Ratio Basics and Operations

What a Ratio Is

A ratio compares two quantities of the same kind by asking how many times one is of the other. The ratio of a to b is written a : b (read "a is to b") or as the fraction $\frac{a}{b}$. The first term is called the antecedent and the second the consequent.

If a class has 24 boys and 18 girls, boys : girls = 24 : 18 = 4 : 3. For every 4 boys there are 3 girls.

Three ground rules.

• Same kind. You can compare 40 minutes with 2 hours, but not 5 kg with 3 metres.
• Same units. Convert first. 40 minutes : 2 hours is 40 : 120 = 1 : 3, not 40 : 2.
• No units in the answer. The units cancel, so a ratio is a pure number. Write 1 : 3, not "1 : 3 minutes".

Order matters too. Boys : girls = 4 : 3 means girls : boys = 3 : 4, and mixing the two up is an easy slip to make in the exam.

Simplest form. Divide both terms by their HCF. For 84 : 126 the HCF is 42, so the ratio is 2 : 3. A ratio is in its lowest terms when the HCF of its terms is 1.

Equivalent ratios. Multiplying or dividing both terms by the same non-zero number does not change a ratio: 2 : 3 = 4 : 6 = 10 : 15 = 2k : 3k. This is the idea behind most questions. A ratio tells you only the relative sizes, so if boys : girls = 4 : 3, the actual numbers are 4k and 3k for some k. With k = 6 there are 24 boys and 18 girls; with k = 10 there are 40 and 30.

It also means the total must be a multiple of 4 + 3 = 7. A class with boys and girls in the ratio 4 : 3 can have 42 or 56 students, but never 50. SSC sometimes asks exactly this: "which of these cannot be the total?"

Part to whole and whole to part. If A : B = 4 : 3, the whole is 7 parts. So

• A is $\frac{4}{7}$ of the total and B is $\frac{3}{7}$ of it
• A : total = 4 : 7, and total : B = 7 : 3
• the difference A − B is 1 part, which is $\frac{1}{7}$ of the total

Greater and lesser inequality. A ratio like 5 : 3, where the first term is bigger, is a ratio of greater inequality; 3 : 5 is a ratio of lesser inequality; 4 : 4 is a ratio of equality. The names rarely appear, but the idea matters when you compare ratios.

Key Point: A ratio a : b means the two quantities are ak and bk. Bringing in k turns almost every ratio question into a one-line equation.

Simplifying Ratios: Fractions, Decimals and Mixed Units

A ratio should end up as whole numbers with no common factor. Three kinds of terms need some work first.

Fractions. Multiply every term by the LCM of the denominators.

• $\frac{2}{3} : \frac{3}{4} : \frac{5}{6}$: the LCM of 3, 4 and 6 is 12, which gives 8 : 9 : 10.
• $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$: multiply by 12 to get 6 : 4 : 3.
• Mixed numbers become improper fractions first: $1\frac{1}{2} : 2\frac{1}{3} = \frac{3}{2} : \frac{7}{3}$, and multiplying by 6 gives 9 : 14.

For two terms there is a quicker way: $\frac{a}{b} : \frac{c}{d}$ = ad : bc. So $\frac{3}{5} : \frac{4}{7}$ = 21 : 20.

A ratio of fractions can be checked backwards. Divide each term of your answer by the matching fraction; you should get the same number every time. For 8 : 9 : 10 against $\frac{2}{3}, \frac{3}{4}, \frac{5}{6}$: 8 ÷ $\frac{2}{3}$ = 12, 9 ÷ $\frac{3}{4}$ = 12 and 10 ÷ $\frac{5}{6}$ = 12, so it is right.

Decimals. Multiply by 10, 100 or 1000 to clear the decimal points, then simplify.

• 0.75 : 1.2 = 75 : 120 = 5 : 8
• 0.4 : 0.06 : 1.2 = 40 : 6 : 120 = 20 : 3 : 60
• 2.5 : 0.125 = 2500 : 125 = 20 : 1

Use the same multiplier for every term. Clearing 0.4 by 10 and 0.06 by 100 separately is the usual mistake.

Mixed units. Bring both terms to the smaller unit.

The trap answers come from skipping the conversion. Writing 75 paise : ₹3 as 75 : 3 = 25 : 1 says the 75 paise is the bigger amount, which is plainly wrong. A quick "which one is bigger?" check catches this.

Square roots and powers. Simplify each term first. $\sqrt{12} : \sqrt{27} = 2\sqrt{3} : 3\sqrt{3}$, which is 2 : 3. And $2^5 : 4^2$ = 32 : 16 = 2 : 1.

Comparing and Ordering Ratios

To compare two ratios, compare them as fractions.

Cross-multiplying. For a : b and c : d, compare a × d with b × c. If a × d is bigger, a : b is the larger ratio.

$$\frac{5}{7} \text{ vs } \frac{7}{10}: \quad 5 \times 10 = 50, \quad 7 \times 7 = 49$$

Since 50 > 49, 5 : 7 is larger than 7 : 10. Two ratios are equal only when the cross products match.

Decimals. With four options, turning each into a decimal is usually fastest. Take 3 : 5, 5 : 8, 7 : 11 and 4 : 7:

So 7 : 11 is the largest and 4 : 7 the smallest. In ascending order they read 4 : 7, 3 : 5, 5 : 8, 7 : 11. The fraction-to-percentage table in the Percentage chapter makes most of these values instant.

Equal first or second terms. Make one term the same and compare the other. For 3 : 4 and 5 : 7, make the antecedents 15: 15 : 20 and 15 : 21. The one with the smaller consequent is larger, so 3 : 4 is the larger ratio.

The "closer to 1" idea. Adding the same positive number to both terms of a ratio pulls it towards 1.

• A ratio below 1 goes up: 3 : 5 = 0.6, but 4 : 6 = 0.667.
• A ratio above 1 comes down: 5 : 3 = 1.667, but 6 : 4 = 1.5.

So among 5 : 7, 7 : 9 and 9 : 11 (each consequent is 2 more than the antecedent), the one with the biggest terms, 9 : 11, is the largest. Among 7 : 5, 9 : 7 and 11 : 9 it goes the other way: 7 : 5 is the largest.

Comparing after an operation. If a question asks which is larger, the duplicate ratio of 2 : 3 or the ratio 3 : 5, first find the duplicate ratio (4 : 9, as explained in the next part) and then compare: 4 × 5 = 20 and 9 × 3 = 27, so 3 : 5 is larger.

Ratios from a table. The same method answers "in which year was the ratio of boys to girls the highest?" in a table. Work out each year's ratio as a decimal and pick the biggest; never judge by the raw numbers.

[SSC Tip] Do not compare ratios by the size of their terms. 7 : 12 has bigger numbers than 3 : 5 but is the smaller ratio (0.583 against 0.6). Always compare the value of a ÷ b.

Compound, Duplicate, Triplicate and Inverse Ratios

These are names for operations on ratios. SSC asks them on their own and also mixes two or three in one question.

"Duplicate" means squared, not doubled. The duplicate ratio of 3 : 5 is 9 : 25; 6 : 10 is just 3 : 5 again. Students fall into this trap easily. In the same way, the sub-duplicate ratio takes square roots, not halves.

Compound ratio. Multiply the antecedents together and the consequents together:

$$\text{compound of } a : b,\ c : d,\ e : f = ace : bdf$$

The compound ratio of 3 : 4, 8 : 9 and 15 : 16 is (3 × 8 × 15) : (4 × 9 × 16) = 360 : 576 = 5 : 8. Cancelling first saves work: $\frac{3 \times 8 \times 15}{4 \times 9 \times 16} = \frac{5}{8}$.

A duplicate ratio is simply the compound of a ratio with itself, and the compound of a ratio with its inverse is always 1 : 1.

Inverse of three terms. The inverse of a : b : c is $\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$, which is not c : b : a. For 2 : 3 : 4 it is $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$ = 6 : 4 : 3. Just reversing the order to 4 : 3 : 2 gives a wrong answer; reversing works only for two terms.

Mixed operations. Work out each piece first, then compound them. For the compound ratio of the duplicate ratio of 2 : 3, the sub-triplicate ratio of 27 : 8 and the inverse ratio of 4 : 9:

• the duplicate of 2 : 3 is 4 : 9
• the sub-triplicate of 27 : 8 is 3 : 2
• the inverse of 4 : 9 is 9 : 4

Compounding: (4 × 3 × 9) : (9 × 2 × 4) = 108 : 72 = 3 : 2.

Working backwards. If the compound ratio of 5 : 8 and the duplicate ratio of x : 3 is 10 : 9, then

$$\frac{5x^2}{72} = \frac{10}{9} \Rightarrow x^2 = 16 \Rightarrow x = 4$$

With surds. The duplicate ratio of $2\sqrt{2} : 3\sqrt{5}$ is 8 : 45, because $(2\sqrt{2})^2 = 8$ and $(3\sqrt{5})^2 = 45$.

Combining Ratios: A : B and B : C

This type comes up often in recent CGL papers. Two ratios share a term, and you need one ratio for all three quantities.

The rule: make the common term equal. Say A : B = 3 : 5 and B : C = 2 : 3. B is 5 in the first ratio and 2 in the second. The LCM of 5 and 2 is 10, so scale both:

• A : B = 3 : 5 = 6 : 10 (multiply by 2)
• B : C = 2 : 3 = 10 : 15 (multiply by 5)

Now B is 10 in both, so A : B : C = 6 : 10 : 15.

The quick pattern. For A : B = a : b and B : C = c : d,

$$A : B : C = ac : bc : bd$$

Multiply the first ratio by c (the B of the second) and the second ratio by b (the B of the first). With the numbers above: 3 × 2 : 5 × 2 : 5 × 3 = 6 : 10 : 15. Divide by the HCF at the end if all three terms share a factor.

A : C directly. A : C = ac : bd, the compound of the two ratios. Here A : C = 6 : 15 = 2 : 5. You do not need the three-term ratio for this.

When the common term is not in the middle.

• A : B = 3 : 4 and A : C = 5 : 6. Here A is common. Make A equal to 15: A : B = 15 : 20 and A : C = 15 : 18. So A : B : C = 15 : 20 : 18, and B : C = 20 : 18 = 10 : 9.
• A : B = 5 : 6 and C : B = 4 : 9. Here B is common but is the second term of both. Make B = 18: A : B = 15 : 18 and C : B = 8 : 18. So A : B : C = 15 : 18 : 8.

Write each ratio with its letters before scaling. Most wrong answers come from mixing up which term is which.

Ratios given as fractions. A : B = $\frac{1}{2} : \frac{1}{3}$ and B : C = $\frac{1}{4} : \frac{1}{5}$. Clean each one first: A : B = 3 : 2 and B : C = 5 : 4. Then B = 10, and A : B : C = 15 : 10 : 8.

With a total. If A : B = 3 : 4, B : C = 6 : 7 and A + B + C = ₹2,450, then A : B : C = 18 : 24 : 28 = 9 : 12 : 14. The parts add up to 35, so one part is ₹70, and the shares are ₹630, ₹840 and ₹980.

Other ratios from the same three. Once you have A : B : C, anything else follows. With 6 : 10 : 15, (A + B) : (B + C) = 16 : 25, and C : (A + B + C) = 15 : 31.

[SSC Tip] Never put two ratios together until the common term matches. A : B = 3 : 5 and B : C = 2 : 3 do not give A : B : C = 3 : 5 : 3 or 3 : 2 : 3, which is what you get by just writing the numbers side by side; both turn up as wrong options.

Longer Chains: Three and Four Links

Here A : B, B : C and C : D are given, and you need A : D or the full ratio A : B : C : D.

A : D directly. Multiply straight through, just like a compound ratio:

$$\frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D}$$

If A : B = 3 : 5, B : C = 2 : 3 and C : D = 6 : 5, then A : D = (3 × 2 × 6) : (5 × 3 × 5) = 36 : 75 = 12 : 25. The B and C in the middle cancel out.

A : B : C : D in full. Build it one link at a time.

1. A : B : C = 6 : 10 : 15, as worked out in Combining Ratios above.
2. C is 15 here and 6 in C : D = 6 : 5. The LCM of 15 and 6 is 30, so double the first: 12 : 20 : 30. Multiply C : D by 5: 30 : 25.
3. A : B : C : D = 12 : 20 : 30 : 25.

For A : B = a : b, B : C = c : d and C : D = e : f, the pattern is

$$A : B : C : D = ace : bce : bde : bdf$$

Each term takes the consequent of every link to its left and the antecedent of every link to its right. With the numbers above this gives 36 : 60 : 90 : 75, which is 12 : 20 : 30 : 25 after dividing by 3.

Reversed links. Sometimes a link is written the other way round. If A : B = 3 : 4, C : B = 2 : 5 and D : C = 3 : 4, flip the reversed ones first: B : C = 5 : 2 and C : D = 4 : 3. Then A : D = (3 × 5 × 4) : (4 × 2 × 3) = 60 : 24 = 5 : 2.

Links in a jumbled order. If A : B = 4 : 3, B : D = 6 : 7 and C : D = 2 : 5, arrange them so each link shares a letter with the next: A : B, B : D, D : C. With D : C = 5 : 2, the chain gives A : B : D = 8 : 6 : 7, and then D = 35 gives A : B : D : C = 40 : 30 : 35 : 14. So A : B : C : D = 40 : 30 : 14 : 35.

Fractions in the links. A : B = $\frac{2}{3} : \frac{4}{5}$, B : C = $\frac{1}{2} : \frac{2}{3}$ and C : D = $\frac{3}{4} : \frac{1}{2}$. Clean each link: 5 : 6, 3 : 4 and 3 : 2. Then A : B : C = 5 : 6 : 8, and making C = 24 gives A : B : C : D = 15 : 18 : 24 : 16.

With a difference. Once the full ratio is known, the rest is dividing an amount. If A : B : C : D = 12 : 20 : 30 : 25 and D gets ₹1,040 more than A, then 13 parts = ₹1,040, one part is ₹80, and B gets 20 × 80 = ₹1,600.

Equal Products: 2A = 3B = 4C and A's 1/3 = B's 1/4

Many questions give the relation as an equality instead of a ratio. There are three forms.

Form 1: pA = qB = rC. Set each equal to k. Then A = $\frac{k}{p}$, B = $\frac{k}{q}$ and C = $\frac{k}{r}$, so

$$A : B : C = \frac{1}{p} : \frac{1}{q} : \frac{1}{r}$$

For 2A = 3B = 4C: A : B : C = $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$, and multiplying by 12 gives 6 : 4 : 3. The numbers flip: the quantity with the biggest multiplier is the smallest. The trap option is 2 : 3 : 4. To check, put the numbers back: 2 × 6 = 12, 3 × 4 = 12 and 4 × 3 = 12, all equal.

For two terms the flip is direct: 5A = 8B gives A : B = 8 : 5.

A short way for three terms is A : B : C = qr : pr : pq. For 2A = 3B = 4C that is 12 : 8 : 6 = 6 : 4 : 3.

Form 2: A/p = B/q = C/r. Now there is no flip:

$$\frac{A}{3} = \frac{B}{4} = \frac{C}{5} = k \Rightarrow A : B : C = 3 : 4 : 5$$

If A + B + C = 96 as well, then 12k = 96, so k = 8, and A = 24, B = 32, C = 40.

Form 3: a fraction of A equals a fraction of B. "$\frac{1}{3}$ of A = $\frac{1}{4}$ of B" means $\frac{A}{3} = \frac{B}{4}$, so A : B = 3 : 4. In general,

$$\frac{m}{n} \text{ of } A = \frac{p}{q} \text{ of } B \Rightarrow \frac{A}{B} = \frac{p}{q} \div \frac{m}{n}$$

For $\frac{2}{3}$ of A = $\frac{4}{5}$ of B: $\frac{A}{B} = \frac{4}{5} \times \frac{3}{2} = \frac{6}{5}$, so A : B = 6 : 5. The quantity with the smaller fraction is the bigger one, which is a handy quick check.

With three people, "$\frac{1}{2}$ of A = $\frac{2}{3}$ of B = $\frac{3}{4}$ of C" works the same way. Put each equal to k: A = 2k, B = $\frac{3k}{2}$ and C = $\frac{4k}{3}$. Multiplying by 6 gives A : B : C = 12 : 9 : 8.

Mixed forms. "4A = 5B and 3B = 7C" gives A : B = 5 : 4 and B : C = 7 : 3. B is 4 and 7, so make it 28: A : B : C = 35 : 28 : 12. This is the same combining method as in Combining Ratios.

The percentage version, such as 20% of A = 30% of B, works exactly like Form 3.

Dividing an Amount in a Given Ratio

To split an amount in the ratio a : b : c, add the parts and give each person their share of the total:

$$\text{A's share} = \frac{a}{a + b + c} \times \text{Total}$$

Two parts. ₹1,800 in the ratio 4 : 5: there are 9 parts, one part is ₹200, so the shares are ₹800 and ₹1,000.

Three parts. ₹5,400 in the ratio 2 : 3 : 4: the parts total 9, so one part is ₹600 and the shares are ₹1,200, ₹1,800 and ₹2,400. Add the shares back to check the total.

Ratio given as fractions. Clear the fractions first. ₹3,900 in the ratio $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$ is the same as 6 : 4 : 3. The parts total 13, so one part is ₹300 and the shares are ₹1,800, ₹1,200 and ₹900. Dividing in 2 : 3 : 4 by mistake is the usual trap.

Inversely proportional to numbers. "Divide ₹4,400 in inverse proportion to 4, 5 and 10" means the ratio $\frac{1}{4} : \frac{1}{5} : \frac{1}{10}$ = 5 : 4 : 2. There are 11 parts of ₹400, so the shares are ₹2,000, ₹1,600 and ₹800. The person with the biggest number gets the smallest share.

"A gets x more than B". The difference of the shares is the difference of the parts. If an amount is split in the ratio 5 : 8 and the second person gets ₹1,260 more, then 3 parts = ₹1,260, one part = ₹420, and the total is 13 × 420 = ₹5,460.

One share given. If C's share in the ratio 4 : 5 : 6 is ₹1,500, then one part is 1500 ÷ 6 = ₹250, and the total is 15 × 250 = ₹3,750.

A share as a fraction of the others. "₹7,200 is divided among A, B and C so that A gets $\frac{1}{3}$ of what B and C together get, and B gets $\frac{2}{7}$ of what A and C together get. Find C's share."

A = $\frac{1}{3}$(B + C) means A : (B + C) = 1 : 3, so A gets $\frac{1}{4}$ of the total, which is ₹1,800. B = $\frac{2}{7}$(A + C) means B gets $\frac{2}{9}$ of the total, which is ₹1,600. So C gets 7200 − 1800 − 1600 = ₹3,800.

The rule: if X gets $\frac{p}{q}$ of what the others get together, X's share is $\frac{p}{p + q}$ of the total.

An equal-product condition. "Divide ₹2,220 among A, B and C so that 4 times A's share = 5 times B's share = 6 times C's share." The ratio is $\frac{1}{4} : \frac{1}{5} : \frac{1}{6}$ = 15 : 12 : 10, which is 37 parts of ₹60. So A gets ₹900, B ₹720 and C ₹600. Check: 4 × 900 = 5 × 720 = 6 × 600 = 3600.

Before dividing, check that the total is a multiple of the sum of the parts. If 2220 ÷ 37 had not come out exact, the ratio you wrote would almost certainly be wrong.

Finding Numbers, and Changing a Ratio by Adding a Number

Numbers from their ratio. If two numbers are in the ratio a : b, call them ak and bk and use the other fact to find k. For the ratio 3 : 5:

For products and squares k comes from a square root, so take the positive root unless the question says otherwise. Three numbers work the same way with ak, bk and ck.

With the HCF or LCM. If the ratio a : b is in lowest terms, k is the HCF of the two numbers and their LCM is abk. Two numbers in the ratio 4 : 7 with LCM 336 have 28k = 336, so k = 12 and the numbers are 48 and 84.

[Tier 2] Sums of pairs. If (A + B) : (B + C) : (C + A) = 6 : 7 : 8 and A + B + C = 63, add the three pairs: they make 2(A + B + C) = 126, which is 21 parts, so one part is 6. Then A + B = 36, B + C = 42 and C + A = 48, which gives C = 63 − 36 = 27, A = 63 − 42 = 21 and B = 63 − 48 = 15.

Adding the same number to both terms. What number x must be added to both terms of a : b to make it c : d?

$$\frac{a + x}{b + x} = \frac{c}{d} \Rightarrow x = \frac{bc - ad}{d - c}$$

To change 5 : 9 into 2 : 3: x = (9 × 2 − 5 × 3) ÷ (3 − 2) = 3. Check: 8 : 12 = 2 : 3.

Subtracting from both terms. Solve $\frac{a - x}{b - x} = \frac{c}{d}$ in the same way. To turn 13 : 19 into 1 : 2: 2(13 − x) = 19 − x, so 26 − 2x = 19 − x and x = 7. Check: 6 : 12 = 1 : 2.

A number added to each of two numbers. "Two numbers are in the ratio 2 : 5. If 8 is added to each, the ratio becomes 1 : 2. Find the numbers." Write them as 2k and 5k:

$$\frac{2k + 8}{5k + 8} = \frac{1}{2} \Rightarrow 4k + 16 = 5k + 8 \Rightarrow k = 8$$

The numbers are 16 and 40, and 24 : 48 is indeed 1 : 2.

[SSC Tip] In "add x to both terms" questions, checking the four options takes seconds: add each option to both terms and see which one gives the new ratio.

Solved Examples

Question 1

The ratio of 2.4 kg to 960 g is: (a) 1 : 400 (b) 5 : 2 (c) 2 : 5 (d) 25 : 1

Answer. Both must be in the same unit, so I change 2.4 kg to 2400 g. The ratio is 2400 : 960, and dividing both by 480 gives 5 : 2. The answer is (b) 5 : 2. Writing 2.4 : 960 without converting gives 1 : 400, which would make the 2.4 kg the lighter one.

Question 2

$\frac{2}{5} : \frac{3}{4} : \frac{5}{6}$ in its simplest form is: (a) 2 : 3 : 5 (b) 50 : 45 : 24 (c) 8 : 15 : 25 (d) 24 : 45 : 50

Answer. The denominators are 5, 4 and 6, and their LCM is 60. Multiplying every term by 60: $\frac{2}{5} \times 60 = 24$, $\frac{3}{4} \times 60 = 45$ and $\frac{5}{6} \times 60 = 50$. The three numbers have no common factor, so the answer is (d) 24 : 45 : 50.

Question 3

Which of the following ratios is the largest? (a) 4 : 7 (b) 5 : 9 (c) 7 : 12 (d) 9 : 16

Answer. I turn each into a decimal: 4 ÷ 7 ≈ 0.571, 5 ÷ 9 ≈ 0.556, 7 ÷ 12 ≈ 0.583 and 9 ÷ 16 = 0.5625. The biggest is 0.583, so the answer is (c) 7 : 12. 9 : 16 has the biggest numbers but not the biggest value.

Question 4

The compound ratio of the duplicate ratio of 4 : 5, the sub-duplicate ratio of 225 : 64 and the inverse ratio of 3 : 2 is: (a) 4 : 5 (b) 9 : 5 (c) 1 : 1 (d) 5 : 4

Answer. I work out each piece first. The duplicate of 4 : 5 is 16 : 25. The sub-duplicate of 225 : 64 is 15 : 8 (square roots). The inverse of 3 : 2 is 2 : 3. Compounding means multiplying the first terms together and the second terms together: (16 × 15 × 2) : (25 × 8 × 3) = 480 : 600 = 4 : 5. The answer is (a) 4 : 5.

Question 5

The inverse ratio of 4 : 5 : 6 is: (a) 6 : 5 : 4 (b) 10 : 12 : 15 (c) 15 : 12 : 10 (d) 5 : 4 : 3

Answer. The inverse ratio is $\frac{1}{4} : \frac{1}{5} : \frac{1}{6}$. The LCM of 4, 5 and 6 is 60, and multiplying by 60 gives 15 : 12 : 10. The answer is (c) 15 : 12 : 10. Simply reversing the order to 6 : 5 : 4 looks right but isn't; that trick only works for a ratio of two terms.

Question 6

If the compound ratio of 2 : 5 and the triplicate ratio of x : 2 is 27 : 20, then x = ? (a) 9 (b) 3 (c) 6 (d) 2

Answer. The triplicate ratio of x : 2 is $x^3 : 8$. Compounding it with 2 : 5 gives $2x^3 : 40$. So $\frac{2x^3}{40} = \frac{27}{20}$, which means $2x^3 = 54$ and $x^3 = 27$. So x = 3. The answer is (b) 3.

Question 7

If A : B = 3 : 5 and B : C = 7 : 4, then A : B : C = ? (a) 3 : 5 : 4 (b) 21 : 35 : 28 (c) 15 : 35 : 20 (d) 21 : 35 : 20

Answer. B is 5 in the first ratio and 7 in the second, so I make it 35. A : B = 3 : 5 = 21 : 35 (times 7) and B : C = 7 : 4 = 35 : 20 (times 5). So A : B : C = 21 : 35 : 20. The answer is (d) 21 : 35 : 20.

Question 8

A sum of ₹2,870 is divided among A, B and C so that A : B = 5 : 4 and B : C = 6 : 7. C's share is: (a) ₹980 (b) ₹840 (c) ₹1,050 (d) ₹1,120

Answer. B is 4 in one ratio and 6 in the other, so I make it 12. A : B = 15 : 12 and B : C = 12 : 14, so A : B : C = 15 : 12 : 14. That is 41 parts, and 2870 ÷ 41 = ₹70. C gets 14 × 70 = ₹980. The answer is (a) ₹980.

Question 9

If A : B = 3 : 4 and A : C = 2 : 5, then B : C = ? (a) 4 : 5 (b) 8 : 15 (c) 15 : 8 (d) 3 : 5

Answer. Here A is the common term. It is 3 in one ratio and 2 in the other, so I make it 6: A : B = 6 : 8 and A : C = 6 : 15. Now B : C = 8 : 15. The answer is (b) 8 : 15. Reading off the second terms, 4 : 5, ignores the fact that the two A's were different.

Question 10

If A : B = 2 : 3, B : C = 5 : 7 and C : D = 14 : 15, then A : D = ? (a) 2 : 15 (b) 28 : 45 (c) 10 : 21 (d) 4 : 9

Answer. For A : D I multiply straight through: $\frac{A}{D} = \frac{2}{3} \times \frac{5}{7} \times \frac{14}{15} = \frac{140}{315} = \frac{4}{9}$. The answer is (d) 4 : 9.

Question 11

If A : B = 3 : 4, B : C = 6 : 5 and C : D = 2 : 3, then A : B : C : D = ? (a) 9 : 12 : 10 : 15 (b) 3 : 4 : 5 : 6 (c) 9 : 12 : 15 : 10 (d) 18 : 24 : 20 : 15

Answer. First A : B : C. B is 4 and 6, so I make it 12: A : B = 9 : 12 and B : C = 12 : 10, giving A : B : C = 9 : 12 : 10. Now C is 10 here and 2 in C : D = 2 : 3, so C : D = 10 : 15. That gives A : B : C : D = 9 : 12 : 10 : 15. The answer is (a) 9 : 12 : 10 : 15. Quick check: 9 : 12 = 3 : 4, 12 : 10 = 6 : 5 and 10 : 15 = 2 : 3.

Question 12

If P : Q = 3 : 7, R : Q = 2 : 5 and R : S = 4 : 9, then P : S = ? (a) 8 : 105 (b) 21 : 10 (c) 10 : 21 (d) 4 : 15

Answer. The middle link is written backwards, so I flip it first: Q : R = 5 : 2. Now P : S = $\frac{3}{7} \times \frac{5}{2} \times \frac{4}{9} = \frac{60}{126} = \frac{10}{21}$. The answer is (c) 10 : 21. Using R : Q without flipping gives $\frac{3 \times 2 \times 4}{7 \times 5 \times 9} = \frac{24}{315} = \frac{8}{105}$, which is wrong.

Question 13

If 3A = 4B = 9C, then A : B : C = ? (a) 3 : 4 : 9 (b) 12 : 9 : 4 (c) 9 : 4 : 3 (d) 4 : 9 : 12

Answer. I put each equal to k: A = $\frac{k}{3}$, B = $\frac{k}{4}$ and C = $\frac{k}{9}$. So A : B : C = $\frac{1}{3} : \frac{1}{4} : \frac{1}{9}$, and multiplying by 36 gives 12 : 9 : 4. The answer is (b) 12 : 9 : 4. Check: 3 × 12 = 4 × 9 = 9 × 4 = 36.

Question 14

₹2,940 is divided among A, B and C so that $\frac{2}{3}$ of A's share = $\frac{3}{4}$ of B's share = $\frac{4}{5}$ of C's share. C's share is: (a) ₹1,080 (b) ₹960 (c) ₹840 (d) ₹900

Answer. I put each equal to k. Then A = $\frac{3k}{2}$, B = $\frac{4k}{3}$ and C = $\frac{5k}{4}$. The LCM of 2, 3 and 4 is 12, so A : B : C = 18 : 16 : 15. That is 49 parts, and 2940 ÷ 49 = 60. C gets 15 × 60 = ₹900. The answer is (d) ₹900. A gets ₹1,080 and B gets ₹960. Check: $\frac{2}{3}$ of 1080 = 720, $\frac{3}{4}$ of 960 = 720 and $\frac{4}{5}$ of 900 = 720.

Question 15

If $\frac{A}{3} = \frac{B}{4}$ and 2B = 3C, then A : B : C = ? (a) 9 : 12 : 8 (b) 3 : 4 : 2 (c) 3 : 4 : 6 (d) 8 : 12 : 9

Answer. $\frac{A}{3} = \frac{B}{4}$ gives A : B = 3 : 4 directly. 2B = 3C gives B : C = 3 : 2, because the numbers flip. B is 4 and 3, so I make it 12: A : B = 9 : 12 and B : C = 12 : 8. So A : B : C = 9 : 12 : 8. The answer is (a) 9 : 12 : 8.

Question 16

₹6,200 is divided among A, B and C in the ratio $\frac{1}{2} : \frac{1}{3} : \frac{1}{5}$. B's share is: (a) ₹1,860 (b) ₹3,000 (c) ₹2,000 (d) ₹1,200

Answer. First a clean ratio. The LCM of 2, 3 and 5 is 30, so $\frac{1}{2} : \frac{1}{3} : \frac{1}{5}$ = 15 : 10 : 6. That is 31 parts, and 6200 ÷ 31 = 200. B gets 10 × 200 = ₹2,000. The answer is (c) ₹2,000.

Question 17

₹5,850 is divided among A, B and C in inverse proportion to 2, 3 and 4. C gets: (a) ₹2,600 (b) ₹1,350 (c) ₹2,700 (d) ₹1,800

Answer. Inverse proportion means the ratio $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$, which is 6 : 4 : 3. That is 13 parts, and 5850 ÷ 13 = 450. C gets 3 × 450 = ₹1,350. The answer is (b) ₹1,350. Splitting in 2 : 3 : 4 instead gives C ₹2,600, the biggest share, but C has the biggest number, so C must get the smallest share.

Question 18

A sum of money is divided between A and B in the ratio 7 : 4. If A gets ₹1,350 more than B, the total sum is: (a) ₹3,150 (b) ₹4,050 (c) ₹5,400 (d) ₹4,950

Answer. A has 7 parts and B has 4, so A has 3 parts more. 3 parts = ₹1,350, so 1 part = ₹450. The total is 11 parts, which is 11 × 450 = ₹4,950. The answer is (d) ₹4,950.

Question 19

Two numbers are in the ratio 5 : 7 and their product is 2240. The sum of the numbers is: (a) 96 (b) 64 (c) 84 (d) 120

Answer. I call the numbers 5k and 7k. Their product is $35k^2 = 2240$, so $k^2 = 64$ and k = 8. The numbers are 40 and 56, and their sum is 96. The answer is (a) 96.

Question 20

What number must be added to each term of the ratio 7 : 13 so that it becomes 2 : 3? (a) 3 (b) 6 (c) 5 (d) 7

Answer. I need $\frac{7 + x}{13 + x} = \frac{2}{3}$. Cross-multiplying: 21 + 3x = 26 + 2x, so x = 5. Check: 12 : 18 = 2 : 3. The answer is (c) 5.

Question 21

Two numbers are in the ratio 4 : 7. If 6 is added to each, the ratio becomes 5 : 8. The larger number is: (a) 24 (b) 35 (c) 48 (d) 42

Answer. I take the numbers as 4k and 7k. Then $\frac{4k + 6}{7k + 6} = \frac{5}{8}$, so 32k + 48 = 35k + 30. That gives 3k = 18 and k = 6. The numbers are 24 and 42, so the larger is 42. The answer is (d) 42. Check: 30 : 48 = 5 : 8.