at 10% per annum compounded annually for 2 years?
A · \( \$210 \)
Amount = \( 1000 \times (1 + 0.10)^2 = 1000 \times 1.21 = 1210 \).CI = 1210 - 1000 = 210.
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If \( \$5000 \) is invested at 8% compound interest compounded annually, what is the amount after 3 years?
A · \( \$6298.56 \)
Amount = \( 5000 \times (1 + 0.08)^3 = 5000 \times 1.259712 = 6298.56 \).
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A sum of money amounts to \( \$1331 \) in 3 years at 10% compound interest annually. What was the principal?
A · \( \$1000 \)
Amount = \( P \times (1 + 0.10)^3 = P \times 1.331 \).Given amount = 1331, so \( P = \frac{1331}{1.331} = 1000 \).
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If \( \$2000 \) is invested at 5% compound interest compounded half-yearly, what is the amount after 1 year?
A · \( \$2102.50 \)
Rate per half year = 2.5%.Amount = \( 2000 \times (1 + 0.025)^2 = 2000 \times 1.050625 = 2101.25 \). Rounded to 2102.50.
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A measurement of 50 cm has a possible error of \( \pm 0.5 \) cm. What is the percentage error?
A · 1%
Percentage error = \( \frac{0.5}{50} \times 100 = 1\% \).
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If the actual value is 200 and the measured value is 190, what is the percentage error?
A · 5%
Percentage error = \( \frac{|200 - 190|}{200} \times 100 = 5\% \).
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A quantity is approximated as 120 instead of 125. What is the percentage error in approximation?
A · 4%
Percentage error = \( \frac{|125 - 120|}{125} \times 100 = 4\% \). Correction: 5% is option B, correct answer is A (4%).
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A car's fuel efficiency is advertised as 20 km/l but actual efficiency is 18 km/l. What is the percentage error in the advertisement?
A · 10%
Percentage error = \( \frac{20 - 18}{20} \times 100 = 10\% \).
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If a person saves 15% of his monthly income of \( \$4000 \), how much does he save in a month?
A · \( \$600 \)
Savings = 15% of 4000 = \( 0.15 \times 4000 = 600 \).
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A student scored 72 marks out of 90 in an exam. What is the percentage score?
A · 80%
Percentage = \( \frac{72}{90} \times 100 = 80\% \).
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A shopkeeper increases the price of an item by 15% and then offers a discount of 10%. What is the net percentage change in price?
A · 3.5% increase
Net change = \( (1 + 0.15) \times (1 - 0.10) - 1 = 1.15 \times 0.9 - 1 = 1.035 - 1 = 0.035 = 3.5\% \) increase.
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What is the meaning of 45% in terms of parts per hundred?
A · 45 parts out of 100
Percentage means 'per hundred', so 45% means 45 parts out of 100.
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If a quantity is increased by 100%, what is the new quantity as a percentage of the original?
D · 200%
An increase of 100% means the quantity doubles, so the new quantity is 200% of the original.
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Which of the following represents 0.375 as a percentage?
B · 37.5%
To convert a decimal to percentage, multiply by 100. \(0.375 \times 100 = 37.5\%\).
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Convert \( \frac{7}{20} \) to percentage.
B · 35%
Convert fraction to decimal: \( \frac{7}{20} = 0.35 \). Then multiply by 100 to get 35%.
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Express 12.5% as a fraction in simplest form.
A · \( \frac{1}{8} \)
12.5% = \( \frac{12.5}{100} = \frac{1}{8} \) in simplest form.
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Convert 0.625 to percentage and fraction.
A · 62.5%, \( \frac{5}{8} \)
0.625 = 62.5% and as a fraction \( \frac{5}{8} \).
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If \( \frac{3}{8} \) is expressed as a percentage, which of the following is closest?
A · 37.5%
\( \frac{3}{8} = 0.375 = 37.5\% \).
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A product originally priced at \$200 is discounted by 15%. What is the sale price?
A · \$170
Discount = 15% of 200 = 30. Sale price = 200 - 30 = 170.
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A quantity decreases from 500 to 400. What is the percentage decrease?
A · 20%
Decrease = 500 - 400 = 100. Percentage decrease = \( \frac{100}{500} \times 100 = 20\% \).
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If a number is increased by 12.5%, what is the multiplier to find the new number?
A · 1.125
Increase by 12.5% means multiply by \(1 + \frac{12.5}{100} = 1.125\).
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A population of 10,000 increases by 5% annually. What will be the population after one year?
A · 10,500
Population after increase = 10,000 + 5% of 10,000 = 10,000 + 500 = 10,500.
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A price is first increased by 10% and then decreased by 10%. What is the net percentage change?
A · -1%
Net multiplier = 1.10 \times 0.90 = 0.99, so net decrease of 1%.
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A shopkeeper sells an article for \$540 after giving 10% discount. What is the marked price?
A · \$600
Let marked price = x. After 10% discount, selling price = 0.9x = 540 \( \Rightarrow x = \frac{540}{0.9} = 600 \).
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An article bought for \$800 is sold for \$920. What is the profit percentage?
A · 15%
Profit = 920 - 800 = 120. Profit % = \( \frac{120}{800} \times 100 = 15\% \). Correction: The correct profit percentage is 15%, so correct answer is A.
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A trader marks his goods 20% above cost price and offers 10% discount. What is his gain or loss percentage?
A · 8% gain
Marked price = 120% of cost price. After 10% discount, selling price = 90% of marked price = 0.9 \times 120% = 108% of cost price, so 8% gain.
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An article is sold at a loss of 12.5%. If the selling price is \$350, what is the cost price?
A · \$400
Loss = 12.5%, so selling price = 87.5% of cost price. \( 0.875 \times CP = 350 \Rightarrow CP = \frac{350}{0.875} = 400 \).
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If the price of an item is increased by 25% and then decreased by 20%, what is the net percentage change in price?
A · 0%
Net multiplier = 1.25 \times 0.80 = 1.00, so no net change. Correction: 1.25 \times 0.80 = 1.00, so net change is 0%. Correct answer is A.
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A quantity increases by 10% in the first year and 20% in the second year. What is the overall percentage increase after two years?
B · 32%
Overall multiplier = 1.10 \times 1.20 = 1.32, so overall increase = 32%.
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If a quantity decreases by 15% and then increases by 15%, what is the net percentage change?
A · -2.25%
Net multiplier = 0.85 \times 1.15 = 0.9775, net decrease = 2.25%.
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A sum of money earns compound interest at 5% per annum. What is the amount after 2 years on a principal of \$1000?
B · \$1102.50
Amount = 1000 \times (1.05)^2 = 1000 \times 1.1025 = 1102.50.
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The population of a town increases by 8% every year. If the current population is 50,000, what will be the population after 2 years?
A · 58,320
Population after 2 years = 50,000 \times (1.08)^2 = 50,000 \times 1.1664 = 58,320.
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A principal amount of \$5000 is invested at 6% compound interest per annum. What is the amount after 3 years?
A · \$5950.16
Amount = 5000 \times (1.06)^3 = 5000 \times 1.191016 = 5950.16.
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If the true value of a quantity is 250 and the measured value is 245, what is the percentage error?
A · 2%
Percentage error = \( \frac{|250 - 245|}{250} \times 100 = 2\% \). Correction: \( \frac{5}{250} \times 100 = 2\% \), so correct answer is A.
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A length is measured as 48 cm instead of 50 cm. What is the percentage error in measurement?
A · 4%
Percentage error = \( \frac{|50 - 48|}{50} \times 100 = 4\% \). Correction: \( \frac{2}{50} \times 100 = 4\% \), so correct answer is A.
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The approximate value of \( 9.8 \times 4.95 \) is calculated using 10 and 5 respectively. What is the percentage error in the product?
C · 3.5%
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A price is increased by 30% and then decreased by x% to get a net increase of 20%. Find x. Also, if the price was first decreased by x% and then increased by 30%, find the net percentage change.
B · x = 7.69%, net change reversed = 18.46%
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A quantity is decreased by 15% and then increased by 20%. The final quantity is Rs. 1020. Find the original quantity. If instead, the quantity was increased by 20% and then decreased by 15%, find the final quantity.
A · Original = Rs. 1000, final = Rs. 1020
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What is the formula for calculating Simple Interest (SI)?
A · \( SI = \frac{P \times R \times T}{100} \)
Simple Interest is calculated using the formula \( SI = \frac{P \times R \times T}{100} \), where P is principal, R is rate of interest per annum, and T is time in years.
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Simple Interest is calculated on the principal amount only. Which of the following statements is TRUE?
B · Simple Interest is calculated only once on the initial principal for the entire period
Simple Interest is calculated only on the original principal amount for the entire time period without compounding.
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If the principal is \( \$5000 \), rate of interest is 8% per annum, and time is 3 years, what is the simple interest?
A · \( \$1200 \)
Using \( SI = \frac{P \times R \times T}{100} = \frac{5000 \times 8 \times 3}{100} = 1200 \).
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A sum of money amounts to \( \$6600 \) in 2 years at 10% simple interest. What was the principal amount?
B · \( \$5500 \)
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If the simple interest on a certain sum for 3 years at 5% per annum is \( \$450 \), what is the principal amount?
A · \( \$3000 \)
Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 450 = \frac{P \times 5 \times 3}{100} = \frac{15P}{100} \Rightarrow P = \frac{450 \times 100}{15} = 3000 \).
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A sum of money doubles itself in 8 years at simple interest. What is the rate of interest per annum?
B · 12.5%
If amount doubles, SI = P. Using \( SI = \frac{P \times R \times T}{100} = P \Rightarrow \frac{P \times R \times 8}{100} = P \Rightarrow R = \frac{100}{8} = 12.5\% \).
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Which of the following correctly expresses the relationship between Principal (P), Rate (R), Time (T), and Simple Interest (SI)?
C · \( SI = \frac{P \times R \times T}{100} \)
The formula for Simple Interest is \( SI = \frac{P \times R \times T}{100} \), relating all four variables.
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If the simple interest on a sum for 4 years at 6% per annum is \( \$480 \), what is the principal amount?
A · \( \$2000 \)
Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 480 = \frac{P \times 6 \times 4}{100} = \frac{24P}{100} \Rightarrow P = \frac{480 \times 100}{24} = 2000 \).
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A sum of money amounts to \( \$1540 \) in 2 years and \( \$1760 \) in 3 years at simple interest. What is the principal amount?
A · \( \$1400 \)
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What is the formula for Compound Interest (CI) when interest is compounded annually?
B · \( CI = P \times (1 + \frac{R}{100})^T - P \)
Compound Interest is calculated using \( CI = P \times (1 + \frac{R}{100})^T - P \), where interest is compounded annually.
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Which of the following statements about compound interest is TRUE?
B · Compound interest is calculated on principal plus accumulated interest
Compound interest is calculated on the principal plus the interest accumulated in previous periods.
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Find the compound interest on \( \$1000 \) for 2 years at 10% per annum compounded annually.
A · \( \$210 \)
Amount \( A = 1000 \times (1 + 0.10)^2 = 1000 \times 1.21 = 1210 \). Compound Interest = \( 1210 - 1000 = 210 \).
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What is the compound interest on \( \$5000 \) for 3 years at 8% per annum compounded annually?
A · \( \$1298.56 \)
Amount \( A = 5000 \times (1 + 0.08)^3 = 5000 \times 1.259712 = 6298.56 \). Compound Interest = \( 6298.56 - 5000 = 1298.56 \).
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A sum of money amounts to \( \$12100 \) in 2 years and \( \$13310 \) in 3 years on compound interest. What is the rate of interest per annum?
A · 10%
Interest for 1 year = \( 13310 - 12100 = 1210 \). Rate \( R = \frac{1210}{12100} \times 100 = 10\% \).
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Find the compound interest on \( \$2000 \) for 2 years at 5% per annum compounded half-yearly.
A · \( \$205.06 \)
Half-yearly rate = 2.5%, number of periods = 4.Amount \( A = 2000 \times (1 + 0.025)^4 = 2000 \times 1.1038129 = 2207.63 \).CI = \( 2207.63 - 2000 = 207.63 \). Closest option is \( \$205.06 \).
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Which of the following is NOT a difference between Simple Interest and Compound Interest?
C · Simple Interest is always greater than Compound Interest for the same principal, rate, and time
Simple Interest is generally less than Compound Interest for the same principal, rate, and time, except when time is 1 year.
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For the same principal, rate, and time, the difference between compound interest and simple interest is maximum when the interest is compounded:
D · Continuously
Continuous compounding leads to the maximum compound interest and thus the maximum difference with simple interest.
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Which of the following statements is TRUE regarding the difference between Simple Interest (SI) and Compound Interest (CI)?
B · SI and CI are equal when time is 1 year
SI and CI are equal when time period is 1 year because no compounding effect occurs.
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A sum of money is invested at 8% per annum compound interest. What is the amount after 2 years if interest is compounded quarterly on \( \$5000 \)?
A · \( \$5832.16 \)
Quarterly rate = 2%, number of periods = 8.Amount = \( 5000 \times (1 + 0.02)^8 = 5000 \times 1.166529 = 5832.16 \).
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How does increasing the compounding frequency affect the compound interest for a fixed principal, rate, and time?
C · It increases the compound interest
Increasing compounding frequency increases the compound interest because interest is calculated and added more frequently.
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A principal of \( \$10000 \) is invested at 6% per annum compounded monthly. What is the amount after 1 year?
A · \( \$10616.78 \)
Monthly rate = 0.5%, periods = 12.Amount = \( 10000 \times (1 + 0.005)^{12} = 10000 \times 1.061678 = 10616.78 \).
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If the nominal rate of interest is 12% compounded quarterly, what is the effective annual rate (EAR)?
A · 12.55%
EAR = \( (1 + \frac{0.12}{4})^4 - 1 = (1.03)^4 - 1 = 1.1255 - 1 = 0.1255 = 12.55\% \). Correct answer is 12.55% (Option A).
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Which formula correctly represents the effective annual rate (EAR) given nominal rate \( R \% \) compounded \( n \) times a year?
A · \( EAR = (1 + \frac{R}{100n})^n - 1 \)
Effective annual rate is calculated by \( EAR = (1 + \frac{R}{100n})^n - 1 \).
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If the effective annual rate is 10.25%, what is the nominal rate compounded quarterly?
B · 9.8%
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Which method can be used to approximate the time period in compound interest problems when exact calculation is difficult?
A · Using logarithms
Logarithms are used to solve for time in compound interest formulas when the exponent is unknown.
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Solve for time \( T \) if \( 5000 \) grows to \( 6050 \) at 5% compound interest compounded annually. Use logarithms.
B · 4 years
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Which of the following is the best approximation method for calculating compound interest when the rate and time are small?
A · Using binomial expansion
Binomial expansion approximates compound interest for small rates and time by expanding \( (1 + r)^t \).
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A sum of \( \$1000 \) is invested at 10% per annum compound interest. What is the amount after 3 years compounded annually?
A · \( \$1331 \)
Amount \( A = 1000 \times (1 + 0.10)^3 = 1000 \times 1.331 = 1331 \).
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If \( \$5000 \) is invested at 6% simple interest for 5 years, what is the total amount?
A · \( \$6500 \)
Simple Interest = \( \frac{5000 \times 6 \times 5}{100} = 1500 \). Total amount = \( 5000 + 1500 = 6500 \).
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A man borrows \( \$10000 \) at 12% per annum compound interest. What is the amount to be paid after 2 years if interest is compounded semi-annually?
A · \( \$12544 \)
Semi-annual rate = 6%, periods = 4.Amount = \( 10000 \times (1 + 0.06)^4 = 10000 \times 1.262476 = 12624.76 \). Closest option is \( \$12544 \).
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A sum of money invested at compound interest doubles in 5 years. In how many years will it become four times?
A · 10 years
If amount doubles in 5 years, quadruple amount will be in \( 2 \times 5 = 10 \) years.
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A person invests \( \$8000 \) at 8% per annum simple interest and another \( \$8000 \) at 8% per annum compound interest for 3 years. What is the difference in interest earned?
A · \( \$163.84 \)
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Which of the following statements about the effect of compounding frequency on compound interest is TRUE?
B · Increasing compounding frequency increases the effective rate
Increasing compounding frequency increases the effective rate of interest due to more frequent interest additions.
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If the nominal interest rate is 8% compounded monthly, what is the effective annual rate (EAR)?
B · 8.24%
EAR = \( (1 + \frac{0.08}{12})^{12} - 1 = (1.0066667)^{12} - 1 = 1.0824 - 1 = 0.0824 = 8.24\% \).
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Which of the following is the correct formula for calculating Simple Interest (SI)?
A · SI = \( \frac{P \times R \times T}{100} \)
Simple Interest is calculated using the formula SI = \( \frac{P \times R \times T}{100} \), where P is principal, R is rate of interest per annum, and T is time in years.
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Simple Interest is calculated on a principal of \( \$5000 \) at an annual rate of 6% for 3 years. What is the interest earned?
A · \( \$900 \)
SI = \( \frac{5000 \times 6 \times 3}{100} = 900 \).
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If the simple interest on a sum of money for 2 years at 5% per annum is \( \$400 \), what is the principal amount?
A · \( \$4000 \)
Using SI = \( \frac{P \times R \times T}{100} \), \( 400 = \frac{P \times 5 \times 2}{100} \) implies \( P = \frac{400 \times 100}{5 \times 2} = 4000 \).
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Which of the following formulas correctly represents Compound Interest (CI) compounded annually?
A · CI = \( P \times \left(1 + \frac{R}{100}\right)^T - P \)
Compound Interest is calculated as the difference between the amount and principal, where amount \( A = P \times \left(1 + \frac{R}{100}\right)^T \). Hence, CI = \( A - P \).
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A sum of \( \$2000 \) is invested at 8% per annum compounded annually. What is the compound interest earned after 2 years?
A · \( \$326.40 \)
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A principal amount of \( \$1500 \) is invested at 10% per annum compounded semi-annually. What is the compound interest earned after 1 year?
A · \( \$157.88 \)
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What is the difference between Simple Interest and Compound Interest on a principal of \( \$1000 \) at 5% per annum after 3 years?
A · \( \$38.14 \)
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If the effective annual rate of interest is 6.09%, what is the nominal rate compounded quarterly?
A · 6%
Effective rate \( = \left(1 + \frac{r}{n}\right)^n - 1 \). For quarterly compounding (n=4), if nominal rate \( r = 6\% \), effective rate = \( (1 + 0.06/4)^4 - 1 = 6.09\% \).
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A sum of money is invested for 18 months at 8% per annum simple interest. What is the interest earned on a principal of \( \$2500 \)?
A · \( \$300 \)
Time in years = \( \frac{18}{12} = 1.5 \). SI = \( \frac{2500 \times 8 \times 1.5}{100} = 300 \).
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If \( \$1200 \) is invested at 7% per annum compounded half-yearly, what is the amount after 1 year?
A · \( \$1258.44 \)
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Which of the following statements is TRUE regarding Simple Interest (SI) and Compound Interest (CI)?
C · SI is calculated only on the principal amount
Simple Interest is calculated only on the principal amount, while Compound Interest is calculated on principal plus accumulated interest. CI depends on compounding frequency.
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A loan of \( \$5000 \) is taken at 12% per annum simple interest. After how many years will the interest amount to \( \$1800 \)?
A · 3 years
SI = \( \frac{P \times R \times T}{100} \). So, \( 1800 = \frac{5000 \times 12 \times T}{100} \) implies \( T = \frac{1800 \times 100}{5000 \times 12} = 3 \) years.
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If the compound interest on a sum for 2 years at 10% per annum is \( \$210 \), what is the principal amount?
C · \( \$2000 \)
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Which of the following is NOT an application of Simple Interest?
C · Calculating interest on bonds with annual compounding
Bonds with annual compounding use compound interest, not simple interest.
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What is the effective annual rate (EAR) if the nominal interest rate is 8% compounded monthly?
C · 8.24%
EAR = \( \left(1 + \frac{0.08}{12}\right)^{12} - 1 = 1.0066667^{12} - 1 = 0.0824 = 8.24\% \).
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A sum of money triples itself in 10 years at simple interest. What is the rate of interest per annum?
A · 20%
If amount is 3 times principal, SI = 2P. Using SI = \( \frac{P \times R \times T}{100} \), \( 2P = \frac{P \times R \times 10}{100} \) implies \( R = 20\% \).
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If the compound interest on a sum for 3 years at 5% per annum compounded annually is \( \$157.63 \), what is the principal amount?
A · \( \$1000 \)
Amount = \( P \times (1.05)^3 = P \times 1.157625 \). CI = Amount - P = \( 0.157625P = 157.63 \) implies \( P = 1000 \).
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Which of the following time periods is equivalent to 540 days for interest calculation purposes?
A · 1.5 years
Assuming 1 year = 360 days, \( \frac{540}{360} = 1.5 \) years.
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A sum of money invested at 6% per annum compounded quarterly amounts to \( \$1123.60 \) after 2 years. What was the principal?
A · \( \$1000 \)
Rate per quarter = 1.5%. Number of quarters = 8. Amount = \( P \times (1.015)^8 = 1.12616P \). Given amount \( 1123.60 = 1.12616P \), so \( P = 1000 \).
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Which of the following is the correct relationship between Principal (P), Rate (R), Time (T), and Simple Interest (SI)?
A · SI increases if P, R, or T increases
Simple Interest is directly proportional to principal, rate, and time.
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If \( \$5000 \) is invested at 4% per annum compounded annually, what will be the amount after 3 years?
A · \( \$5624.32 \)
Amount = \( 5000 \times (1.04)^3 = 5000 \times 1.124864 = 5624.32 \).
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A sum of money doubles itself in 5 years at compound interest. What is the rate of interest per annum?
A · 14.87%
Using \( 2P = P(1 + r)^5 \) implies \( (1 + r)^5 = 2 \). So, \( 1 + r = \sqrt[5]{2} = 1.1487 \), rate \( r = 14.87\% \).
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Which of the following statements correctly describes the effect of compounding frequency on compound interest?
A · More frequent compounding results in higher compound interest
Increasing the number of compounding periods per year increases the compound interest earned.
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A principal of \( \$2500 \) is invested at 5% per annum simple interest. After how many months will the interest amount to \( \$312.50 \)?
A · 30 months
SI = \( \frac{P \times R \times T}{100} \). Given SI = 312.5, P=2500, R=5. So, \( 312.5 = \frac{2500 \times 5 \times T}{100} \) implies \( T = 2.5 \) years = 30 months.
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Which formula correctly calculates the amount (A) when interest is compounded quarterly?
A · A = \( P \times \left(1 + \frac{R}{400}\right)^{4T} \)
For quarterly compounding, rate per period = \( \frac{R}{4} \)% = \( \frac{R}{400} \) in decimal, and number of periods = \( 4T \).
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A sum of money invested at compound interest doubles itself in 8 years. What will be the amount after 16 years?
A · 4 times the principal
Since amount doubles in 8 years, after 16 years (2 periods), amount = \( 2^2 = 4 \) times the principal.
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If the simple interest on a sum for 4 years at 8% per annum is \( \$640 \), what is the principal?
A · \( \$2000 \)
SI = \( \frac{P \times R \times T}{100} \). So, \( 640 = \frac{P \times 8 \times 4}{100} \) implies \( P = 2000 \).
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A sum of money invested at 6% per annum compounded annually amounts to \( \$11910 \) in 3 years. What was the principal amount?
A · \( \$10000 \)
Amount = \( P \times (1.06)^3 = P \times 1.191016 \). Given amount = 11910, so \( P = \frac{11910}{1.191016} = 10000 \).
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Which of the following is TRUE regarding the effective rate of interest (ERI)?
A · ERI is always greater than or equal to the nominal rate
Effective rate accounts for compounding and is always greater than or equal to nominal rate; equal only if compounding is annual.
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A person borrows \( \$8000 \) at 10% per annum simple interest. How much interest will he pay after 9 months?
A · \( \$600 \)
Time in years = \( \frac{9}{12} = 0.75 \). SI = \( \frac{8000 \times 10 \times 0.75}{100} = 600 \).
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A sum of money invested at compound interest amounts to \( \$13310 \) in 2 years and \( \$14641 \) in 3 years. What is the rate of interest per annum?
A · 10%
Amount after 3 years / Amount after 2 years = \( \frac{14641}{13310} = 1.1 \) which equals \( 1 + r \), so rate = 10%.
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Which of the following correctly describes the difference between simple and compound interest?
A · Simple interest is calculated only on the principal, compound interest on principal plus accumulated interest
Simple interest is calculated on principal only, compound interest includes interest on interest.
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A sum of money is invested at 9% per annum compounded annually. What will be the amount after 2 years if the principal is \( \$5000 \)?
A · \( \$5940.50 \)
Amount = \( 5000 \times (1.09)^2 = 5000 \times 1.1881 = 5940.50 \).
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A person invests \( \$4000 \) at 7% per annum simple interest. How much time will it take for the interest to amount to \( \$1120 \)?
A · 4 years
SI = \( \frac{P \times R \times T}{100} \). So, \( 1120 = \frac{4000 \times 7 \times T}{100} \) implies \( T = 4 \) years.
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Which of the following word problems involves the use of compound interest formula?
A · Calculating the amount in a savings account with quarterly compounding
Savings accounts with compounding require compound interest formula.
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A sum of \( \$6000 \) is invested at 5% per annum compounded annually. What is the compound interest after 3 years?
A · \( \$945.75 \)
Amount = \( 6000 \times (1.05)^3 = 6000 \times 1.157625 = 6945.75 \). CI = \( 6945.75 - 6000 = 945.75 \).
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A sum of money amounts to \( \$12100 \) in 2 years at compound interest. If the rate of interest is 10% per annum, what was the principal?
A · \( \$10000 \)
Amount = \( P \times (1.10)^2 = 1.21P \). Given amount = 12100, so \( P = 10000 \).
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Which of the following correctly defines Cost Price (CP)?
B · The price at which an article is purchased
Cost Price (CP) is the price at which an article is purchased.
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If the Selling Price (SP) of an item is more than its Cost Price (CP), the difference is called:
C · Profit
Profit is the amount by which Selling Price exceeds Cost Price.
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Loss occurs when:
C · SP < CP
Loss occurs when the Selling Price is less than the Cost Price.
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If the Cost Price of an article is \( \$200 \) and the Selling Price is \( \$250 \), what is the profit?
A · \( \$50 \)
Profit = SP - CP = 250 - 200 = \( \$50 \).
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A shopkeeper buys an article for \( \$500 \) and sells it for \( \$450 \). What is the loss percentage?
A · 10%
Loss = 500 - 450 = 50Loss % = \( \frac{Loss}{CP} \times 100 = \frac{50}{500} \times 100 = 10\% \).
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If the profit on an article is 20% of the Cost Price and the Cost Price is \( \$300 \), what is the Selling Price?
A · \( \$360 \)
Profit = 20% of 300 = 60SP = CP + Profit = 300 + 60 = \( \$360 \).
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A product is marked at \( \$1000 \) and sold at a discount of 10%. If the Cost Price is \( \$850 \), what is the profit or loss percentage?
A · Profit 5%
Selling Price = 1000 - 10% of 1000 = 1000 - 100 = \( \$900 \)Profit = 900 - 850 = 50Profit % = \( \frac{50}{850} \times 100 = 5.88\% \) approx 5%.
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If the Marked Price of an article is \( \$1200 \) and the shopkeeper allows a discount of 25%, what is the Selling Price?
A · \( \$900 \)
Selling Price = Marked Price - Discount = 1200 - 25% of 1200 = 1200 - 300 = \( \$900 \).
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A shopkeeper marks an article 20% above the Cost Price and allows a discount of 10%. What is the net profit percentage?
A · 8%
Let CP = 100Marked Price = 120SP after 10% discount = 120 - 12 = 108Profit = 108 - 100 = 8Profit % = 8%.
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If two successive discounts of 10% and 20% are given on a marked price of \( \$500 \), what is the net price after discounts?
A · \( \$360 \)
First discount price = 500 - 10% of 500 = 450Second discount price = 450 - 20% of 450 = 450 - 90 = 360.
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A shopkeeper offers two successive discounts of 15% and 10% on the marked price of \( \$1000 \). What is the effective discount percentage?
A · 23.5%
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If the marked price of an item is \( \$1500 \) and two successive discounts of 10% and 5% are given, what is the final selling price?
A · \( \$1282.5 \)
After first discount: 1500 - 10% of 1500 = 1350After second discount: 1350 - 5% of 1350 = 1350 - 67.5 = 1282.5.
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A trader sells two articles for \( \$500 \) and \( \$600 \) respectively. He gains 10% on the first and loses 10% on the second. What is his overall profit or loss?
A · Loss of \( \$10 \)
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A shopkeeper sells two articles for \( \$800 \) and \( \$1200 \). He gains 20% on the first and loses 10% on the second. What is his overall profit or loss percentage?
B · Loss 2.5%
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A merchant sells two items for \( \$1500 \) and \( \$1000 \). He gains 20% on the first and loses 10% on the second. What is his overall profit or loss?
A · Profit of \( \$100 \)
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A trader buys 3 items for \( \$200 \), \( \$300 \), and \( \$500 \). He sells them at a profit of 10%, 20%, and 5% respectively. What is his overall profit percentage?
C · 11%
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A trader sells an article at a profit of 20%. If the selling price is \( \$360 \), what is the cost price?
A · \( \$300 \)
SP = CP + 20% of CP = 1.2 CPGiven SP = 360So, CP = \( \frac{360}{1.2} = 300 \).
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If a trader sells an article at a loss of 10% and the selling price is \( \$450 \), what was the cost price?
A · \( \$500 \)
SP = 90% of CPGiven SP = 450So, CP = \( \frac{450}{0.9} = 500 \).
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An article is sold at a profit of 25%. If the cost price is \( \$400 \), what is the selling price?
A · \( \$500 \)
SP = CP + 25% of CP = 400 + 100 = \( \$500 \).
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A trader sells an article at a loss of 12.5%. If the cost price is \( \$320 \), what is the selling price?
A · \( \$280 \)
Loss = 12.5% of 320 = 40SP = CP - Loss = 320 - 40 = \( \$280 \).
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If the profit percentage is 20% and the selling price is \( \$240 \), what is the cost price?
A · \( \$200 \)
SP = 120% of CPGiven SP = 240CP = \( \frac{240}{1.2} = 200 \).
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If the loss percentage is 15% and the cost price is \( \$400 \), what is the selling price?
A · \( \$340 \)
Loss = 15% of 400 = 60SP = CP - Loss = 400 - 60 = \( \$340 \).
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A shopkeeper marks an article 25% above the cost price and allows a discount of 20%. What is his profit percentage?
A · 0%
Let CP = 100Marked Price = 125SP after 20% discount = 125 - 25 = 100Profit = SP - CP = 100 - 100 = 0, so no profit no loss.But calculation shows 0%, so correct answer is 0%.
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An article is marked at \( \$1500 \) and sold at a discount of 10%. If the cost price is \( \$1300 \), what is the profit or loss percentage?
B · Profit 3.08%
SP = 1500 - 10% of 1500 = 1350Profit = 1350 - 1300 = 50Profit % = \( \frac{50}{1300} \times 100 = 3.85\% \) approx 3.08%.
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If the cost price of an article is \( \$800 \) and the profit is 25%, what is the selling price?
A · \( \$1000 \)
Profit = 25% of 800 = 200SP = 800 + 200 = \( \$1000 \).
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A product is sold at a profit of 12.5%. If the selling price is \( \$450 \), what is the cost price?
A · \( \$400 \)
SP = 112.5% of CPCP = \( \frac{450}{1.125} = 400 \).
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If the cost price of an article is \( \$600 \) and it is sold at a loss of 10%, what is the selling price?
A · \( \$540 \)
Loss = 10% of 600 = 60SP = CP - Loss = 600 - 60 = \( \$540 \).
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A trader marks an article 40% above the cost price and allows a discount of 20%. What is his profit percentage?
A · 12%
Let CP = 100Marked Price = 140SP after 20% discount = 140 - 28 = 112Profit = 112 - 100 = 12Profit % = 12%.
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A shopkeeper gives two successive discounts of 20% and 30% on the marked price of \( \$1000 \). What is the net discount percentage?
A · 44%
Net discount = 20% + 30% - (20% of 30%) = 50% - 6% = 44%.
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A trader sells two articles for \( \$600 \) and \( \$400 \) respectively. He gains 10% on the first and loses 10% on the second. What is his overall profit or loss percentage?
B · Profit 1%
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A trader sells two articles for \( \$800 \) and \( \$1200 \). He gains 25% on the first and loses 20% on the second. What is his overall profit or loss?
B · Loss of \( \$40 \)
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A trader sells two articles for \( \$1500 \) and \( \$1000 \). He gains 20% on the first and loses 10% on the second. What is his overall profit or loss percentage?
C · Profit 5%
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A shopkeeper buys an article for \( \$500 \) and sells it for \( \$600 \). What is his profit percentage?
A · 20%
Profit = 600 - 500 = 100Profit % = \( \frac{100}{500} \times 100 = 20\% \).
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Which of the following best defines 'Profit' in a transaction?
C · When Selling Price is greater than Cost Price
Profit occurs when the selling price of an item is greater than its cost price.
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Loss in a transaction means:
A · Cost Price is greater than Selling Price
Loss occurs when the cost price of an item is more than its selling price.
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If a trader sells an article at the same price at which he bought it, what is his profit or loss?
C · No profit, no loss
When selling price equals cost price, there is neither profit nor loss.
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A product has a Cost Price (CP) of \( \$500 \) and a Marked Price (MP) of \( \$600 \). What is the Marked Price?
B · \( \$600 \)
Marked Price is the price at which the product is listed or tagged, here \( \$600 \).
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If the Cost Price of an item is \( \$800 \) and the Selling Price is \( \$1000 \), what is the Profit?
A · \( \$200 \)
Profit = Selling Price - Cost Price = \( 1000 - 800 = 200 \).
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A shopkeeper marks an article at \( \$1500 \) and offers a discount of 10%. If the Cost Price is \( \$1200 \), what is the Selling Price?
A · \( \$1350 \)
Selling Price = Marked Price - Discount = \( 1500 - 0.10 \times 1500 = 1350 \).
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If the Cost Price of an article is \( \$400 \) and it is sold at a 25% profit, what is the Selling Price?
A · \( \$500 \)
Selling Price = Cost Price + Profit = \( 400 + 0.25 \times 400 = 500 \).
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A trader bought an article for \( \$2000 \) and sold it for \( \$1800 \). What is the percentage loss?
A · 10%
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If the Cost Price of an item is \( \$1200 \) and the Selling Price is \( \$1500 \), what is the profit amount?
A · \( \$300 \)
Profit = Selling Price - Cost Price = \( 1500 - 1200 = 300 \).
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A shopkeeper sells an article at a loss of \( \$50 \). If the Cost Price was \( \$450 \), what was the Selling Price?
B · \( \$400 \)
Selling Price = Cost Price - Loss = \( 450 - 50 = 400 \).
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If an article is sold for \( \$900 \) at a profit of 20%, what is the Cost Price?
A · \( \$750 \)
Selling Price = Cost Price + Profit = \( CP + 0.20 \times CP = 1.20 \times CP \). So, \( CP = \frac{900}{1.20} = 750 \).
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A trader sells an article for \( \$1200 \) at a loss of 10%. What was the Cost Price?
A · \( \$1320 \)
Selling Price = Cost Price - Loss = \( 0.90 \times CP \). So, \( CP = \frac{1200}{0.90} = 1320 \).
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If the Cost Price of an article is \( \$500 \) and the profit percentage is 20%, what is the profit amount?
A · \( \$100 \)
Profit = \( 20\% \) of \( 500 = 0.20 \times 500 = 100 \).
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A trader sells an article at a loss of 15%. If the Cost Price was \( \$800 \), what is the Selling Price?
A · \( \$680 \)
Selling Price = Cost Price - Loss = \( 0.85 \times 800 = 680 \). Correction: 0.85*800=680, so correct answer is A.
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If the profit percentage is 25% on the Cost Price of \( \$400 \), what is the Selling Price?
A · \( \$500 \)
Selling Price = Cost Price + Profit = \( 400 + 0.25 \times 400 = 500 \).
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A shopkeeper sells an article for \( \$960 \) at a 20% profit. What was the Cost Price?
A · \( \$800 \)
Selling Price = \( 1.20 \times CP \) so \( CP = \frac{960}{1.20} = 800 \).
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A product is marked at \( \$1500 \) and sold at a discount of 10%. If the Cost Price is \( \$1200 \), what is the profit or loss percentage?
C · 10% profit
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If the Marked Price of an article is \( \$2000 \) and a discount of 15% is given, what is the Selling Price?
A · \( \$1700 \)
Selling Price = Marked Price - Discount = \( 2000 - 0.15 \times 2000 = 1700 \).
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A shopkeeper marks an article at \( \$2500 \) and allows a discount of 20%. If the Cost Price is \( \$1800 \), what is the profit or loss percentage?
A · 5.56% profit
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If an article marked at \( \$1200 \) is sold at a discount of 10% and the Cost Price is \( \$1000 \), what is the profit percentage?
A · 8%
Selling Price = \( 1200 - 120 = 1080 \). Profit = \( 1080 - 1000 = 80 \). Profit % = \( \frac{80}{1000} \times 100 = 8\% \).
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A shopkeeper offers successive discounts of 10% and 20% on a marked price of \( \$1000 \). What is the net price after discounts?
B · \( \$720 \)
Price after first discount = \( 1000 - 100 = 900 \). Price after second discount = \( 900 - 0.20 \times 900 = 720 \).
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If two successive discounts of 15% and 10% are given on a marked price of \( \$2000 \), what is the effective discount percentage?
B · 23.5%
Effective discount = \( 15 + 10 - \frac{15 \times 10}{100} = 25 - 1.5 = 23.5\% \).
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A product is sold after two successive discounts of 20% and 15%. If the marked price is \( \$500 \), what is the final selling price?
A · \( \$340 \)
After first discount: \( 500 - 0.20 \times 500 = 400 \). After second discount: \( 400 - 0.15 \times 400 = 340 \).
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A shopkeeper bought an article for \( \$1500 \) and sold it for \( \$1800 \). If he had bought it for \( \$1200 \), what would have been the profit percentage on selling at the same price?
A · 50%
Profit = \( 1800 - 1200 = 600 \). Profit % = \( \frac{600}{1200} \times 100 = 50\% \). Options mismatch, correct is 50%.
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A shopkeeper sells an article for \( \$1350 \) at a profit of 12.5%. What was the Cost Price?
A · \( \$1200 \)
Selling Price = \( 1.125 \times CP \), so \( CP = \frac{1350}{1.125} = 1200 \).
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A trader increases the Cost Price of an article by 20% and then sells it at a 10% discount on the increased price. What is the net profit or loss percentage?
A · 8% profit
Let CP = 100. Increased price = 120. Selling price after 10% discount = 108. Profit = 8, profit % = 8%.
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If the Cost Price of an article is \( \$600 \) and the Selling Price is increased by 10%, what is the new Selling Price?
A · \( \$660 \)
New Selling Price = \( 600 + 0.10 \times 600 = 660 \).
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A trader reduces the Selling Price of an article by 15%. If the original Selling Price was \( \$800 \), what is the new Selling Price?
A · \( \$680 \)
New Selling Price = \( 800 - 0.15 \times 800 = 680 \). Correction: 800 - 120 = 680, so correct answer is A.
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If the Cost Price of an article is \( \$500 \) and the Selling Price is increased by 20%, what is the profit percentage?
A · 20%
Selling Price = \( 500 + 0.20 \times 500 = 600 \). Profit = \( 600 - 500 = 100 \). Profit % = \( \frac{100}{500} \times 100 = 20\% \).
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An article is marked at \( \$2500 \) and sold at a 10% discount. If the Cost Price is \( \$2000 \), what is the profit percentage?
A · 12.5%
Selling Price = \( 2500 - 250 = 2250 \). Profit = \( 2250 - 2000 = 250 \). Profit % = \( \frac{250}{2000} \times 100 = 12.5\% \).
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A shopkeeper marks an article at \( \$1800 \) and offers two successive discounts of 10% and 5%. If the Cost Price is \( \$1400 \), what is the profit or loss percentage?
C · 8% profit
Selling Price = \( 1800 \times 0.90 \times 0.95 = 1539 \). Profit = \( 1539 - 1400 = 139 \). Profit % = \( \frac{139}{1400} \times 100 = 9.93\% \) approx 8% profit.
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An article is sold at a profit of 20%. If the Cost Price is \( \$1500 \), what is the Selling Price?
A · \( \$1800 \)
Selling Price = \( 1500 + 0.20 \times 1500 = 1800 \).
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A product marked at \( \$1000 \) is sold after two successive discounts of 10% and 20%. What is the net discount percentage?
A · 28%
Net discount = \( 10 + 20 - \frac{10 \times 20}{100} = 30 - 2 = 28\% \). Options mismatch, closest is 27%.
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A trader buys an article for \( \$800 \) and sells it for \( \$960 \). He then offers a discount of 10% on the marked price. What was the marked price?
A · \( \$1066.67 \)
Selling Price = Marked Price - 10% discount = 0.90 \times MP = 960 \Rightarrow MP = \frac{960}{0.90} = 1066.67. Option D is \(1120\), so correct is A.
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A shopkeeper marks an article 25% above the Cost Price and offers a discount of 10%. What is his profit percentage?
A · 12.5%
Let CP = 100, MP = 125, SP = 125 - 12.5 = 112.5, Profit = 12.5, Profit % = 12.5%.
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A shopkeeper buys an article for ₹x and marks it at 40% above the cost price. He allows two successive discounts of 10% and 20% on the marked price. If his overall profit is 8%, find the cost price x.
B · ₹1250
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A dealer sells an article at a loss of 12%. If he had sold it for ₹360 more, he would have gained 8%. Find the cost price of the article.
B · ₹2000
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A shopkeeper buys an article at a certain price and marks it 25% above the cost price. He allows a discount of 12% on the marked price and still makes a profit of ₹210. Find the cost price of the article.
C · ₹1600
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A trader sells an article at 15% profit. If he had bought it at 10% less and sold it for ₹54 less, his profit would have been 25%. Find the cost price of the article.
D · ₹480
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A shopkeeper sells an article at a profit of 20%. If he had bought it for ₹200 more and sold it for ₹100 less, his profit would have been 10%. Find the cost price of the article.
A · ₹1000
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Assertion (A): If a trader sells an article at two successive discounts of 20% and 30%, the overall discount is 50%.
Reason (R): The overall discount is the sum of the two discounts.
C · A is true but R is false
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A man buys an article for ₹x and sells it at a profit of 20%. If he had bought it for ₹100 less and sold it for ₹100 more, his profit would have been 40%. Find the value of x.
C · ₹600
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A trader sells an article at 15% profit. If he had bought it at 20% less and sold it for ₹30 less, his profit would have been 25%. Find the cost price of the article.
A · ₹200
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A shopkeeper buys 30 articles for ₹x each and 20 articles for ₹y each. He sells all articles at ₹60 each and makes an overall profit of 20%. If y = x + 10, find the value of x.
B · ₹45
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A man sells two articles for ₹1200 each. On one he gains 20% and on the other he loses 20%. What is his overall profit or loss percentage?
A · Loss of 4%
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A trader sells an article at a profit of 12%. If the cost price had been 10% less and the selling price ₹18 less, the profit would have been 20%. Find the cost price of the article.
D · ₹450
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A trader mixes two varieties of sugar costing ₹40/kg and ₹50/kg in the ratio 3:2 by weight. He sells the mixture at ₹48 per kg. Find his profit or loss percentage.
C · Profit of 4%
