Ratio and Proportion — Study Notes for Assam TET Paper I

Overview

Ratio and Proportion forms a fundamental building block of primary mathematics and appears consistently in Assam TET Paper I. This topic tests your ability to compare quantities, find missing values in proportions, and apply the unitary method to solve real-life problems involving cost, time, work, and distribution.

For the exam, you need mastery at two levels: solving problems quickly and accurately, and understanding how to teach these concepts to Classes I–V students. Questions typically involve finding equivalent ratios, determining whether quantities are in proportion, and applying the unitary method to word problems set in familiar contexts like markets, farms, and households of Assam.

The topic connects directly to fractions, multiplication, division, and percentages. A strong grasp here simplifies later topics like speed-distance-time and work problems. Expect 2–4 questions from this area, often presented as word problems requiring careful reading.

Key Concepts

• **Ratio** is a comparison of two quantities of the same kind expressed as a:b or a/b. The first term is the antecedent, the second is the consequent.

• **Equivalent ratios** are obtained by multiplying or dividing both terms by the same non-zero number. For example, 2:3 = 4:6 = 6:9.

• **Simplest form** of a ratio has no common factor other than 1 between its terms. Always reduce ratios to simplest form before comparing.

• **Proportion** states that two ratios are equal. If a:b = c:d, we write a:b :: c:d and read it as "a is to b as c is to d."

• **Cross-product rule**: In a proportion a:b :: c:d, the product of extremes equals the product of means, i.e., a × d = b × c.

• **Unitary method** finds the value of one unit first, then scales to the required quantity. It works for both direct and inverse relationships at the primary level.

• **Direct proportion**: When one quantity increases, the other increases proportionally (more items cost more money).

• **Continued proportion**: Three quantities a, b, c are in continued proportion if a:b = b:c. Here b is the mean proportional and b² = a × c.

Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Ratio of a to b | a : b = a/b (both quantities must have the same unit) | | Simplifying ratio | Divide both terms by their HCF | | Equivalent ratio | Multiply/divide both terms by same number | | Proportion test | a : b :: c : d if a × d = b × c | | Finding missing term | If a : b :: c : x, then x = (b × c) / a | | Mean proportional | If a : b :: b : c, then b = √(a × c) | | Unitary method (direct) | Value of n units = (Value of 1 unit) × n | | Dividing in ratio a : b | Parts are (a × Total)/(a+b) and (b × Total)/(a+b) |

**Key fact**: Ratios have no units. Convert quantities to the same unit before forming a ratio. For example, to find ratio of 50 paise to ₹2, convert ₹2 to 200 paise first, giving ratio 50:200 = 1:4.

Worked Examples

### Example 1: Simplifying and Comparing Ratios

**Problem**: Express the ratio of 45 minutes to 1 hour 30 minutes in simplest form.

**Solution**:

• Convert to same unit: 1 hour 30 minutes = 90 minutes
• Ratio = 45 : 90
• HCF of 45 and 90 is 45
• Divide both terms: 45 ÷ 45 : 90 ÷ 45 = 1 : 2
• **Answer**: 1 : 2

---

### Example 2: Finding Missing Term in Proportion

**Problem**: If 3 : 5 :: 12 : x, find x.

**Solution**:

• Using cross-product: 3 × x = 5 × 12
• 3x = 60
• x = 60 ÷ 3 = 20
• **Verification**: 3 × 20 = 60 and 5 × 12 = 60 ✓
• **Answer**: x = 20

---

### Example 3: Unitary Method

**Problem**: If 8 kg of rice costs ₹320, what is the cost of 15 kg of rice?

**Solution**:

• Cost of 8 kg = ₹320
• Cost of 1 kg = 320 ÷ 8 = ₹40
• Cost of 15 kg = 40 × 15 = ₹600
• **Answer**: ₹600

---

### Example 4: Dividing a Quantity in a Given Ratio

**Problem**: Divide ₹630 between Rina and Mina in the ratio 4 : 5.

**Solution**:

• Total parts = 4 + 5 = 9
• Value of 1 part = 630 ÷ 9 = ₹70
• Rina's share = 4 × 70 = ₹280
• Mina's share = 5 × 70 = ₹350
• **Verification**: 280 + 350 = 630 ✓
• **Answer**: Rina gets ₹280, Mina gets ₹350

---

### Example 5: Checking Proportion

**Problem**: Are 6, 8, 9, 12 in proportion?

**Solution**:

• Check if 6 : 8 :: 9 : 12
• Product of extremes = 6 × 12 = 72
• Product of means = 8 × 9 = 72
• Since both products are equal, the numbers are in proportion.
• **Answer**: Yes, they are in proportion.

Common Mistakes

1. **Mixing units without conversion** → Always convert both quantities to the same unit before forming a ratio. Students write 2 hours : 30 minutes as 2:30, which is wrong. Correct: 120:30 = 4:1.

2. **Reversing the order of ratio** → Ratio is order-sensitive. The ratio of boys to girls (3:2) is different from girls to boys (2:3). Read the question carefully.

3. **Not simplifying to lowest terms** → Leaving answer as 12:18 instead of 2:3 loses marks. Always find HCF and reduce.

4. **Confusing proportion check** → Students add terms instead of using cross-multiplication. Remember: multiply extremes and means, do not add.

5. **Unitary method direction error** → In direct proportion, if quantity increases, value increases. Students sometimes divide when they should multiply. Think: "More items, more cost" or "Less items, less cost."

6. **Forgetting that ratios are unitless** → Writing "The ratio is 3:5 kg" is incorrect. Ratios have no units—just 3:5.

Quick Reference

• **Ratio a:b** means a/b; both quantities must be in same units before comparison.

• **Cross-multiply to check proportion**: a × d must equal b × c.

• **Unitary method**: Find value of ONE unit first, then multiply for required quantity.

• **To divide amount in ratio a:b**: Each part = Total ÷ (a+b), then multiply by a and b separately.

• **Simplify ratios** by dividing both terms by their HCF.

• **Mean proportional of a and c** = √(a × c).