MathsCBSE Class 8 · Proportional Reasoning-1

Ratios and Proportion Word Problems for Class 8

Updated 17 October 2026 · Fact-checked

A ratio a : b compares two quantities in the same unit. Two ratios are proportional (a : b :: c : d) when their simplest forms are equal, that is, when both terms change by the same multiplying factor, not by the same difference. Class 8 problems ask for equivalent ratios, missing terms, unit rates for best buys, scaled figures and age problems whose ratio changes with time.

Multiplying both terms by the same factor keeps the ratio 3 : 2, but subtracting the same amount changes the ratio and distorts the shape.

Illustration drawn for QuestionBuddy.

Understand Ratios and Proportion Word Problems

Proportional change is multiplicative. Look at digital images of a tiger. An image changed by the same factor in width and height (for example both halved) looks similar; one where both width and height are reduced by the same amount (subtraction) looks distorted. So two quantities change proportionally when each is multiplied by the same factor.

Ratio. The ratio of width to height of an image $60$ mm by $40$ mm is $60 : 40$ . The numbers 60 and 40 are the terms. In a ratio $a : b$ , for every $a$ units of the first quantity there are $b$ units of the second. A ratio compares quantities in the same unit (convert first: 1.5 m to 90 cm is $150 : 90$ ).

Simplest form. Divide both terms by their HCF: $60 : 40$ has HCF 20, so it is $3 : 2$ . Two ratios are proportional (written $a : b :: c : d$ ) when their simplest forms are the same. For example $72 : 96$ has HCF 24 and becomes $3 : 4$ , so it is proportional to $3 : 4$ .

Finding missing terms. Find the factor that carries one known term to the other known term, and apply it to the unknown. If $6 : 10$ is the ratio of glasses to spoons of sugar and there are to be 18 glasses, the factor is $18 \div 6 = 3$ , so the sugar is $10 \times 3 = 30$ spoons. This is the unitary method or the rule of three: find the value for one unit, then multiply.

Unit rates for comparison. To compare prices or speeds, find the quantity per one unit (price per gram, distance per minute) for each, then compare. In the chapter's tea example, 200 g for 200 rupees costs 1 rupee per gram, while 1 kg for 800 rupees costs 0.8 rupee per gram.

Ratios that are not constant. A situation can look like a ratio but change with time. If Neelima is 3 and her mother is 30, the ratio of ages is $1 : 10$ ; years later it is different, because the age difference stays the same (a constant difference, not a constant factor).

Always check the units and the order. The ratio of teachers to students (5 : 170) is not the ratio of students to teachers (170 : 5).

Sharing in a ratio, mixtures and unit conversions are on the next topic page of this chapter.

Ratios and Proportion Word Problems: formulas to remember

Simplest form: $a : b = \frac{a}{HCF} : \frac{b}{HCF}$; Divide both terms by the HCF.
Proportion: $a : b :: c : d ⟺ \frac{a}{b} = \frac{c}{d} ⟺ a d = b c$; Equal simplest forms; the product of extremes equals the product of means.
Equivalent ratio: $a : b = k a : k b$; Multiply both terms by the same factor k.
Unit rate: $rate = \frac{total quantity}{number of units}$; Used for best-buy comparisons.
Missing term: $\frac{a}{b} = \frac{c}{x} \Rightarrow x = \frac{b c}{a}$; The rule of three.

How to solve Ratios and Proportion Word Problems questions

A routine for proportion word problems.

1Identify the two quantities and their units; convert to the same unit if you must compare.
2Write the ratio in the order asked (first : second).
3Reduce to the simplest form by the HCF.
4To find an unknown, find the factor that links the known terms and apply it to the unknown (or find the value of one unit).
5For comparisons (best buy, speed), compute the value per unit for each option and compare.
6For age or time problems, remember that differences stay constant while ratios change.
7Check that the answer keeps the original ratio.

Ratios and Proportion Word Problems in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Scaling a recipe

Suppose a cook uses rice and water in the ratio 1 : 2 for a pot of pulao, and today she needs 4.5 cups of rice.

How much water does she need, and what is the ratio of water to the total in the pot?

Try it first, then show the solutionSolution

Water $= 4.5 \times 2 = 9$ cups.
Total $= 4.5 + 9 = 13.5$ cups, and water : total $= 9 : 13.5 = 2 : 3$ .

Answer: 9 cups of water; water to total is 2 : 3.

Memory hook: Same factor on both terms keeps the taste.

Unit rate and best buy

Imagine two shops selling the same oil: 750 g for 90 rupees and 1.2 kg for 138 rupees.

Which is cheaper per kilogram?

Try it first, then show the solutionSolution

First: $90 \div 750 = 0.12$ rupee per gram, i.e. 120 rupees per kg.
Second: $138 \div 1200 = 0.115$ rupee per gram, i.e. 115 rupees per kg.

Answer: The 1.2 kg pack is cheaper by 5 rupees per kg.

Memory hook: Compare price per one unit, never total price.

Tricks to remember Ratios and Proportion Word Problems

Short memory aids. Say each one aloud once, then use it on a question the same day.

Same factor, not same difference
Proportional means multiply both by the same number; subtracting the same amount from both distorts the ratio.
Use it when: Deciding whether two changes are proportional.
HCF to simplify
Divide both terms by their HCF to reach the simplest form in one step.
Use it when: Comparing ratios.
Unit first
Convert both quantities to one unit before writing the ratio.
Use it when: Ratios like 1.5 m to 90 cm.
Per one, then times
Find the value of one unit, then multiply by the number of units asked.
Use it when: Rule of three and unitary method.

Ratios and Proportion Word Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Equivalent ratios · 1 mark · MCQBoard level

Which ratio is proportional to 14 : 21? (a) 20 : 28 (b) 28 : 42 (c) 35 : 56 (d) 42 : 60

Try it first, then show the solutionSolution

$14 : 21 = 2 : 3$ (divide by 7).
$28 : 42$ has HCF 14 and simplifies to $2 : 3$ . The others simplify to $5 : 7$ , $5 : 8$ , $7 : 10$ .

Answer: (b) 28 : 42

Example 2: Simplest form with units · 2 marks · Short answerBoard level

Write in the simplest form: (i) the ratio 84 : 140 (ii) the ratio of 1.5 m to 90 cm.

Try it first, then show the solutionSolution

(i) HCF of 84 and 140 is 28, so $84 : 140 = 3 : 5$ . (1 mark)
(ii) $1.5 m = 150 cm$ , so the ratio is $150 : 90$ ; HCF 30 gives $5 : 3$ . (1 mark)

Answer: (i) 3 : 5 (ii) 5 : 3

Example 3: Distance in a different time · 3 marks · Short answerHarder board level

A cyclist covers 18 km in 45 minutes at a steady speed. How far does she travel in 2 hours 20 minutes?

Try it first, then show the solutionSolution

$2 h 20 min = 140$ minutes. (1 mark)
Distance per minute $= 18 \div 45 = 0.4$ km. (1 mark)
Distance $= 0.4 \times 140 = 56$ km. Check: $18 : 45 = 2 : 5$ and $56 : 140 = 2 : 5$ , so the two ratios are proportional. (1 mark)

Answer: 56 km

Example 4: Best buy · 3 marks · Short answerHarder board level

Brand P sells 750 g for 90 rupees and Brand Q sells 1.2 kg for 138 rupees. Find the price per kg of each, which is cheaper, and by how much per kg.

Try it first, then show the solutionSolution

P: $\frac{90}{750} = 0.12$ rupee per g $= 120$ rupees per kg. (1 mark)
Q: $\frac{138}{1200} = 0.115$ rupee per g $= 115$ rupees per kg. (1 mark)
Q is cheaper by $120 - 115 = 5$ rupees per kg. (1 mark)

Answer: P: 120 rupees per kg, Q: 115 rupees per kg; Q is cheaper by 5 rupees per kg.

Example 5: Ratio of ages over time · 4 marks · Long answerHarder board level

A father is 36 years old and his son is 9. After how many years will their ages be in the ratio 5 : 2? What was the ratio 3 years ago?

Try it first, then show the solutionSolution

After $x$ years the ages are $36 + x$ and $9 + x$ , with $\frac{36 + x}{9 + x} = \frac{5}{2}$ . (1 mark)
Cross-multiply: $2 (36 + x) = 5 (9 + x)$ , so $72 + 2 x = 45 + 5 x$ and $27 = 3 x$ , giving $x = 9$ . (1 mark)
Check: ages $45$ and $18$ , and $45 : 18 = 5 : 2$ . (1 mark)
Three years ago: $33$ and $6$ , so $33 : 6 = 11 : 2$ . (1 mark)

Answer: After 9 years (ages 45 and 18); 3 years ago the ratio was 11 : 2.

Where students lose marks in Ratios and Proportion Word Problems

The slips examiners see most often, and what to write instead.

Mistake: Writing a ratio of unlike units
A ratio compares quantities in the same unit.
Wrong working
"1.5 m : 90 cm = 1.5 : 90 = 1 : 60"
Right working
$150 cm : 90 cm = 5 : 3$ .
Do this: Convert first, then compare.
Usually costs: The full answer.
Mistake: Subtracting the same amount to scale a figure
Equal differences do not preserve the ratio.
Wrong working
"90 mm x 60 mm reduced by 30 mm each gives 60 mm x 30 mm, which is still in proportion."
Right working
$90 : 60 = 3 : 2$ but $60 : 30 = 2 : 1$ , so they are not proportional.
Do this: Check that both terms are multiplied by the same factor.
Usually costs: 1 to 2 marks.
Mistake: Reversing the order of the ratio
The order matters: a : b is not b : a.
Wrong working
"Ratio of teachers to students = 170 : 5."
Right working
Teachers to students $= 5 : 170 = 1 : 34$ .
Do this: Underline "first to second" in the question.
Usually costs: 1 mark.
Mistake: Comparing total prices of packs of different sizes
Only the price per unit allows a fair comparison.
Wrong working
"Pack Q costs more (138 rupees) than P (90 rupees), so P is the better buy."
Right working
Per kg, P costs 120 and Q costs 115; Q is cheaper.
Do this: Always divide by the quantity first.
Usually costs: The answer.

Exam tips

Convert to one unit and write the ratio in the order asked before simplifying.
Show the HCF in the working; a mark is often allocated for it.
For best-buy questions, give price per one unit for every option, then compare.
In age problems, form the equation using the new ages and cross-multiply; check by substituting.
Round only at the end and state the unit.

Practice questions on Ratios and Proportion Word Problems

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): The ratios 60 : 40 and 40 : 20 are proportional.
Reason (R): Two ratios are proportional when their simplest forms are equal.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Simplify both ratios.
Answer
(d). $60 : 40 = 3 : 2$ and $40 : 20 = 2 : 1$ , which are different, so A is false. R is true.
Q2 · 2 marks · Very short answerBoard level
Fill in the blanks so that each ratio is proportional to 14 : 21: (i) 6 : __ (ii) __ : 30.
Hint
Reduce 14 : 21 first.
Answer
$14 : 21 = 2 : 3$ . (i) $6 : 9$ (multiply by 3). (ii) $20 : 30$ (multiply by 10).
Q3 · 3 marks · Short answerHarder board level
A 24 cm by 16 cm photograph is enlarged in proportion so that its width becomes 36 cm. Find the new height and the ratio of the area of the original to that of the enlarged photograph.
Hint
Scale factor is 36 divided by 24; areas scale by the square.
Answer
Scale factor $= 36 \div 24 = 1.5$ , so new height $= 16 \times 1.5 = 24$ cm. Areas: $24 \times 16 = 384$ and $36 \times 24 = 864$ square cm, ratio $384 : 864 = 4 : 9$ , which is the square of the length ratio $2 : 3$ .
Q4 · 4 marks · Long answerHOTS / competency
The present ages of Sam and Tara are in the ratio 3 : 5. Eight years from now the ratio will be 5 : 7. Find their present ages, and explain why the ratio changes even though the age difference does not.
Hint
Let the ages be 3k and 5k.
Answer
Ages $3 k$ and $5 k$ . After 8 years: $\frac{3 k + 8}{5 k + 8} = \frac{5}{7}$ , so $7 (3 k + 8) = 5 (5 k + 8)$ , $21 k + 56 = 25 k + 40$ , $4 k = 16$ , $k = 4$ . Present ages: 12 and 20. Check: after 8 years, 20 and 28, ratio $5 : 7$ . The difference stays $8$ , but each age grows by the same amount, not the same factor, so the ratio of the ages changes.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: three shops sell the same milk. Shop A: 500 mL for 28 rupees. Shop B: 1.5 L for 81 rupees. Shop C: 2 L for 104 rupees.
(i) Find the price per litre at each shop. (1 mark)
(ii) Which shop is cheapest per litre? (1 mark)
(iii) How much does a family save on 6 L by buying from the cheapest shop rather than from Shop A? (2 marks)
Hint
Convert 500 mL to 0.5 L.
Answer
(i) A: $28 \div 0.5 = 56$ rupees per L. B: $81 \div 1.5 = 54$ rupees per L. C: $104 \div 2 = 52$ rupees per L. (ii) Shop C. (iii) Saving per litre $= 56 - 52 = 4$ rupees, so for 6 L the saving is $6 \times 4 = 24$ rupees.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.When are two ratios proportional?Show answer
When their simplest forms are equal.
2.How do you reduce a ratio to simplest form?Show answer
Divide both terms by their HCF.
3.What does the unitary method find first?Show answer
The value of one unit.
4.Do equal differences preserve a ratio?Show answer
No; equal factors do.
5.Is 5 : 170 the same as 170 : 5?Show answer
No, the order matters.
6.Why does the ratio of two ages change with time?Show answer
Both ages grow by the same amount, not by the same factor.

Ratios and Proportion Word Problems: frequently asked questions

What is the difference between ratio and proportion?

A ratio compares two quantities; a proportion says two ratios are equal.

Can the terms of a ratio be decimals?

Yes, but it is usually cleaner to multiply both terms to clear the decimals, such as 1.5 : 2.5 = 3 : 5.

Can I use the cross-product rule to check a proportion?

Yes: a : b :: c : d if and only if ad = bc, which is the same as equal simplest forms.

Why are the images A, C and D in the chapter similar?

Because their widths and heights were multiplied by the same factor, so the ratio of width to height is the same.