In this chapter, you will learn
- —Understand what a ratio is and how to express it in different forms
- —Simplify and compare ratios
- —Understand the concept of proportion and its properties
- —Apply unitary method to solve problems
- —Understand direct and indirect variation
- —Solve real-life problems using ratios and proportions
What is a Ratio?
Ratio is a comparison of two quantities expressed in the same units.
Definition: The ratio of a to b is written as a : b or a/b (b ≠ 0).
The first quantity is called the antecedent and the second is called the consequent.
Key Properties:
- Ratio is a comparison of quantities of the same kind
- Ratio has no units (it's a pure number)
- Order matters: 3:5 ≠ 5:3
- Ratio remains unchanged if both terms are multiplied or divided by the same non-zero number
Examples:
- If there are 5 boys and 3 girls, ratio = 5:3
- If length is 10 m and width is 6 m, ratio = 10:6 = 5:3 (simplified)
Exam Tip
Remember: ratios are used to compare quantities. Always express in simplest form by dividing by the HCF of both terms.
Common Mistake
Students often forget to simplify ratios. 12:8 ≠ 12:8; it should be simplified to 3:2.
Simplifying and Comparing Ratios
Simplifying Ratios: Divide both terms by their HCF (Highest Common Factor).
Example: Simplify 24:36
HCF(24, 36) = 12
24:36 = (24÷12):(36÷12) = 2:3
Comparing Ratios: Convert to fractions and compare.
- Ratio a:b = a/b
- To compare: convert to same denominator or convert to decimals
- Example: Compare 3:4 and 5:6 → 3/4 = 0.75, 5/6 ≈ 0.833 → 5:6 is greater
Equivalent Ratios: Ratios with different terms but same value.
- 2:3 = 4:6 = 6:9 = 8:12 (all equal to 2/3)
- Generate by multiplying or dividing both terms by the same number
Exam Tip
Always simplify ratios to their lowest terms. Use cross-multiplication to compare ratios quickly.
Common Mistake
Mixing up HCF and LCM. Use HCF to simplify, not LCM.
Unitary Method
Unitary Method: Finding the value of one unit first, then the value of the required quantity.
Steps:
- Find the value of one unit
- Multiply by the required quantity
Example:
If 5 pencils cost Rs. 40, what is the cost of 12 pencils?
- Cost of 1 pencil = 40 ÷ 5 = Rs. 8
- Cost of 12 pencils = 8 × 12 = Rs. 96
Another Example:
If 3 workers build a wall in 12 days, how many days for 9 workers (same work)?
- Total work = 3 × 12 = 36 worker-days
- Days for 9 workers = 36 ÷ 9 = 4 days
Exam Tip
Unitary method is essential for solving word problems. Always find the unit value first.
Common Mistake
Forgetting that if quantity increases, time/cost per unit should decrease in certain contexts.
Proportion and Its Properties
Proportion: A statement that two ratios are equal.
a : b = c : d or a/b = c/d
Read as: "a is to b as c is to d"
Properties of Proportion:
1. Cross-multiplication: If a:b = c:d, then ad = bc
Example: If 2:3 = x:6, then 2×6 = 3×x → x = 4
2. In proportion a:b = c:d:
- a and d are called extremes
- b and c are called means
- Product of extremes = Product of means (ad = bc)
Exam Tip
Cross-multiplication is the quickest way to check if two ratios are proportional.
Common Mistake
Writing proportions with wrong order. Remember: order of terms matters in a proportion.
Direct and Indirect Variation
Direct Variation: When one quantity increases, the other increases proportionally.
Formula: y = kx (where k is constant)
Examples:
- More work hours → More work done
- More speed → More distance in same time
- Cost is directly proportional to quantity
Indirect Variation (Inverse): When one quantity increases, the other decreases proportionally.
Formula: y = k/x (where k is constant) or xy = k
Examples:
- More workers → Less time to complete work
- More speed → Less time to cover distance
- More price per unit → Less quantity you can buy with fixed money
Key Difference:
- Direct: Ratio of quantities is constant (y/x = k)
- Indirect: Product of quantities is constant (xy = k)
Exam Tip
Identify from word problem context: 'more...more' = direct; 'more...less' = indirect variation.
Common Mistake
Confusing direct and indirect variation. Think: more workers = less time (inverse).
Chapter Summary
Ratio and Proportion are fundamental concepts in mathematics with wide applications. Key points:
- Ratio: Comparison of two quantities, expressed as a:b or a/b
- Simplification: Divide by HCF to get simplest form
- Unitary Method: Find value of one unit, then multiply
- Proportion: Two ratios are equal; cross-product rule: ad = bc
- Direct Variation: y = kx (both increase together)
- Indirect Variation: xy = k (one increases, other decreases)
Exam Focus: Simplifying ratios, solving proportions using cross-multiplication, unitary method applications, identifying and solving direct/indirect variation problems.