Ratio and Proportion

Direct, inverse, compound proportion

199 practice questionsPage updated 31 Aug 2026Reviewed by AISEA Editorial DeskPractice questions

A ratio compares quantities of the same kind in a specified order. A proportion states that two ratios or rates are equal. In aptitude exams, these ideas appear in sharing, chained ratios, direct and inverse variation, ages, mixtures, work, wages, rates and geometric scaling. Keep units aligned, identify which quantity changes, and use a common middle term before combining ratios.

Key formulas and rules

Ratio of A to B = A : B = A / B, with the order stated by the question.

Equivalent ratio: a : b = ka : kb for any non-zero common factor k.

Proportion: a : b = c : d means a / b = c / d and ad = bc.

Share of S in the ratio a : b : c = S × a / (a + b + c), with analogous formulas for the other shares.

Direct proportion: y = kx, so y₁ / y₂ = x₁ / x₂.

Inverse proportion: y = k / x, so x₁y₁ = x₂y₂.

Compound ratio of a : b, c : d and e : f = ace : bdf.

Mean proportional between a and b = √(ab); fourth proportional to a, b and c = bc / a.

For equal work and efficiency, workers × days × hours per day remains constant.

Key concepts

Keep the ratio's order and units

A : B is not the same as B : A. Put the requested quantity first and compare like units: rupees with rupees, litres with litres and workers with workers. Convert units before forming a ratio. A ratio may be simplified, but its meaning and order must stay unchanged.

Simplify and scale ratios

Divide both terms by their greatest common factor to simplify a ratio. Multiplying every term by the same non-zero factor produces an equivalent ratio. Never add or subtract the same number from both terms to simplify; that generally changes the ratio.

Worked examples

Example 1

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C and A : C.

Solution

Step 1: The two values of B are 3 and 4. Their least common multiple is 12.
Step 2: Scale 2 : 3 to 8 : 12 and 4 : 5 to 12 : 15.
Step 3: Therefore A : B : C = 8 : 12 : 15.
Step 4: A : C = 8 : 15.
Answer: A : B : C = 8 : 12 : 15 and A : C = 8 : 15.

Example 2

Divide ₹800 between A and B in the ratio 3 : 5.

Solution

Step 1: Total parts = 3 + 5 = 8.
Step 2: One part = ₹800 / 8 = ₹100.
Step 3: A's share = 3 × ₹100 = ₹300.
Step 4: B's share = 5 × ₹100 = ₹500.
Step 5: Check: ₹300 + ₹500 = ₹800.
Answer: A receives ₹300 and B receives ₹500.

Example 3

Eight workers complete a job in 12 days. How many days will 16 workers take for the same job if all workers have equal efficiency?

Solution

Step 1: This is inverse proportion because the amount of work is fixed.
Step 2: Workers × days = constant, so 8 × 12 = 16 × d.
Step 3: d = 96 / 16 = 6 days.
Step 4: Doubling workers halves the time, which confirms the direction.
Answer: 6 days.

Example 4

The present ages of A and B are in the ratio 5 : 7. Four years later, the ratio becomes 3 : 4. Find A's present age.

Solution

Step 1: Let the present ages be 5x and 7x.
Step 2: After four years, (5x + 4) / (7x + 4) = 3 / 4.
Step 3: Cross-multiply: 4(5x + 4) = 3(7x + 4).
Step 4: 20x + 16 = 21x + 12, so x = 4.
Step 5: A's present age = 5 × 4 = 20 years.
Answer: 20 years.

Example 5

Three pencils cost ₹15. Assuming the unit price is unchanged, how much will seven pencils cost?

Solution

Step 1: Cost per pencil = ₹15 / 3 = ₹5.
Step 2: Seven pencils cost 7 × ₹5 = ₹35.
Step 3: The cost-to-pencil ratio is 15 : 3 = 5 : 1, so the same result follows by direct proportion.
Answer: ₹35.

Example 6

Find the compound ratio of 2 : 3, 4 : 5 and 6 : 7.

Solution

Step 1: Multiply the antecedents: 2 × 4 × 6 = 48.
Step 2: Multiply the consequents: 3 × 5 × 7 = 105.
Step 3: Compound ratio = 48 : 105.
Step 4: Divide both terms by 3 to simplify.
Answer: 16 : 35.

Example 7

Five workers complete a job in 24 days by working eight hours per day. How many days will eight workers need if they work six hours per day and have the same efficiency?

Solution

Step 1: Total work = 5 × 24 × 8 = 960 worker-hours.
Step 2: New work completed per day = 8 × 6 = 48 worker-hours.
Step 3: Required days = 960 / 48 = 20.
Answer: 20 days.

Example 8

The sides of two squares are in the ratio 3 : 5. Find the ratio of their areas.

Solution

Step 1: Area of a square is side × side, so area scales with the square of the side ratio.
Step 2: Area ratio = 3² : 5² = 9 : 25.
Answer: 9 : 25.

Example 9

A mixture contains milk and water in the ratio 5 : 2. If 14 litres of water is added, the ratio becomes 5 : 4. Find the original quantity of milk.

Solution

Step 1: Let the original quantities be milk = 5x and water = 2x.
Step 2: After adding water, 5x / (2x + 14) = 5 / 4.
Step 3: Cross-multiply: 20x = 10x + 70, so x = 7.
Step 4: Original milk = 5 × 7 = 35 litres.
Answer: 35 litres.

Solve Proportions and their ApplicationsOpenStax · Checked 1 Sep 2026Supports: ratio definitions, equivalent proportions, cross-products and matching units

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Solve Applications with Rational EquationsOpenStax · Checked 1 Sep 2026Supports: direct variation, inverse variation and work applications

Ratio and Proportion

Key formulas and rules

Key concepts

Keep the ratio's order and units

Simplify and scale ratios

Join ratios through a common term

Divide a total into parts

Recognise direct proportion

Recognise inverse proportion

Handle quantities that are added or subtracted

Use ratios in mixtures and weighted values

Apply scale factors to geometry

Cross-check every proportion

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Key formulas and rules

Key concepts

Keep the ratio's order and units

Simplify and scale ratios

Join ratios through a common term

Divide a total into parts

Recognise direct proportion

Recognise inverse proportion

Handle quantities that are added or subtracted

Use ratios in mixtures and weighted values

Apply scale factors to geometry

Cross-check every proportion

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Keep building the same skill family.