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Ratio and Proportion

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Introduction to Ratio and Proportion

In everyday life, we often compare quantities to understand their relationship. For example, if you have 3 apples and 2 oranges, you might want to express how many apples there are compared to oranges. This comparison is called a ratio.

A ratio is a way to compare two quantities of the same kind by division. It tells us how many times one quantity contains another. When two such ratios are equal, we say they are in proportion.

Understanding ratio and proportion is important not only in daily activities like cooking or sharing money but also in solving problems in competitive exams. This chapter will guide you through these concepts step-by-step, with examples and tips to master them.

Ratio

A ratio compares two quantities of the same kind by showing how many times one quantity is to the other. We write a ratio using a colon (:), for example, 3:2.

Ratios can be of two main types:

  • Part-to-Part Ratio: Compares one part to another part. For example, in a class with 12 boys and 8 girls, the ratio of boys to girls is 12:8.
  • Part-to-Whole Ratio: Compares one part to the total. Using the same example, the ratio of boys to the total students (12 + 8 = 20) is 12:20.

Ratios can be simplified just like fractions, by dividing both terms by their highest common factor (HCF).

3 parts 2 parts

Figure: A bar divided into 3 blue units and 2 red units representing the ratio 3:2

How to Simplify Ratios

To simplify a ratio, divide both terms by their HCF. For example, to simplify 18:24:

  • Find HCF of 18 and 24, which is 6.
  • Divide both terms by 6: \( \frac{18}{6} : \frac{24}{6} = 3:4 \).

So, 18:24 simplifies to 3:4.

Proportion

A proportion states that two ratios are equal. For example, if \( \frac{a}{b} = \frac{c}{d} \), then the two ratios \( a:b \) and \( c:d \) are in proportion.

Proportions are useful to find unknown quantities when three values are known.

graph TD    A[Identify the two ratios] --> B[Set up the proportion \( \frac{a}{b} = \frac{c}{d} \)]    B --> C[Cross multiply: \( a \times d = b \times c \)]    C --> D[Solve for the unknown variable]

Flowchart: Steps to solve a proportion problem

Properties of Proportion

  • If \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \) (cross multiplication).
  • The product of the means equals the product of the extremes.
  • If three terms of a proportion are known, the fourth can be found.

Direct and Inverse Proportion

Sometimes quantities change in relation to each other. This change can be of two types:

Aspect Direct Proportion Inverse Proportion
Definition Both quantities increase or decrease together. One quantity increases while the other decreases.
Formula \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \) \( x_1 \times y_1 = x_2 \times y_2 \)
Example If 5 kg sugar costs INR 250, 8 kg costs INR ? If 6 workers take 10 days, 10 workers take ? days.

Direct Proportion Example

If the quantity of sugar increases, the cost increases proportionally.

Inverse Proportion Example

If the number of workers increases, the time taken decreases proportionally.

Worked Examples

Example 1: Simplifying a Ratio Easy
Simplify the ratio 18:24 to its lowest terms.

Step 1: Find the highest common factor (HCF) of 18 and 24.

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

HCF is 6.

Step 2: Divide both terms by 6.

\( \frac{18}{6} : \frac{24}{6} = 3 : 4 \)

Answer: The simplified ratio is 3:4.

Example 2: Solving a Proportion Problem Easy
Find the value of \( x \) in the proportion \( \frac{3}{4} = \frac{x}{12} \).

Step 1: Write the proportion as \( \frac{3}{4} = \frac{x}{12} \).

Step 2: Cross multiply: \( 3 \times 12 = 4 \times x \).

\( 36 = 4x \)

Step 3: Solve for \( x \):

\( x = \frac{36}{4} = 9 \)

Answer: \( x = 9 \)

Example 3: Direct Proportion Application Medium
If 5 kg of sugar costs INR 250, find the cost of 8 kg of sugar.

Step 1: Let the cost of 8 kg sugar be \( x \).

Step 2: Since cost and quantity are directly proportional, set up the proportion:

\( \frac{5}{250} = \frac{8}{x} \)

Step 3: Cross multiply:

\( 5 \times x = 250 \times 8 \)

\( 5x = 2000 \)

Step 4: Solve for \( x \):

\( x = \frac{2000}{5} = 400 \)

Answer: The cost of 8 kg sugar is INR 400.

Example 4: Inverse Proportion Application Medium
If 6 workers complete a task in 10 days, how many days will 10 workers take to complete the same task?

Step 1: Let the number of days taken by 10 workers be \( x \).

Step 2: Since workers and days are inversely proportional, use the formula:

\( 6 \times 10 = 10 \times x \)

Step 3: Calculate:

\( 60 = 10x \)

Step 4: Solve for \( x \):

\( x = \frac{60}{10} = 6 \)

Answer: 10 workers will take 6 days to complete the task.

Example 5: Mixture Problem Using Ratio and Proportion Hard
Two solutions are mixed in the ratio 3:2 to make 50 litres of mixture. Find the quantity of each solution used.

Step 1: Total parts = 3 + 2 = 5 parts.

Step 2: Each part represents \( \frac{50}{5} = 10 \) litres.

Step 3: Quantity of first solution = \( 3 \times 10 = 30 \) litres.

Step 4: Quantity of second solution = \( 2 \times 10 = 20 \) litres.

Answer: 30 litres of first solution and 20 litres of second solution are mixed.

Formula Bank

Ratio
\[ \text{Ratio} = \frac{\text{Quantity 1}}{\text{Quantity 2}} \]
where: Quantity 1, Quantity 2 are values being compared
Proportion
\[ \frac{a}{b} = \frac{c}{d} \implies ad = bc \]
where: a, b, c, d are terms of the ratios
Direct Proportion
\[ \frac{x_1}{y_1} = \frac{x_2}{y_2} \]
where: \( x_1, y_1 \) and \( x_2, y_2 \) are corresponding quantities
Inverse Proportion
\[ x_1 \times y_1 = x_2 \times y_2 \]
where: \( x_1, y_1 \) and \( x_2, y_2 \) are corresponding quantities
Mean Proportion
\[ b = \sqrt{ac} \]
where: a, b, c are quantities in continued proportion \( a:b = b:c \)
Compound Ratio
\[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]
where: a, b, c, d are terms of the ratios

Tips & Tricks

Tip: Always simplify ratios before solving proportion problems to reduce calculation errors.

When to use: When given complex ratios or large numbers.

Tip: Use cross multiplication method to quickly solve proportion equations.

When to use: When finding unknown terms in proportion.

Tip: In inverse proportion problems, remember that the product of the two quantities remains constant.

When to use: When quantities vary inversely, such as work-time or speed-distance problems.

Tip: Convert all quantities to the same unit (metric) before forming ratios.

When to use: When dealing with measurement units in problems.

Tip: For mixture problems, use the ratio to divide the total quantity directly instead of solving equations.

When to use: When total quantity and mixing ratio are given.

Common Mistakes to Avoid

❌ Confusing part-to-part ratio with part-to-whole ratio.
✓ Remember that part-to-part compares two parts, while part-to-whole compares a part to the total.
Why: Students often misinterpret the denominator in ratios, leading to wrong answers.
❌ Not simplifying ratios before solving proportion problems.
✓ Always simplify ratios to lowest terms to avoid calculation errors.
Why: Simplifying reduces complexity and chances of mistakes.
❌ Incorrectly applying direct proportion formula to inverse proportion problems.
✓ Identify the type of proportion first; use product constant formula for inverse proportion.
Why: Misunderstanding the nature of variation leads to wrong formula application.
❌ Ignoring units or mixing different units in ratio calculations.
✓ Convert all quantities to the same unit system before forming ratios.
Why: Different units distort ratio values.
❌ Forgetting to cross multiply both sides correctly in proportion problems.
✓ Multiply numerator of one ratio with denominator of the other and vice versa carefully.
Why: Incorrect cross multiplication leads to wrong answers.
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