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Fractions and decimals

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Introduction to Fractions and Decimals

Fractions and decimals are fundamental concepts in mathematics that help us represent parts of a whole. Whether you are measuring ingredients in a recipe, calculating discounts during shopping, or working with currency like Indian Rupees (INR), fractions and decimals are everywhere.

Understanding these concepts is essential for competitive exams, as they form the basis of many quantitative aptitude problems. In this chapter, we will explore fractions and decimals from the ground up, learn how to perform operations on them, and see how they connect to percentages, ratios, and averages.

For example, when you buy 2.5 kg of apples priced at Rs.120 per kg, you use decimals to calculate the total cost. Similarly, if a tank is \(\frac{3}{4}\) full of water, fractions help describe the quantity precisely. This chapter will equip you with the skills to handle such problems confidently.

Definition and Types of Fractions

A fraction represents a part of a whole. It is written as \(\frac{a}{b}\), where:

  • Numerator (\(a\)) is the number of parts considered.
  • Denominator (\(b\)) is the total number of equal parts the whole is divided into.

For example, \(\frac{3}{5}\) means 3 parts out of 5 equal parts.

There are three main types of fractions:

  • Proper Fraction: Numerator is less than denominator (e.g., \(\frac{2}{7}\)). It represents a quantity less than 1.
  • Improper Fraction: Numerator is greater than or equal to denominator (e.g., \(\frac{9}{4}\)). It represents a quantity equal to or greater than 1.
  • Mixed Fraction: A combination of a whole number and a proper fraction (e.g., \(2 \frac{1}{3}\)).
Proper: 3/5 Improper: 9/4 Mixed: 2 1/3 (Whole parts not shown)

Decimal Representation and Conversion

A decimal is another way to represent fractions, especially those with denominators that are powers of 10 (like 10, 100, 1000). Decimals use a decimal point to separate the whole number part from the fractional part.

For example, 0.75 means 75 parts out of 100, which is \(\frac{75}{100}\) or simplified \(\frac{3}{4}\).

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal as a fraction with denominator 10, 100, 1000, etc., depending on the number of decimal places, and then simplify.

Fraction Decimal
\(\frac{1}{2}\)0.5
\(\frac{3}{4}\)0.75
\(\frac{2}{5}\)0.4
\(\frac{7}{10}\)0.7
\(\frac{1}{8}\)0.125

Operations on Fractions

Performing operations on fractions requires understanding how to handle numerators and denominators carefully. Let's look at addition, subtraction, multiplication, and division step-by-step.

graph TD    A[Start] --> B{Operation?}    B --> C[Addition/Subtraction]    B --> D[Multiplication/Division]    C --> E[Find LCM of denominators]    E --> F[Convert fractions to equivalent fractions with LCM denominator]    F --> G[Add/Subtract numerators]    G --> H[Simplify the result]    D --> I[Multiply numerators and denominators (for multiplication)]    I --> J[Simplify the result]    D --> K[Invert divisor and multiply (for division)]    K --> J

Addition and Subtraction: To add or subtract fractions, first find the Least Common Multiple (LCM) of the denominators to get a common denominator. Then convert each fraction to an equivalent fraction with this denominator, add or subtract the numerators, and simplify.

Multiplication: Multiply the numerators together and the denominators together. Simplify the resulting fraction.

Division: To divide by a fraction, multiply by its reciprocal (flip numerator and denominator) and then multiply as usual.

Operations on Decimals

Decimals are operated on similarly to whole numbers, but alignment of the decimal point is crucial.

  • Addition and Subtraction: Align the decimal points vertically before adding or subtracting. If necessary, add zeros to equalize decimal places.
  • Multiplication: Multiply as if the numbers were whole numbers, ignoring the decimal points initially. Then, count the total number of decimal places in both numbers and place the decimal point in the product accordingly.
  • Division: To divide decimals, move the decimal point in the divisor to the right to make it a whole number. Move the decimal point in the dividend the same number of places. Then divide as usual.

Conversion between Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are closely related and often interchangeable.

To convert:

  • Fraction to Decimal: Divide numerator by denominator.
  • Decimal to Fraction: Write decimal as fraction with denominator \(10^n\), where \(n\) is the number of decimal places, then simplify.
  • Fraction to Percentage: Multiply fraction by 100.
  • Decimal to Percentage: Multiply decimal by 100.
  • Percentage to Fraction/Decimal: Divide by 100.
Fraction Decimal Percentage
\(\frac{1}{2}\)0.550%
\(\frac{3}{4}\)0.7575%
\(\frac{1}{5}\)0.220%
\(\frac{7}{10}\)0.770%

Worked Examples

Example 1: Adding Fractions with Different Denominators Easy
Add \(\frac{2}{3}\) and \(\frac{3}{5}\).

Step 1: Find the LCM of denominators 3 and 5. LCM(3,5) = 15.

Step 2: Convert each fraction to an equivalent fraction with denominator 15.

\(\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}\)

\(\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}\)

Step 3: Add the numerators: \(10 + 9 = 19\).

Step 4: Write the sum as \(\frac{19}{15}\), which is an improper fraction.

Step 5: Convert to mixed fraction: \(\frac{19}{15} = 1 \frac{4}{15}\).

Answer: \(1 \frac{4}{15}\)

Example 2: Multiplying Decimals in Currency Calculations Medium
Calculate the total cost if 2.5 kg of apples cost Rs.120 per kg.

Step 1: Multiply the quantity by the price per kg.

\(2.5 \times 120\)

Step 2: Multiply as whole numbers ignoring decimals: \(25 \times 120 = 3000\).

Step 3: Since 2.5 has one decimal place, place the decimal point one place from the right in the product.

Total cost = Rs.300.0 or Rs.300

Answer: Rs.300

Example 3: Converting Recurring Decimals to Fractions Hard
Convert the recurring decimal \(0.\overline{6}\) (i.e., 0.666...) into a fraction.

Step 1: Let \(x = 0.666...\)

Step 2: Multiply both sides by 10 to shift the decimal point:

\(10x = 6.666...\)

Step 3: Subtract the original equation from this:

\(10x - x = 6.666... - 0.666...\)

\(9x = 6\)

Step 4: Solve for \(x\):

\(x = \frac{6}{9} = \frac{2}{3}\)

Answer: \(0.\overline{6} = \frac{2}{3}\)

Example 4: Percentage Calculation using Fractions Medium
Find 25% of 80 liters of milk using fraction conversion.

Step 1: Convert 25% to fraction: \(25\% = \frac{25}{100} = \frac{1}{4}\).

Step 2: Multiply the fraction by the total quantity:

\(\frac{1}{4} \times 80 = 20\) liters.

Answer: 20 liters of milk

Example 5: Solving Ratio Problems with Fractions and Decimals Medium
If the ratio of sugar to flour is 3:5, find the quantity of sugar needed for 2.5 kg of flour.

Step 1: The ratio sugar:flour = 3:5.

Step 2: For 5 parts flour, quantity = 2.5 kg.

Step 3: Find the value of one part:

\(1 \text{ part} = \frac{2.5}{5} = 0.5\) kg.

Step 4: Quantity of sugar (3 parts) = \(3 \times 0.5 = 1.5\) kg.

Answer: 1.5 kg of sugar

Formula Bank

Addition of Fractions
\[\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\]
where: \(a,b,c,d\) are integers; \(b,d eq 0\)
Subtraction of Fractions
\[\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\]
where: \(a,b,c,d\) are integers; \(b,d eq 0\)
Multiplication of Fractions
\[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]
where: \(a,b,c,d\) are integers; \(b,d eq 0\)
Division of Fractions
\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\]
where: \(a,b,c,d\) are integers; \(b,c,d eq 0\)
Fraction to Decimal Conversion
Decimal = \(\frac{\text{Numerator}}{\text{Denominator}}\)
Numerator and Denominator are integers; Denominator \( eq 0\)
Decimal to Fraction Conversion
Fraction = \(\frac{\text{Decimal} \times 10^n}{10^n}\)
Decimal is the decimal number; \(n\) is number of decimal places
Percentage from Fraction
Percentage = \(\left(\frac{\text{Numerator}}{\text{Denominator}}\right) \times 100\)
Numerator and Denominator are integers; Denominator \( eq 0\)

Tips & Tricks

Tip: Always simplify fractions at every step to avoid large numbers.

When to use: During fraction operations to reduce calculation complexity.

Tip: Convert fractions to decimals for easier comparison.

When to use: When comparing sizes of fractions quickly.

Tip: Use cross multiplication to compare fractions without converting to decimals.

When to use: To quickly determine which fraction is larger.

Tip: Remember that multiplying or dividing numerator and denominator by the same number keeps the fraction equivalent.

When to use: To simplify fractions or find equivalent fractions.

Tip: For recurring decimals, use algebraic methods to convert to fractions.

When to use: When dealing with repeating decimal problems.

Common Mistakes to Avoid

❌ Adding fractions by simply adding numerators and denominators.
✓ Find common denominator before adding numerators.
Why: Students confuse fraction addition with whole number addition.
❌ Ignoring place value when adding or subtracting decimals.
✓ Align decimal points before performing operations.
Why: Misalignment leads to incorrect sums or differences.
❌ Not simplifying fractions after operations.
✓ Always reduce fractions to simplest form.
Why: Leads to unnecessarily complex answers and may cause errors.
❌ Confusing recurring decimals with terminating decimals.
✓ Identify recurring patterns and apply correct conversion methods.
Why: Misinterpretation leads to wrong fraction equivalents.
❌ Using incorrect conversion factor when converting decimals to fractions.
✓ Count decimal places accurately and use \(10^n\) accordingly.
Why: Incorrect denominator leads to wrong fraction.
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