Ratio and Proportion

1. Introduction to Ratio and Proportion

Ratio and proportion are important concepts in mathematics that help us compare quantities and solve real-life problems involving sharing, scaling, and mixing.

2. Ratio

A ratio is a way to compare two or more quantities of the same kind by division. It shows how many times one quantity is contained in another.

• The ratio of a to b is written as a : b.
• Example: If there are 2 apples and 3 oranges, the ratio of apples to oranges is 2 : 3.
• Ratios can be simplified just like fractions. For example, 6 : 9 = 2 : 3.

2.1 Properties of Ratios

• Both quantities must be in the same units.
• The order of terms in a ratio is important (2 : 3 is not the same as 3 : 2).
• Ratios can be scaled up or down by multiplying or dividing both terms by the same number.

3. Proportion

A proportion states that two ratios are equal. It is used to solve problems where two ratios are compared.

• If a : b = c : d, then a, b, c, d are in proportion.
• It is written as a : b :: c : d or a/b = c/d.
• Example: If 2 : 3 = 4 : 6, then 2, 3, 4, 6 are in proportion.

3.1 Properties of Proportion

• In a proportion, the product of the extremes equals the product of the means: a × d = b × c.
• If three numbers a, b, c are in continued proportion, then a : b = b : c.

4. Solving Ratio and Proportion Problems

• To find a missing term in a proportion, use cross-multiplication.
• To divide a quantity in a given ratio, add the parts of the ratio and distribute accordingly.

Examples

1. Ratio Example: There are 12 boys and 8 girls in a class. What is the ratio of boys to girls?
   Solution: 12 : 8 = 3 : 2 (after dividing both by 4)
2. Proportion Example: If 3 pencils cost ₹15, how much do 7 pencils cost?
   Solution: Let the cost of 7 pencils be ₹x.
   3 : 7 = 15 : x ⇒ 3/x = 15/7 ⇒ x = (7 × 15) / 3 = ₹35
3. Dividing in a Ratio: Divide ₹120 between A and B in the ratio 2 : 3.
   Solution: Total parts = 2 + 3 = 5
   A gets (2/5) × 120 = ₹48, B gets (3/5) × 120 = ₹72

5. Practice Exercises

1. Write the ratio of 18 to 24 in simplest form.
2. If 5 kg of rice costs ₹200, what is the cost of 8 kg of rice?
3. Divide 90 sweets between two children in the ratio 4 : 5.
4. Are the numbers 8, 12, 18, 27 in proportion?
5. If 7 notebooks cost ₹84, how much do 10 notebooks cost?

6. Summary

• Ratio compares two quantities of the same kind.
• Proportion shows that two ratios are equal.
• Cross-multiplication helps solve proportion problems.
• Ratios and proportions are useful in daily life for sharing, mixing, and scaling.