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REET L1+L2 · Maths

Ratio and Proportion

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This page brings together every formula, type and word-problem technique for ratio and proportion you need for REET. The Arithmetic section brings a word problem from this topic almost every year.

  • 1 question in REET 2024
  • Class 5–7 level
  • 8 min read
Verified 17 September 2026

How REET asks this topic

REET 2024 carried a word problem based on ratio, where you had to use the 'parts' of a ratio and a given difference to find a variable, and from there the total. The pattern is typical: rather than simply simplifying a ratio, you're expected to build an equation from a real situation and solve it.

Definition — Ratio and Proportion

A ratio compares two quantities of the same unit, written as a : b. Here a is called the antecedent and b the consequent.

A proportion is the equality of two ratios, written as a : b :: c : d. Here a and d are called the extremes, and b and c the means. The condition for a proportion: product of the extremes = product of the means, i.e. a × d = b × c — this is called cross multiplication.

Related Terms and Formulas

Mean proportional
If a : b = b : c, then b is the mean proportional; b² = a × c, i.e. b = √(ac)
Third proportional
In a : b = b : c, c is called the third proportional of a and b; c = b²/a
Fourth proportional
In a : b = c : d, d is called the fourth proportional; d = (b × c) / a
Compounded ratio
Combining a:b and c:d gives a compounded ratio of ac : bd

Direct and Inverse Proportion

In direct proportion, when one quantity increases, the other increases in the same ratio (for example, total cost increasing as the number of items increases). In inverse proportion, when one quantity increases, the other decreases in the same ratio (for example, time decreasing as the number of workers increases). Multiplying or dividing both terms of a ratio by the same non-zero number never changes its value — this is the fundamental property of a ratio.

Solved Examples

Example 1: Two numbers are in the ratio 5:7, and their sum is 144. Find the numbers. Solution: Let the numbers be 5k and 7k. 5k+7k=144 ⇒ 12k=144 ⇒ k=12. The numbers are 60 and 84.

Example 2: The present ages of A and B are in the ratio 4:5. After 5 years, this ratio becomes 5:6. Find their present ages. Solution: Let the ages be 4x and 5x. (4x+5)/(5x+5) = 5/6 ⇒ 6(4x+5)=5(5x+5) ⇒ 24x+30=25x+25 ⇒ x=5. So A's age is 20 years and B's age is 25 years. Check: (20+5):(25+5) = 25:30 = 5:6 ✓.

Practical Examples of Direct and Inverse Proportion

Example of direct proportion: If 5 notebooks cost ₹75, what will 8 notebooks cost? Here, as the number of notebooks increases, the cost increases in the same ratio — 1 notebook = ₹15, so 8 notebooks = ₹120.

Example of inverse proportion: If 6 workers complete a task in 10 days, how many days will 4 workers take to complete the same task? Here, as the number of workers decreases, the time increases — 6×10 = 4×days ⇒ days = 60/4 = 15 days. Identifying whether a question is direct or inverse is always the first step in any ratio word problem.

Example of third proportional: Find the third proportional of 4 and 8. Solution: If a:b = b:c and a=4, b=8, then c = b²/a = 64/4 = 16. So 4, 8 and 16 are in continued proportion (4:8 = 8:16).

Example of fourth proportional: Find the fourth proportional of 2, 3 and 10. Solution: If a:b :: c:d and a=2, b=3, c=10, then d = (b×c)/a = (3×10)/2 = 15. So 2:3 :: 10:15 (check: 2×15=30=3×10 ✓).

How to Teach This to Children

Classroom activity

Use coloured counters or paint-mixing to show ratio — for example, mixing 2 parts blue with 3 parts yellow to make green, then asking how much of each is needed to make 10 parts of green.

Teaching aids

Ratio strips (bar models) and fraction-ratio comparison charts are useful here.

Common misconception

Students often flip the order of a ratio (writing 3:5 as 5:3), or confuse 'how much more' with 'how many times'. In a word problem, it's essential to be clear about which quantity is the 'first term' and which is the 'second term'. Some students also equate the ratio a:b with the fraction a/b, forgetting that a ratio only makes sense for comparing quantities of the same unit.

Assessment idea

Check understanding using a kitchen recipe (like flour:sugar) or the real ratio of boys to girls in class. Real-life examples like a map's scale also strengthen understanding of ratio — for instance, finding the real distance between two cities using a scale of 1 cm = 1 km.

Memory trick / exam tip

In ratio word problems, always let each part of the ratio equal 'k' (for example, treat 3:5 as 3k, 5k) — this instantly turns the given difference or sum into an equation in k, and the answer falls out directly. Don't forget to cross-check the final answer against the original ratio.

REET 2024 Questions (with Solutions)

Question 117

The ratio of female and male employees in a factory is 3 : 5. If the number of females is 40 less than the number of male, then how many total employees are there in the factory ?

  1. A60
  2. B160
  3. C100
  4. D200

Explanation

माना स्त्रियाँ 3k और पुरुष 5k हैं। अन्तर 5k − 3k = 2k = 40, अतः k = 20। कुल कर्मचारी = 8k = 160।

Practice Questions

Question 1

Simplify the ratio 45:60 to its simplest form.

  1. A3:4
  2. B4:5
  3. C9:12
  4. D5:4

Explanation

The HCF of 45 and 60 is 15. Dividing both terms by 15 gives 45:60 = 3:4.

Question 2

If a:b = 2:3 and b:c = 4:5, what is a:b:c?

  1. A8:12:15
  2. B2:3:5
  3. C8:15:12
  4. D6:9:15

Explanation

To make the value of b the same in both ratios, scale a:b=2:3 to 8:12 (×4) and b:c=4:5 to 12:15 (×3). So a:b:c = 8:12:15.

Question 3

₹750 is divided among A, B, C in the ratio 2:3:5. What is B's share?

  1. A₹225
  2. B₹150
  3. C₹375
  4. D₹250

Explanation

Total parts = 2+3+5 = 10. One part = 750/10 = 75. B's share = 3 × 75 = ₹225.

Question 4

If x:y = 3:4, what is (2x+3y):(3x+4y)?

  1. A18:25
  2. B9:16
  3. C6:7
  4. D3:4

Explanation

Let x=3k, y=4k. 2x+3y = 6k+12k = 18k. 3x+4y = 9k+16k = 25k. Ratio = 18:25.

Question 5

What is the mean proportional between 4 and 25?

  1. A10
  2. B14.5
  3. C100
  4. D20

Explanation

Mean proportional = √(4×25) = √100 = 10.

Question 6

If a:b :: b:c and a=9, c=25, find b (b is positive).

  1. A15
  2. B17
  3. C16
  4. D225

Explanation

b² = a×c = 9×25 = 225 ⇒ b = 15.

Question 7

In a school, the ratio of boys to girls is 7:5. If there are 84 boys, how many girls are there?

  1. A60
  2. B70
  3. C56
  4. D48

Explanation

7 parts = 84 ⇒ 1 part = 12. Girls = 5 × 12 = 60.

Question 8

Two numbers are in the ratio 3:4. If 4 is added to each, the ratio becomes 4:5. Find the numbers.

  1. A12 और 16
  2. B9 और 12
  3. C15 और 20
  4. D6 और 8

Explanation

Let the numbers be 3k, 4k. (3k+4)/(4k+4)=4/5 ⇒ 15k+20=16k+16 ⇒ k=4. Numbers = 12, 16. Check: 16/20 = 4/5 ✓.

Question 9

In a recipe, flour and sugar are mixed in the ratio 5:2. If 3 kg of sugar is used, how much flour is needed?

  1. A7.5 किग्रा
  2. B6 किग्रा
  3. C10 किग्रा
  4. D5 किग्रा

Explanation

2 parts of sugar = 3 kg ⇒ 1 part = 1.5 kg. Flour = 5 × 1.5 = 7.5 kg.

Question 10

In the ratio a:b, if both terms are multiplied or divided by the same non-zero number, the value of the ratio:

  1. Aनहीं बदलता
  2. Bबदल जाता है
  3. Cदोगुना हो जाता है
  4. Dआधा हो जाता है

Explanation

This is the fundamental property of a ratio — for example, 2:3, 4:6 and 6:9 all represent the same ratio, so the value doesn't change.

Frequently Asked Questions

How many questions come from ratio-proportion in REET?+

Usually 1 question from this topic appears, often as a word problem; REET 2024 also had 1 question.

What's the difference between ratio and proportion?+

A ratio (a:b) is a comparison between two quantities, while a proportion (a:b :: c:d) is the equality of two ratios.

How do you find the mean proportional?+

If a:b = b:c, the mean proportional b = √(ac).

What happens when you multiply or divide both terms of a ratio by the same number?+

The value of the ratio doesn't change — for example, 2:3 and 4:6 represent the same ratio.

What's the difference between direct and inverse proportion?+

In direct proportion both quantities rise and fall together (like items and price), while in inverse proportion one falls as the other rises (like workers and time).

How is the fourth proportional found?+

If a:b :: c:d, then d is the fourth proportional and is found as d = (b×c)/a — this comes directly from cross multiplication.

What is the third proportional?+

If a:b :: b:c, then c is called the third proportional of a and b, found as c = b²/a.

How is a compounded ratio found?+

Combining two ratios a:b and c:d gives a compounded ratio of ac:bd — that is, corresponding terms of both ratios are multiplied together.

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Linear Equations

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Percentage

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Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked