Ratio and Proportion questions with answers and concepts

Learn ratio and proportion with concept notes and worked examples, then practise focused questions with explained answers. Build accuracy for placement and competitive aptitude tests before trying timed mode.

152 practice questionsPage updated 11 Nov 2026Reviewed by AISEA Editorial DeskPractice questions

Practise this concept

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

9 rules

Ratio of A to B: $A : B = \frac{A}{B}$ , preserving the specified order.

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

Proportion: $a : b = c : d ⟺ \frac{a}{b} = \frac{c}{d} ⟺ a d = b c$ .

Share of total $S$ : In ratio $a : b : c$ , $Share of A = S \times \frac{a}{a + b + c}$ .

Direct proportion: $y = k x ⟹ \frac{y _{1}}{y _{2}} = \frac{x _{1}}{x _{2}}$ .

Inverse proportion: $y = \frac{k}{x} ⟹ x_{1} y_{1} = x_{2} y_{2}$ .

Compound ratio: Of $a : b$ , $c : d$ and $e : f$ is $a ce : b df$ .

Mean proportional: Between $a$ and $b$ is $ab$ .

Fourth proportional: To $a, b, c$ is $x = \frac{b c}{a}$ where $a : b = c : x$ .

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.

Join ratios through a common term

To combine A : B and B : C, make the two B terms equal. For 2 : 3 and 4 : 5, scale them to 8 : 12 and 12 : 15, then read A : B : C as 8 : 12 : 15. Do not multiply or compare the ratios until the shared quantity has the same value in both.

Divide a total into parts

Add the ratio terms to find total parts. If S is divided in a : b : c, the first share is S × a / (a + b + c). Check that all calculated shares add back to S. If one share is known, use its ratio term to find one part before calculating the other shares.

Recognise direct proportion

Two quantities are directly proportional when one is a constant multiple of the other. If the number of pencils increases while unit price stays fixed, total cost increases in the same ratio. Set up matching quantities in the same order and verify that y / x stays constant.

Recognise inverse proportion

Two quantities are inversely proportional when their product stays constant. For the same work and equal efficiency, more workers require fewer days, so workers × days is constant. If working hours also change, use workers × days × hours per day and include every changing factor.

Handle quantities that are added or subtracted

For age or number questions, represent the original quantities as ratio terms such as 5x and 8x, then apply the stated addition or subtraction before forming the new ratio. The change is an absolute number, not a change to the ratio terms themselves. Solve the resulting equation and check that the quantities remain positive.

Use ratios in mixtures and weighted values

A mixture ratio describes amounts of components, not their prices. When liquid is added or removed, write the new amount of each component explicitly. For a value made from different quantities, calculate the weighted total rather than averaging ratios or prices without considering their quantities.

Apply scale factors to geometry

If corresponding lengths are in the ratio a : b, areas are in the ratio a² : b² and volumes are in the ratio a³ : b³. Perimeter scales with length, while area and volume do not. State which geometric measure the question asks for before transferring a side ratio.

Cross-check every proportion

Cross-multiply to verify a solved proportion, restore the units in the final answer and check the direction of change. In a work problem, more workers should not produce more days for the same work. In a share problem, the parts must sum to the stated total. Reject answers that violate these checks even if the arithmetic looks tidy.

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

If $A : B = 2 : 3$ and $B : C = 4 : 5$ , find the compound ratios $A : B : C$ and $A : C$ .

Solution

Step 1: Equalize the common term $B$ using $lcm (3, 4) = 12$ .

A : B = (2 \times 4) : (3 \times 4) = 8 : 12

B : C = (4 \times 3) : (5 \times 3) = 12 : 15

Step 2: Combine ratios.

A : B : C = 8 : 12 : 15

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 ratio units = $3 + 5 = 8$ .

Step 2: Calculate individual shares.

Share of A = \frac{3}{8} \times 800 = 300

Share of B = \frac{5}{8} \times 800 = 500

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: Recognize inverse variation between workers ( $W$ ) and days ( $D$ ).

W_{1} \times D_{1} = W_{2} \times D_{2}

Step 2: Solve for $D_{2}$ .

8 \times 12 = 16 \times D_{2} ⟹ 96 = 16 D_{2} ⟹ D_{2} = \frac{96}{16} = 6 days

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 $5 x$ and $7 x$ .

Step 2: Set up the equation for four years later.

\frac{5 x + 4}{7 x + 4} = \frac{3}{4}

4 (5 x + 4) = 3 (7 x + 4)

20 x + 16 = 21 x + 12 ⟹ x = 4

Step 3: Compute A's present age.

Age of A = 5 x = 5 \times 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: Unitary method:

Cost of 1 pencil = \frac{15}{3} = 5

Cost of 7 pencils = 7 \times 5 = 35

Answer: 35.

Example 6

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

Solution

Step 1: Multiply the antecedents and consequents.

Compound Ratio = \frac{2 \times 4 \times 6}{3 \times 5 \times 7} = \frac{48}{105} = \frac{16}{35}

Answer: 16 : 35.

Example 7

Five workers complete a job in 24 days by working 8 hours per day. How many days will 8 workers need if they work 6 hours per day at the same efficiency?

Solution

Step 1: Apply the work equivalence formula $\frac{W _{1} D _{1} H _{1}}{W _{2} D _{2} H _{2}} = 1$ .

5 \times 24 \times 8 = 8 \times D_{2} \times 6

960 = 48 D_{2} ⟹ D_{2} = \frac{960}{48} = 20 days

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 scales with the square of linear dimensions.

\frac{Area _{1}}{Area _{2}} = (\frac{s _{1}}{s _{2}})^{2} = (\frac{3}{5})^{2} = \frac{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 milk = $5 x$ and water = $2 x$ .

Step 2: Set up the equation with added water.

\frac{5 x}{2 x + 14} = \frac{5}{4}

20 x = 10 x + 70 ⟹ 10 x = 70 ⟹ x = 7

Step 3: Calculate original quantity of milk.

Milk = 5 x = 5 \times 7 = 35 litres

Answer: 35 litres.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

The ratio of incomes of A and B is 7 : 5 and the ratio of their expenditures is 5 : 3. If each saves Rs. 2000, find A's income.

Rs. 14000
Rs. 16000
Rs. 18000
Rs. 20000

Answer: A. Rs. 14000

Explanation

Let A's income = 7x, B's income = 5x
Let A's expenditure = 5y, B's expenditure = 3y
Savings = Income - Expenditure
For A: $7 x - 5 y = 2000$
For B: $5 x - 3 y = 2000$
Solving: $x = 2000$ , $y = 2000$
A's income = 7 x $2000 = R s$ . 14000

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

Two numbers are in the ratio 5 : 8. If 9 is added to each, they become in the ratio 8 : 11. Find the smaller number.

Answer: B. 15

Explanation

Let the numbers be 5x and 8x
$(5 x + 9) / (8 x + 9) = 8/11$
$11 (5 x + 9) = 8 (8 x + 9)$
$55 x + 99 = 64 x + 72$
$99 - 72 = 64 x - 55 x$
$27 = 9 x$
$x = 3$
Smaller number = 5 x $3 = 15$

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

If A : $B = 3$ : 4, B : $C = 5$ : 6, and C : $D = 7$ : 8, find A : D.

35 : 64
105 : 192
15 : 32
21 : 48

Answer: B. 105 : 192

Explanation

A : $B = 3$ : 4, B : $C = 5$ : 6, C : $D = 7$ : 8
$A / B \times B / C \times C / D = A / D$
$(3/4) x (5/6) x (7/8) = (3 \times 5 x 7) / (4 \times 6 x 8) = 105/192$
A : $D = 105$ : $192 = 35$ : 64

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Write the requested ratio in the correct order and convert units before comparing.
Use a common middle term before joining two linked ratios.
For a share, add all ratio parts first and check that the shares return to the original total.
Use direct proportion when the quotient stays constant and inverse proportion when the product stays constant.
Include every changing factor in work problems: workers, days and hours per day.
When a number or age changes by a fixed amount, add or subtract that amount from the algebraic quantity before forming the new ratio.
For geometry, square or cube the length ratio only when the question asks for area or volume.
Cross-multiply, restore units and perform a direction-of-change check before selecting an answer.

Sources and review notes

Reviewed by AISEA Editorial Desk on 1 Sep 2026.

The page was written against the Ratio and Proportion question bank and covers linked ratios, shares, direct and inverse variation, ages, mixtures, work, compound ratios and geometric scaling.
Worked examples were recalculated independently and checked for common units, ratio order, cross-products, changing quantities and direction of variation.
The page separates component ratios from weighted quantities and explicitly checks totals, positivity and geometric scale powers.

Solve Proportions and their ApplicationsOpenStax · Checked 1 Sep 2026Supports: ratio definitions, equivalent proportions, cross-products and matching units
Ratios and ProportionsOpenStax · Checked 1 Sep 2026Supports: using ratios and proportions for scaling and quantitative comparisons
Solve Applications with Rational EquationsOpenStax · Checked 1 Sep 2026Supports: direct variation, inverse variation and work applications

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Next topic: Partnership

Ratio and Proportion questions with answers and concepts

152 practice questionsPage updated 11 Nov 2026Reviewed by AISEA Editorial DeskPractice questions

Practise this concept

Key formulas and rules

9 rules

Ratio of A to B: $A : B = \frac{A}{B}$ , preserving the specified order.

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

Proportion: $a : b = c : d ⟺ \frac{a}{b} = \frac{c}{d} ⟺ a d = b c$ .

Share of total $S$ : In ratio $a : b : c$ , $Share of A = S \times \frac{a}{a + b + c}$ .

Direct proportion: $y = k x ⟹ \frac{y _{1}}{y _{2}} = \frac{x _{1}}{x _{2}}$ .

Inverse proportion: $y = \frac{k}{x} ⟹ x_{1} y_{1} = x_{2} y_{2}$ .

Compound ratio: Of $a : b$ , $c : d$ and $e : f$ is $a ce : b df$ .

Mean proportional: Between $a$ and $b$ is $ab$ .

Fourth proportional: To $a, b, c$ is $x = \frac{b c}{a}$ where $a : b = c : x$ .

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

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

If $A : B = 2 : 3$ and $B : C = 4 : 5$ , find the compound ratios $A : B : C$ and $A : C$ .

Solution

Step 1: Equalize the common term $B$ using $lcm (3, 4) = 12$ .

A : B = (2 \times 4) : (3 \times 4) = 8 : 12

B : C = (4 \times 3) : (5 \times 3) = 12 : 15

Step 2: Combine ratios.

A : B : C = 8 : 12 : 15

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 ratio units = $3 + 5 = 8$ .

Step 2: Calculate individual shares.

Share of A = \frac{3}{8} \times 800 = 300

Share of B = \frac{5}{8} \times 800 = 500

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: Recognize inverse variation between workers ( $W$ ) and days ( $D$ ).

W_{1} \times D_{1} = W_{2} \times D_{2}

Step 2: Solve for $D_{2}$ .

8 \times 12 = 16 \times D_{2} ⟹ 96 = 16 D_{2} ⟹ D_{2} = \frac{96}{16} = 6 days

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 $5 x$ and $7 x$ .

Step 2: Set up the equation for four years later.

\frac{5 x + 4}{7 x + 4} = \frac{3}{4}

4 (5 x + 4) = 3 (7 x + 4)

20 x + 16 = 21 x + 12 ⟹ x = 4

Step 3: Compute A's present age.

Age of A = 5 x = 5 \times 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: Unitary method:

Cost of 1 pencil = \frac{15}{3} = 5

Cost of 7 pencils = 7 \times 5 = 35

Answer: 35.

Example 6

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

Solution

Step 1: Multiply the antecedents and consequents.

Compound Ratio = \frac{2 \times 4 \times 6}{3 \times 5 \times 7} = \frac{48}{105} = \frac{16}{35}

Answer: 16 : 35.

Example 7

Five workers complete a job in 24 days by working 8 hours per day. How many days will 8 workers need if they work 6 hours per day at the same efficiency?

Solution

Step 1: Apply the work equivalence formula $\frac{W _{1} D _{1} H _{1}}{W _{2} D _{2} H _{2}} = 1$ .

5 \times 24 \times 8 = 8 \times D_{2} \times 6

960 = 48 D_{2} ⟹ D_{2} = \frac{960}{48} = 20 days

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 scales with the square of linear dimensions.

\frac{Area _{1}}{Area _{2}} = (\frac{s _{1}}{s _{2}})^{2} = (\frac{3}{5})^{2} = \frac{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 milk = $5 x$ and water = $2 x$ .

Step 2: Set up the equation with added water.

\frac{5 x}{2 x + 14} = \frac{5}{4}

20 x = 10 x + 70 ⟹ 10 x = 70 ⟹ x = 7

Step 3: Calculate original quantity of milk.

Milk = 5 x = 5 \times 7 = 35 litres

Answer: 35 litres.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

The ratio of incomes of A and B is 7 : 5 and the ratio of their expenditures is 5 : 3. If each saves Rs. 2000, find A's income.

Rs. 14000
Rs. 16000
Rs. 18000
Rs. 20000

Answer: A. Rs. 14000

Explanation

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

Two numbers are in the ratio 5 : 8. If 9 is added to each, they become in the ratio 8 : 11. Find the smaller number.

Answer: B. 15

Explanation

Let the numbers be 5x and 8x
$(5 x + 9) / (8 x + 9) = 8/11$
$11 (5 x + 9) = 8 (8 x + 9)$
$55 x + 99 = 64 x + 72$
$99 - 72 = 64 x - 55 x$
$27 = 9 x$
$x = 3$
Smaller number = 5 x $3 = 15$

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

If A : $B = 3$ : 4, B : $C = 5$ : 6, and C : $D = 7$ : 8, find A : D.

35 : 64
105 : 192
15 : 32
21 : 48

Answer: B. 105 : 192

Explanation

A : $B = 3$ : 4, B : $C = 5$ : 6, C : $D = 7$ : 8
$A / B \times B / C \times C / D = A / D$
$(3/4) x (5/6) x (7/8) = (3 \times 5 x 7) / (4 \times 6 x 8) = 105/192$
A : $D = 105$ : $192 = 35$ : 64

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Write the requested ratio in the correct order and convert units before comparing.
Use a common middle term before joining two linked ratios.
For a share, add all ratio parts first and check that the shares return to the original total.
Use direct proportion when the quotient stays constant and inverse proportion when the product stays constant.
Include every changing factor in work problems: workers, days and hours per day.
When a number or age changes by a fixed amount, add or subtract that amount from the algebraic quantity before forming the new ratio.
For geometry, square or cube the length ratio only when the question asks for area or volume.
Cross-multiply, restore units and perform a direction-of-change check before selecting an answer.

Sources and review notes

Reviewed by AISEA Editorial Desk on 1 Sep 2026.

The page was written against the Ratio and Proportion question bank and covers linked ratios, shares, direct and inverse variation, ages, mixtures, work, compound ratios and geometric scaling.
Worked examples were recalculated independently and checked for common units, ratio order, cross-products, changing quantities and direction of variation.
The page separates component ratios from weighted quantities and explicitly checks totals, positivity and geometric scale powers.

Solve Proportions and their ApplicationsOpenStax · Checked 1 Sep 2026Supports: ratio definitions, equivalent proportions, cross-products and matching units
Ratios and ProportionsOpenStax · Checked 1 Sep 2026Supports: using ratios and proportions for scaling and quantitative comparisons
Solve Applications with Rational EquationsOpenStax · Checked 1 Sep 2026Supports: direct variation, inverse variation and work applications

Ready to test your understanding?

Work through 152 questions with explanations after each answer.

Start practice

Next topic: Partnership

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

Ratios and ProportionsOpenStax · Checked 1 Sep 2026Supports: using ratios and proportions for scaling and quantitative comparisons

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

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

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Example 5

Solution

Example 6

Solution

Example 7

Solution

Example 8

Solution

Example 9

Solution

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.

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

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Example 5

Solution

Example 6

Solution

Example 7

Solution

Example 8

Solution

Example 9

Solution

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.