Ratio and Proportion — Extra Practice

Competitive Mathematics · Arithmetic

6 Part12 worked examples130 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

When making tea, a milk : water ratio of 2 : 1 means 1 cup of water for every 2 cups of milk — whether you make 6 cups or 600. A ratio compares two like quantities: how many times one is of the other. The ratio $a : b$ is also written as the fraction $\frac{a}{b}$ ; $a$ is the antecedent and $b$ the consequent. Both quantities must be in the same unit — ₹100 and 500 apples have no ratio.

Proportion means two ratios are equal: $a : b :: c : d$ i.e. $\frac{a}{b} = \frac{c}{d}$ . Every question here needs two tools: the one-part value and cross-multiplication ( $a d = b c$ ).

Ratio

Kinds · comparing · sharing

The kinds of ratio, comparing two ratios, and dividing a sum in a ratio.

Kinds of ratio and comparison

Duplicate · triplicate · compound

The rule for each kind, and which of two ratios is bigger.

Kind	Of $a : b$	Example $3 : 4$
Duplicate	$a^{2} : b^{2}$	9 : 16
Sub-duplicate	$a : b$	$4 : 9 \to 2 : 3$
Triplicate	$a^{3} : b^{3}$	27 : 64
Sub-triplicate	$3 a : 3 b$	$8 : 27 \to 2 : 3$
Inverse	$b : a$	4 : 3
Compound	$a ce : b df$	$(2 : 3), (4 : 5) \to 8 : 15$

Comparing: $a : b > c : d$ when $a d > b c$ (cross-multiply). Or write both as fractions over a common denominator. For $2 : 3$ and $4 : 5$ : $2 \times 5 = 10 < 3 \times 4 = 12$ ⟹ $2 : 3 < 4 : 5$ .

Example 1 · Basic

Which is greater,

5 : 7

8 : 11

Solution, step by step

1Cross-multiply: $5 \times 11 = 55$ and $7 \times 8 = 56$ .
2 $55 < 56$ ⟹ $5 : 7 < 8 : 11$

Answer 8 : 11

Example 2 · Standard

Find the compound ratio of

2 : 3

5 : 4

and

3 : 7

, and the sub-duplicate ratio of

4 : 9

Solution, step by step

1Compound: $\frac{2 \times 5 \times 3}{3 \times 4 \times 7} = \frac{30}{84} = 5 : 14$ .
2Sub-duplicate: $4 : 9 = 2 : 3$

Answer 5 : 14 and 2 : 3

BasicTry it yourself first

Write the ratio 36 : 60 in its simplest form.

BasicTry it yourself first

Write the ratio 56 : 72 in its simplest form.

BasicTry it yourself first

Which is the greater ratio: 3 : 5 or 5 : 8?

StandardTry it yourself first

Which is the greater ratio: 7 : 9 or 11 : 14?

AdvancedTry it yourself first

Which is the greater ratio: 13 : 17 or 19 : 25?

BasicTry it yourself first

What is the duplicate ratio of 3 : 5?

BasicTry it yourself first

What is the triplicate ratio of 2 : 7?

BasicTry it yourself first

What is the sub-duplicate ratio of 4 : 9?

StandardTry it yourself first

What is the sub-triplicate ratio of 8 : 27?

StandardTry it yourself first

What is the inverse ratio of 5 : 8?

StandardTry it yourself first

What is the compound ratio of 2 : 3, 4 : 5 and 3 : 7?

StandardTry it yourself first

What is the compound ratio of 3 : 5, 10 : 11 and 7 : 9?

AdvancedTry it yourself first

What is the compound ratio of 5 : 6, 8 : 15 and 9 : 14?

Sharing a sum and three-term ratios

Total parts · one part

Sharing a sum in a ratio, using the difference of parts, and getting $A : B : C$ from $A : B$ and $B : C$ .

Add up the total parts. For $a : b : c$ the total is $a + b + c$ parts.
One part $= \frac{total amount}{total parts}$ .
The required share $=$ its parts $\times$ one part.

Trick $A : B : C$ from $A : B$ and $B : C$ : equalise the middle term

B

— take the LCM of its two values and scale

A

and

C

to match. For

A : B = 2 : 3

and

B : C = 4 : 5

: LCM of

B

is 12 ⟹

A : B = 8 : 12

B : C = 12 : 15

⟹

A : B : C = 8 : 12 : 15

Example 3 · Basic

Divide ₹9,000 in the ratio 2 : 3 : 4.

Solution, step by step

1Total parts $= 9$ ; one part $= \frac{9000}{9} = 1000$ .
2Shares: ₹2,000, ₹3,000, ₹4,000

Answer ₹2,000, ₹3,000, ₹4,000

Example 4 · Standard

The sum of three numbers is 195. The ratio of the first to the second is 3 : 4 and of the second to the third 2 : 3. What is the third?

Solution, step by step

1LCM of $B$ is 4 ⟹ $A : B = 3 : 4$ and $B : C = 4 : 6$ ⟹ $A : B : C = 3 : 4 : 6$ .
2Total parts 13; one part $= 15$ ⟹ the third $= 6 \times 15 = 90$ .

Answer 90

BasicTry it yourself first

₹7,200 is divided among A, B in the ratio 3 : 5. What is B's share?

BasicTry it yourself first

₹4,500 is divided among A, B, C in the ratio 1 : 2 : 3. What is C's share?

StandardTry it yourself first

₹12,600 is divided among A, B, C in the ratio 2 : 3 : 4. What is B's share?

StandardTry it yourself first

₹8,400 is divided among A, B in the ratio 5 : 7. What is A's share?

AdvancedTry it yourself first

₹20,100 is divided among A, B, C, D in the ratio 3 : 5 : 7 : 5. What is D's share?

BasicTry it yourself first

A sum is shared in the ratio 5 : 3 and the shares differ by ₹1,200. What is the bigger share?

StandardTry it yourself first

A sum is shared in the ratio 7 : 4 and the shares differ by ₹2,700. What is the bigger share?

AdvancedTry it yourself first

A sum is shared in the ratio 11 : 7 and the shares differ by ₹3,600. What is the bigger share?

BasicTry it yourself first

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

StandardTry it yourself first

If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.

AdvancedTry it yourself first

If A : B = 5 : 6 and B : C = 8 : 9, find A : B : C.

StandardTry it yourself first

The sum of three numbers is 140. The ratio of the first to the second is 2 : 3 and of the second to the third 4 : 5. What is the second number?

StandardTry it yourself first

The sum of three numbers is 345. The ratio of the first to the second is 3 : 5 and of the second to the third 10 : 7. What is the third number?

AdvancedTry it yourself first

The sum of three numbers is 330. The ratio of the first to the second is 4 : 7 and of the second to the third 14 : 11. What is the first number?

Proportion

Means · extremes · three proportionals

The rules of $a : b :: c : d$ , third, fourth and mean proportionals, and componendo-dividendo.

Third, fourth and mean proportionals

Product of means = product of extremes

In $a : b :: c : d$ , $a, d$ are the extremes and $b, c$ the means; $a d = b c$ .

Name	Condition	Value
Fourth proportional $d$	$a : b :: c : d$	$d = \frac{b c}{a}$
Third proportional $c$	$a : b :: b : c$	$c = \frac{b ^{2}}{a}$
Mean proportional $m$	$a : m :: m : b$	$m = ab$

Example 5 · Basic

Find the fourth proportional to 3, 7 and 9, and the mean proportional between 16 and 36.

Solution, step by step

1 $d = \frac{7 \times 9}{3} = 21$ .
2 $m = 16 \times 36 = 576 = 24$

Answer 21 and 24

Example 6 · Standard

If 4,

x

and 36 are in continued proportion find

x

and the third proportional to 4 and

x

Solution, step by step

1Continued proportion: $4 : x :: x : 36$ ⟹ $x^{2} = 144$ ⟹ $x = 12$ .
2Third proportional to 4 and 12 $= \frac{1 2 ^{2}}{4} = 36$ ✓

Answer $x = 12$ , third proportional 36

BasicTry it yourself first

What is the fourth proportional to 4, 9 and 12?

StandardTry it yourself first

What is the fourth proportional to 7, 14 and 15?

StandardTry it yourself first

What is the fourth proportional to 12, 18 and 26?

AdvancedTry it yourself first

What is the fourth proportional to 15, 24 and 35?

BasicTry it yourself first

What is the third proportional to 4 and 12?

StandardTry it yourself first

What is the third proportional to 9 and 24?

AdvancedTry it yourself first

What is the third proportional to 14 and 42?

BasicTry it yourself first

What is the mean proportional between 4 and 25?

StandardTry it yourself first

What is the mean proportional between 12 and 27?

AdvancedTry it yourself first

What is the mean proportional between 18 and 98?

Componendo, dividendo and friends

Four ways to rewrite a proportion

If $\frac{a}{b} = \frac{c}{d}$ , four related rules.

Name	Rule
Invertendo	$\frac{b}{a} = \frac{d}{c}$
Alternendo	$\frac{a}{c} = \frac{b}{d}$
Componendo	$\frac{a + b}{b} = \frac{c + d}{d}$
Dividendo	$\frac{a - b}{b} = \frac{c - d}{d}$
Componendo–Dividendo	$\frac{a + b}{a - b} = \frac{c + d}{c - d}$

The most useful is componendo–dividendo: if $\frac{x + y}{x - y} = \frac{5}{3}$ then $\frac{x}{y} = \frac{5 + 3}{5 - 3} = 4$ . Turning “ $\frac{p}{q}$ ” into $\frac{p + q}{p - q}$ gives $\frac{x}{y}$ .

Example 7 · Standard

\frac{a + b}{a - b} = \frac{7}{3}

, find

a : b

Solution, step by step

1By componendo–dividendo $\frac{a}{b} = \frac{7 + 3}{7 - 3} = \frac{10}{4}$ .
2 $a : b = 5 : 2$

Answer 5 : 2

BasicTry it yourself first

x : y = 3 : 4

, find

(1 x + 1 y) : (1 x + 1 y)

StandardTry it yourself first

x : y = 5 : 3

, find

(2 x + 3 y) : (3 x + 1 y)

StandardTry it yourself first

x : y = 4 : 7

, find

(3 x + 2 y) : (5 x + 1 y)

AdvancedTry it yourself first

x : y = 7 : 5

, find

(4 x + 3 y) : (2 x + 3 y)

Practical ratio questions

Changes, savings and coins

Ratio after adding or subtracting, income–expenditure–saving and coin questions.

When the numbers change

Assume

k

, then cross-multiply

The new ratio after adding or subtracting something, or after a percent change.

Take numbers in the ratio $a : b$ as $ak$ and $bk$ . The add / subtract condition gives an equation in $k$ . For a percent change multiply each term directly by $(100 \pm p)$ .

Example 8 · Standard

Two numbers are in the ratio 4 : 7. Adding 6 to each makes the ratio 5 : 8. What is the smaller number?

Solution, step by step

1Numbers $4 k, 7 k$ : $\frac{4 k + 6}{7 k + 6} = \frac{5}{8}$ ⟹ $32 k + 48 = 35 k + 30$ .
2 $3 k = 18$ ⟹ $k = 6$ ; the smaller $= 24$

Answer 24

Example 9 · Standard

Two quantities are in the ratio 5 : 6. The first rises 20% and the second falls 25%. New ratio?

Solution, step by step

1New: $5 \times 1.2 = 6$ and $6 \times 0.75 = 4.5$ .
2Ratio $= 6 : 4.5 = 4 : 3$

Answer 4 : 3

BasicTry it yourself first

Two numbers are in the ratio 3 : 5. When 4 is added to each the ratio becomes 2 : 3. Which is the bigger number?

StandardTry it yourself first

Two numbers are in the ratio 5 : 8. When 9 is subtracted from each the ratio becomes 7 : 13. Which is the bigger number?

StandardTry it yourself first

Two numbers are in the ratio 4 : 9. When 5 is added to each the ratio becomes 33 : 68. Which is the bigger number?

AdvancedTry it yourself first

Two numbers are in the ratio 7 : 12. When 11 is subtracted from each the ratio becomes 24 : 49. Which is the bigger number?

BasicTry it yourself first

Two quantities are in the ratio 2 : 3. The first rises by 50% and the second rises by 0%. What is the new ratio?

StandardTry it yourself first

Two quantities are in the ratio 5 : 6. The first rises by 20% and the second falls by 25%. What is the new ratio?

AdvancedTry it yourself first

Two quantities are in the ratio 7 : 9. The first falls by 10% and the second rises by 20%. What is the new ratio?

Income–expenditure–savings and coins

Different units, one

k

Two ratios with equal savings, and coins of different worth.

Income, expenditure, savings: savings $=$ income $-$ expenditure. If two persons save equally call the incomes $ia$ and expenses $e b$ and form two equations. Coins: turn the ratio of numbers into value — $a$ rupee coins are worth $a$ , $b$ fifty-paise coins $\frac{b}{2}$ , $c$ twenty-five-paise coins $\frac{c}{4}$ .

Example 10 · Standard

The incomes of two persons are in the ratio 5 : 4 and their expenses 4 : 3. If each saves ₹2,000, what are the incomes?

Solution, step by step

1Incomes $5 a, 4 a$ ; expenses $4 b, 3 b$ : $5 a - 4 b = 2000$ and $4 a - 3 b = 2000$ .
2Solving: $a = b = 2000$ ⟹ incomes ₹10,000 and ₹8,000.

Answer ₹10,000 and ₹8,000

Example 11 · Standard

Coins of ₹1, 50 paise and 25 paise are in the ratio 4 : 6 : 8 and total ₹45. How many 25-paise coins are there?

Solution, step by step

1Value $= 4 k + 3 k + 2 k = 9 k = 45$ ⟹ $k = 5$ .
225-paise coins $= 8 k = 40$

Answer 40

BasicTry it yourself first

The incomes of two persons are in the ratio 5 : 4 and their expenses 4 : 3. If each saves ₹2,000, what is the first person’s income?

StandardTry it yourself first

The incomes of two persons are in the ratio 7 : 5 and their expenses 5 : 3. If each saves ₹3,000, what is the first person’s income?

AdvancedTry it yourself first

The incomes of two persons are in the ratio 3 : 2 and their expenses 5 : 3. If each saves ₹1,000, what is the first person’s income?

BasicTry it yourself first

A bag holds ₹1, 50-paise and 25-paise coins in the ratio 2 : 4 : 8. Their total value is ₹54. How many 50-paise coins are there?

StandardTry it yourself first

A bag holds ₹1, 50-paise and 25-paise coins in the ratio 6 : 8 : 12. Their total value is ₹78. How many 50-paise coins are there?

AdvancedTry it yourself first

A bag holds ₹1, 50-paise and 25-paise coins in the ratio 3 : 10 : 16. Their total value is ₹72. How many 50-paise coins are there?

Traps and fast tricks

Slips · tricks

Where marks go in ratio questions.

The commonest mistakes

Six traps

The slips that cost marks.

Trap 1Reversing the terms. “The ratio of A to B” is

A : B

, not

B : A

. Read the wording carefully.

Trap 2Leaving the units different. The ratio of ₹5 to 50 paise is

5 : 0.5 = 10 : 1

, not

5 : 50

Trap 3Forgetting to total the parts first. Add the parts, then find one part.

Trap 4Not equalising the middle term. For

A : B = 2 : 3

B : C = 4 : 5

take the LCM of

B

to get

A : B : C

Trap 5Adding to the ratio terms instead of the numbers. Adding 2 to numbers in the ratio

3 : 5

is not

5 : 7

unless the numbers themselves are 3 and 5.

Trap 6Adding in a compound ratio. In a compound ratio the antecedents are multiplied, not added.

Fast tricks

Assume

k

· LCM · totals

Habits that shorten the work.

Attach a $k$ to every ratio: $a : b$ ⟹ $ak, bk$ . Then turn the condition into an equation.
Divisibility check: the total must be a multiple of the total parts — use it to cut options.
Ratios with a common term: from $a : b$ and $a : c$ take the LCM of $a$ to get $b : c$ .
From percents to ratio: $20%$ of $x$ = $30%$ of $y$ ⟹ $x : y = 30 : 20 = 3 : 2$ .
Ratio and average: for two groups of equal size the ratio of averages equals the ratio of totals.

Example 12 · Standard

If 30% of

x

= 40% of

y

, find

x : y

Solution, step by step

1 $0.3 x = 0.4 y$ ⟹ $\frac{x}{y} = \frac{0.4}{0.3}$ .
2 $x : y = 4 : 3$

Answer 4 : 3

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

BasicTry it yourself first

Write the ratio 35 : 56 in its simplest form.

BasicTry it yourself first

Write the ratio 54 : 66 in its simplest form.

BasicTry it yourself first

Which is the greater ratio: 2 : 5 or 3 : 7?

BasicTry it yourself first

What is the duplicate ratio of 2 : 5?

BasicTry it yourself first

What is the sub-duplicate ratio of 9 : 16?

BasicTry it yourself first

What is the triplicate ratio of 1 : 4?

BasicTry it yourself first

What is the compound ratio of 1 : 2, 3 : 5 and 2 : 3?

BasicTry it yourself first

₹6,300 is divided among A, B in the ratio 2 : 5. What is B's share?

BasicTry it yourself first

₹7,500 is divided among A, B, C in the ratio 1 : 2 : 2. What is C's share?

BasicTry it yourself first

A sum is shared in the ratio 7 : 4 and the shares differ by ₹900. What is the bigger share?

BasicTry it yourself first

If A : B = 1 : 2 and B : C = 3 : 4, find A : B : C.

BasicTry it yourself first

What is the fourth proportional to 5, 8 and 15?

BasicTry it yourself first

What is the third proportional to 5 and 15?

BasicTry it yourself first

What is the mean proportional between 9 and 16?

BasicTry it yourself first

x : y = 2 : 3

, find

(1 x + 2 y) : (2 x + 1 y)

BasicTry it yourself first

Two numbers are in the ratio 2 : 5. When 6 is added to each the ratio becomes 7 : 13. Which is the bigger number?

BasicTry it yourself first

Two quantities are in the ratio 3 : 4. The first rises by 10% and the second rises by 10%. What is the new ratio?

BasicTry it yourself first

A bag holds ₹1, 50-paise and 25-paise coins in the ratio 2 : 4 : 8. Their total value is ₹54. How many 50-paise coins are there?

BasicTry it yourself first

The incomes of two persons are in the ratio 5 : 4 and their expenses 4 : 3. If each saves ₹2,000, what is the first person’s income?

BasicTry it yourself first

The sum of three numbers is 170. The ratio of the first to the second is 1 : 2 and of the second to the third 3 : 4. What is the second number?

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

Write the ratio 182 : 325 in its simplest form.

StandardTry it yourself first

Which is the greater ratio: 11 : 15 or 17 : 23?

StandardTry it yourself first

Which is the greater ratio: 9 : 14 or 23 : 36?

StandardTry it yourself first

What is the triplicate ratio of 3 : 7?

StandardTry it yourself first

What is the sub-duplicate ratio of 25 : 49?

StandardTry it yourself first

What is the sub-triplicate ratio of 27 : 64?

StandardTry it yourself first

What is the compound ratio of 4 : 9, 3 : 8 and 5 : 6?

StandardTry it yourself first

What is the compound ratio of 7 : 10, 4 : 5 and 15 : 14?

StandardTry it yourself first

₹24,000 is divided among A, B, C in the ratio 3 : 4 : 5. What is A's share?

StandardTry it yourself first

₹17,600 is divided among A, B, C, D in the ratio 2 : 3 : 5 : 6. What is C's share?

StandardTry it yourself first

A sum is shared in the ratio 9 : 5 and the shares differ by ₹2,800. What is the bigger share?

StandardTry it yourself first

A sum is shared in the ratio 8 : 3 and the shares differ by ₹4,500. What is the bigger share?

StandardTry it yourself first

If A : B = 4 : 7 and B : C = 14 : 9, find A : B : C.

StandardTry it yourself first

If A : B = 5 : 8 and B : C = 12 : 5, find A : B : C.

StandardTry it yourself first

The sum of three numbers is 310. The ratio of the first to the second is 3 : 4 and of the second to the third 6 : 5. What is the second number?

StandardTry it yourself first

The sum of three numbers is 280. The ratio of the first to the second is 2 : 5 and of the second to the third 15 : 7. What is the third number?

StandardTry it yourself first

What is the fourth proportional to 9, 21 and 33?

StandardTry it yourself first

What is the fourth proportional to 16, 28 and 36?

StandardTry it yourself first

What is the third proportional to 16 and 36?

StandardTry it yourself first

What is the third proportional to 25 and 45?

StandardTry it yourself first

What is the mean proportional between 8 and 50?

StandardTry it yourself first

What is the mean proportional between 20 and 45?

StandardTry it yourself first

x : y = 3 : 5

, find

(5 x + 2 y) : (2 x + 3 y)

StandardTry it yourself first

x : y = 7 : 4

, find

(3 x + 5 y) : (6 x + 1 y)

StandardTry it yourself first

Two numbers are in the ratio 3 : 7. When 5 is added to each the ratio becomes 1 : 2. Which is the bigger number?

StandardTry it yourself first

Two numbers are in the ratio 7 : 10. When 6 is subtracted from each the ratio becomes 11 : 17. Which is the bigger number?

StandardTry it yourself first

Two quantities are in the ratio 4 : 7. The first rises by 25% and the second falls by 20%. What is the new ratio?

StandardTry it yourself first

Two quantities are in the ratio 8 : 9. The first falls by 25% and the second rises by 50%. What is the new ratio?

StandardTry it yourself first

The incomes of two persons are in the ratio 7 : 5 and their expenses 5 : 3. If each saves ₹3,000, what is the first person’s income?

StandardTry it yourself first

A bag holds ₹1, 50-paise and 25-paise coins in the ratio 6 : 8 : 12. Their total value is ₹78. How many 50-paise coins are there?

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

Which is the greater ratio: 29 : 37 or 41 : 52?

AdvancedTry it yourself first

Which is the greater ratio: 47 : 61 or 53 : 69?

AdvancedTry it yourself first

What is the triplicate ratio of 4 : 9?

AdvancedTry it yourself first

What is the sub-duplicate ratio of 64 : 121?

AdvancedTry it yourself first

What is the sub-triplicate ratio of 125 : 216?

AdvancedTry it yourself first

What is the compound ratio of 11 : 13, 26 : 33 and 15 : 22?

AdvancedTry it yourself first

What is the compound ratio of 9 : 14, 35 : 12 and 16 : 15?

AdvancedTry it yourself first

₹38,400 is divided among A, B, C, D in the ratio 3 : 5 : 7 : 9. What is D's share?

AdvancedTry it yourself first

₹52,500 is divided among A, B, C, D in the ratio 4 : 6 : 7 : 8. What is B's share?

AdvancedTry it yourself first

A sum is shared in the ratio 13 : 9 and the shares differ by ₹5,600. What is the bigger share?

AdvancedTry it yourself first

A sum is shared in the ratio 17 : 11 and the shares differ by ₹7,200. What is the bigger share?

AdvancedTry it yourself first

If A : B = 7 : 9 and B : C = 12 : 13, find A : B : C.

AdvancedTry it yourself first

If A : B = 11 : 15 and B : C = 20 : 21, find A : B : C.

AdvancedTry it yourself first

The sum of three numbers is 396. The ratio of the first to the second is 5 : 7 and of the second to the third 14 : 9. What is the first number?

AdvancedTry it yourself first

The sum of three numbers is 363. The ratio of the first to the second is 3 : 8 and of the second to the third 16 : 11. What is the third number?

AdvancedTry it yourself first

What is the fourth proportional to 24, 54 and 84?

AdvancedTry it yourself first

What is the third proportional to 18 and 54?

AdvancedTry it yourself first

What is the mean proportional between 32 and 98?

AdvancedTry it yourself first

What is the mean proportional between 27 and 75?

AdvancedTry it yourself first

x : y = 11 : 7

, find

(5 x + 4 y) : (3 x + 5 y)

AdvancedTry it yourself first

x : y = 13 : 9

, find

(7 x + 6 y) : (4 x + 11 y)

AdvancedTry it yourself first

Two numbers are in the ratio 8 : 11. When 9 is added to each the ratio becomes 65 : 86. Which is the bigger number?

AdvancedTry it yourself first

Two numbers are in the ratio 9 : 13. When 12 is subtracted from each the ratio becomes 15 : 23. Which is the bigger number?

AdvancedTry it yourself first

Two quantities are in the ratio 13 : 17. The first rises by 30% and the second falls by 20%. What is the new ratio?

AdvancedTry it yourself first

The incomes of two persons are in the ratio 3 : 2 and their expenses 5 : 3. If each saves ₹1,000, what is the first person’s income?

AdvancedTry it yourself first

A bag holds ₹1, 50-paise and 25-paise coins in the ratio 3 : 10 : 16. Their total value is ₹72. How many 50-paise coins are there?

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.