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In everyday life, we often compare quantities to understand their relationship. For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, we say the ratio of flour to sugar is 2 to 1. Similarly, when converting currency, mixing paints, or sharing money, the concepts of ratio and proportion help us solve problems efficiently.
Ratio is a way to compare two quantities by division, showing how many times one quantity contains another. Proportion is an equation stating that two ratios are equal. These concepts are fundamental in mathematics and appear frequently in competitive exams, making it essential to understand them clearly.
A ratio compares two quantities of the same kind by division. It tells us how much of one quantity there is compared to another. For example, if there are 3 apples and 2 oranges, the ratio of apples to oranges is 3:2.
Ratios can be written in three ways:
It is important to note that a ratio is not a fraction representing a part of a whole but a comparison between two separate quantities.
Just like fractions, ratios can be simplified by dividing both terms by their greatest common divisor (GCD). Simplifying makes calculations easier and helps in comparing ratios effectively.
Figure: A pie chart representing the ratio 3:2, where the green segment is 3 parts and the blue segment is 2 parts of the whole.
Proportion is an equation that states two ratios are equal. If the ratio of \(a\) to \(b\) is the same as the ratio of \(c\) to \(d\), then we say:
\(\frac{a}{b} = \frac{c}{d}\)
This means the two ratios form a proportion.
The most important property used to verify or solve proportions is cross multiplication. It states that:
\(a \times d = b \times c\)
If this equality holds, the two ratios are in proportion.
graph TD A[Start with two ratios a:b and c:d] B[Cross multiply: Calculate a x d and b x c] C{Are a x d and b x c equal?} D[Yes - Ratios are in proportion] E[No - Ratios are not in proportion] A --> B --> C C -->|Yes| D C -->|No| EWhen three or more quantities are in proportion, such as \(a : b = b : c\), the middle term \(b\) is called the mean proportional between \(a\) and \(c\). This concept is useful in geometric progressions and similar triangles.
Understanding how quantities relate to each other is key in solving many problems. Two common types of proportion are:
| Type | Definition | Formula | Example |
|---|---|---|---|
| Direct Proportion | Two quantities increase or decrease together. | \(\frac{x_1}{y_1} = \frac{x_2}{y_2}\) | If 5 kg rice costs INR 250, 8 kg costs INR ? |
| Inverse Proportion | One quantity increases as the other decreases. | \(x_1 y_1 = x_2 y_2\) | If 6 workers take 10 days, 15 workers take ? days. |
Step 1: Find the greatest common divisor (GCD) of 24 and 36.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The greatest common divisor is 12.
Step 2: Divide both terms by 12.
\(\frac{24}{12} : \frac{36}{12} = 2 : 3\)
Answer: The simplified ratio is 2:3.
Step 1: Write the ratios as fractions:
\(\frac{3}{4}\) and \(\frac{9}{12}\)
Step 2: Cross multiply:
\(3 \times 12 = 36\)
\(4 \times 9 = 36\)
Step 3: Since both cross products are equal, the ratios are in proportion.
Answer: Yes, 3:4 and 9:12 are proportional.
Step 1: Set up the proportion based on direct proportion:
\(\frac{5}{250} = \frac{8}{x}\), where \(x\) is the cost of 8 kg rice.
Step 2: Cross multiply:
\(5 \times x = 8 \times 250\)
\(5x = 2000\)
Step 3: Solve for \(x\):
\(x = \frac{2000}{5} = 400\)
Answer: The cost of 8 kg of rice is INR 400.
Step 1: Understand that the number of workers and days are inversely proportional.
So, \(6 \times 10 = 15 \times x\), where \(x\) is the number of days for 15 workers.
Step 2: Calculate \(x\):
\(60 = 15x\)
\(x = \frac{60}{15} = 4\)
Answer: 15 workers will complete the task in 4 days.
Step 1: Let the initial quantity of milk be \(7x\) litres and water be \(3x\) litres.
Step 2: After adding 20 litres of water, water becomes \(3x + 20\) litres.
Step 3: The new ratio is given as 7:5, so:
\(\frac{7x}{3x + 20} = \frac{7}{5}\)
Step 4: Cross multiply:
\(7x \times 5 = 7 \times (3x + 20)\)
\(35x = 21x + 140\)
Step 5: Solve for \(x\):
\(35x - 21x = 140\)
\(14x = 140\)
\(x = 10\)
Step 6: Find the initial quantity of milk:
\(7x = 7 \times 10 = 70\) litres.
Answer: The initial quantity of milk is 70 litres.
When to use: At the start of any ratio problem.
When to use: When verifying proportion or solving proportion equations.
When to use: When quantities increase or decrease together.
When to use: When one quantity increases as the other decreases.
When to use: When solving complex word problems.
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