How to Use These Questions

There are 36 questions here from all four topics, in no fixed order. A coin question can come right after a partnership question, so first decide the type: a split in a given ratio, a ratio that changes, or capital × time. Once you know the type, the method is usually a single line.

Work each one on paper with a timer before reading the answer, aiming for 36 to 45 seconds. If a question takes more than a minute, find the step that slowed you down (often the LCM when joining ratios, or the months in a partnership) and redo it the next day.

About a fifth of the questions are ratio basics: combining ratios, dividing amounts and finding numbers. Another fifth are on proportion, the three proportionals, expressions and variation. About 30% are applications: income and savings, coins and notes, changing ratios, ratios with percentages and ages. The last 30% are partnership questions, since recent CGL papers have had more of them than earlier years did.

Questions marked [Tier 2] need one more step than a usual Tier 1 question, such as a capital that changes several times in the year, or a cost that is partly fixed. Tier 1 students should try them too, since the setup is the same.

Solved Examples

Question 1

A : B = 2 : 3, B : C = 5 : 4 and C : D = 6 : 7. ₹10,200 is divided among A, B, C and D in this ratio. How much does D get?

Answer. I use the multiply-down chain. A = 2 × 5 × 6 = 60, B = 3 × 5 × 6 = 90, C = 3 × 4 × 6 = 72 and D = 3 × 4 × 7 = 84. Dividing by 6, that's 10 : 15 : 12 : 14, which adds up to 51. So 1 unit = 10,200 ÷ 51 = 200, and D gets 14 × 200 = ₹2,800.

Question 2

A invests ₹36,000 for 8 months, B invests ₹48,000 for 9 months and C invests ₹54,000 for 4 months. B's share of the profit is ₹10,800 more than C's. What is the total profit?

Answer. In thousands: A = 36 × 8 = 288, B = 48 × 9 = 432 and C = 54 × 4 = 216. All three divide by 72, so A : B : C = 4 : 6 : 3. B is 3 units ahead of C, so 3 units = 10,800 and 1 unit = 3,600. The total is 13 × 3,600 = ₹46,800.

Question 3

Men and women in a town are in the ratio 9 : 8. Over a few years the number of men grows by 10% and the number of women by 15%, and the population becomes 38,200. How many women were there at first?

Answer. New ratio = 9 × 1.1 : 8 × 1.15 = 9.9 : 9.2 = 99 : 92. That's 191 units for 38,200 people, so 1 unit = 200 and there are now 92 × 200 = 18,400 women. Before the 15% growth there were 18,400 ÷ 1.15 = 16,000 women. Check: 18,000 men and 16,000 women is 9 : 8, and they grow to 19,800 + 18,400 = 38,200.

Question 4

x is the mean proportional between 18 and 32, y is the third proportional to x and 36, and z is the fourth proportional to x, y and 8. Find z.

Answer. x = 18×32=576\sqrt{18 \times 32} = \sqrt{576} = 24. y is the third proportional to 24 and 36, so y = 36224=129624\frac{36^2}{24} = \frac{1296}{24} = 54. z is the fourth proportional to 24, 54 and 8, so z = 54×824\frac{54 \times 8}{24} = 18. I write each value down before moving on, so I don't mix up the order.

Question 5

A bag has ₹1, 50-paise and 25-paise coins in the ratio 5 : 6 : 8 by number. The total value is ₹210. How many 25-paise coins are there?

Answer. Value ratio = 5 × 1 : 6 × 0.5 : 8 × 0.25 = 5 : 3 : 2, which is 10 units. So 1 unit = ₹21, and the 25-paise coins are worth 2 × 21 = ₹42. Four of them make a rupee, so that's 42 × 4 = 168 coins. Check: 105, 126 and 168 coins are in the ratio 5 : 6 : 8.

Question 6

X starts a business with ₹80,000. Y joins 3 months later with ₹1,20,000. X leaves 2 months before the end of the year. The profit for the year is ₹94,000. Find Y's share.

Answer. X is in for 12 − 2 = 10 months and Y for 12 − 3 = 9 months. In thousands, X = 80 × 10 = 800 and Y = 120 × 9 = 1,080. So X : Y = 800 : 1,080 = 20 : 27, which is 47 units. 1 unit = 94,000 ÷ 47 = 2,000, so Y gets 27 × 2,000 = ₹54,000.

Question 7

₹8,400 is divided among A, B and C so that A gets 23\frac{2}{3} of what B and C get together, and B gets 14\frac{1}{4} of what A and C get together. Find C's share.

Answer. A gets 2 parts for every 3 parts that B and C get, so A has 2 parts out of 5 in the whole. A = 25\frac{2}{5} × 8,400 = 3,360. In the same way B has 1 part out of 5, so B = 1,680. C gets the rest: 8,400 − 3,360 − 1,680 = ₹3,360.

Question 8

The present ages of a man and his nephew are in the ratio 5 : 3. Ten years ago, they were in the ratio 5 : 2. Find their present ages.

Answer. The age gap never changes. Now the gap is 2 units and earlier it was 3 units, so I make both gaps 6: now 15 : 9, then 10 : 4. The man went from 10 units to 15 units in 10 years, so 5 units = 10 years and 1 unit = 2. They are 30 and 18 now. Check: ten years ago they were 20 and 8, and 20 : 8 = 5 : 2.

Question 9

If 5a−3b3a+2b=913\frac{5a - 3b}{3a + 2b} = \frac{9}{13}, find a : b and the value of a2+b2a2−b2\frac{a^2 + b^2}{a^2 - b^2}.

Answer. Cross-multiplying: 13(5a − 3b) = 9(3a + 2b), so 65a − 39b = 27a + 18b. That gives 38a = 57b, and dividing both sides by 19, 2a = 3b. So a : b = 3 : 2. Now I just put in a = 3 and b = 2: 9+49−4=135\frac{9 + 4}{9 - 4} = \mathbf{\frac{13}{5}}.

Question 10

M and N invest ₹1,40,000 and ₹60,000. Each gets 10% interest a year on capital, and the rest of the profit is shared equally. The year's profit is ₹46,000. How much does each get?

Answer. Interest comes out first: M gets 14,000 and N gets 6,000. That uses up 20,000 and leaves 26,000, which is 13,000 each. So M gets 14,000 + 13,000 = ₹27,000 and N gets 6,000 + 13,000 = ₹19,000. If you split the whole ₹46,000 in 7 : 3 by capital, M would get ₹32,200, but that skips the interest step.

Question 11

Men and women in a club are in the ratio 9 : 5. Then 8 men leave and 20 women join, and the ratio becomes 5 : 4. How many members did the club have at first?

Answer. I take them as 9k and 5k. Then (9k − 8) : (5k + 20) = 5 : 4, so 4(9k − 8) = 5(5k + 20). That's 36k − 32 = 25k + 100, so 11k = 132 and k = 12. There were 108 men and 60 women, 168 in all. Check: 100 : 80 = 5 : 4.

Question 12

Find the compound ratio of the duplicate ratio of 2 : 3, the sub-duplicate ratio of 81 : 64 and the inverse ratio of 5 : 6.

Answer. Duplicate of 2 : 3 means square both: 4 : 9. Sub-duplicate of 81 : 64 means square root of both: 9 : 8. Inverse of 5 : 6 is 6 : 5. Compound means multiply them all: (4 × 9 × 6) : (9 × 8 × 5) = 216 : 360 = 3 : 5.

Question 13

A starts a business with ₹36,000. Some months later B joins with ₹54,000. At the end of the year the profit is shared in the ratio 8 : 5. After how many months did B join?

Answer. A's capital-months are 36 × 12 = 432 (in thousands). B's are 58\frac{5}{8} of A's: 432 × 58\frac{5}{8} = 270. With 54, that's 270 ÷ 54 = 5 months in the business. So B joined after 7 months. Saying 5 is the usual slip: that's how long he stayed, not when he came in.

Question 14

The incomes of A and B are in the ratio 4 : 5 and their expenditures in the ratio 3 : 4. Each of them saves ₹7,500. What is B's expenditure?

Answer. Cross-multiplying: ad − bc = 16 − 15 = 1 and d − c = 4 − 3 = 1. So one unit of income is 7,500 × 1 ÷ 1 = ₹7,500. B's income is 5 × 7,500 = 37,500, and B spends 37,500 − 7,500 = ₹30,000. Check: A earns 30,000 and spends 22,500, and 22,500 : 30,000 = 3 : 4.

Question 15

The weight of a metal rod varies jointly as its length and the square of its radius. A rod 2 m long with a radius of 3 cm weighs 18 kg. How much does a rod 3 m long with a radius of 2 cm weigh?

Answer. The weight goes with length × r2r^2. So the new weight = 18 × 32×(23)2\frac{3}{2} \times \left(\frac{2}{3}\right)^2 = 18 × 32×49\frac{3}{2} \times \frac{4}{9} = 12 kg. The new rod is longer but thinner, and the thinness wins because the radius gets squared.

Question 16

A, B and C start a business with ₹1,25,000, ₹1,75,000 and ₹2,00,000 for the same period. The profit is ₹1,10,000. Find C's share.

Answer. Same time for all, so only the capitals matter: 125 : 175 : 200 = 5 : 7 : 8. That's 20 units, so 1 unit = 5,500 and C gets 8 × 5,500 = ₹44,000.

Question 17

₹19,600 is divided among A, B and C so that 34\frac{3}{4} of A's share = 23\frac{2}{3} of B's share = 45\frac{4}{5} of C's share. Find B's share.

Answer. Each share is the same amount divided by its fraction, so A : B : C = 43:32:54\frac{4}{3} : \frac{3}{2} : \frac{5}{4}. Multiplying by 12 gives 16 : 18 : 15, which is 49 units. 1 unit = 19,600 ÷ 49 = 400, so B gets 18 × 400 = ₹7,200. Check: 34\frac{3}{4} of 6,400, 23\frac{2}{3} of 7,200 and 45\frac{4}{5} of 6,000 all come to 4,800.

Question 18

Seats for Hindi, English and Sanskrit in a college are in the ratio 3 : 4 : 7. They are increased by 40%, 75% and 50% respectively. Find the new ratio and the percentage rise in the total number of seats.

Answer. I multiply each term by its own multiplier: 3 × 1.4 : 4 × 1.75 : 7 × 1.5 = 4.2 : 7 : 10.5. Multiplying by 10 to clear the decimals gives 42 : 70 : 105, which is 6 : 10 : 15. The total went from 14 units to 4.2 + 7 + 10.5 = 21.7 units, a rise of 7.7 on 14, which is 55%.

Question 19 [Tier 2]

A, B and C rent a pasture for ₹8,100. A grazes 10 horses for 4 months, B grazes 24 cows for 3 months and C grazes 30 sheep for 5 months. 2 horses eat as much as 3 cows, and 5 sheep eat as much as 1 cow. How much does C pay?

Answer. I change everything into cows. A's 10 horses are 15 cows, so A has 15 × 4 = 60 cow-months. B has 24 × 3 = 72. C's 30 sheep are 6 cows, so C has 6 × 5 = 30. The ratio is 60 : 72 : 30 = 10 : 12 : 5, which is 27 units. 1 unit = 8,100 ÷ 27 = 300, so C pays ₹1,500.

Question 20

What number must be added to each of 3, 11, 9 and 23 so that the results are in proportion?

Answer. Using x = (bc − ad) ÷ ((a + d) − (b + c)): x = 11×9−3×2326−20=99−696\frac{11 \times 9 - 3 \times 23}{26 - 20} = \frac{99 - 69}{6} = 5. Check: the numbers become 8, 16, 14 and 28, and 8 : 16 = 14 : 28 = 1 : 2.

Question 21

The present ages of A, B and C are in the ratio 3 : 5 : 7. In 5 years, the sum of their ages will be 120. What was the ratio of their ages 7 years ago?

Answer. In 5 years each of them gets 5 years older, so the sum goes up by 15. The present sum is 120 − 15 = 105, which is 15 units, so 1 unit = 7. Their ages are 21, 35 and 49. Seven years ago they were 14, 28 and 42, which is 1 : 2 : 3.

Question 22

A and B invest ₹42,000 and ₹56,000. From the profit, 15% is given to charity and the rest is shared in the ratio of their capitals. A gets ₹10,200. What was the total profit?

Answer. The capitals are 42 : 56 = 3 : 4, so A gets 37\frac{3}{7} of what is left after charity. That remainder is 10,200 × 73\frac{7}{3} = 23,800. It is 85% of the profit, so the profit is 23,800 ÷ 0.85 = ₹28,000. Check: charity takes ₹4,200, and 23,800 splits into 10,200 and 13,600.

Question 23

Which of these ratios is the largest: 3 : 7, 5 : 11, 7 : 15, 4 : 9?

Answer. I turn each into a decimal: 3 ÷ 7 is about 0.429, 5 ÷ 11 about 0.455, 7 ÷ 15 about 0.467 and 4 ÷ 9 about 0.444. The largest is 7 : 15. A quicker way here: every ratio is a little under a half. 7 : 15 falls short of a half by only 130\frac{1}{30}, while 3 : 7 is short by 114\frac{1}{14}, 5 : 11 by 122\frac{1}{22} and 4 : 9 by 118\frac{1}{18}. The smallest shortfall wins.

Question 24 [Tier 2]

The salaries of A and B are in the ratio 4 : 5. A gets a 25% raise and B gets a raise of ₹4,000, and the ratio becomes 10 : 11. Find B's salary before the raise.

Answer. I take the salaries as 4k and 5k. After the raises A has 4k × 1.25 = 5k and B has 5k + 4,000. So 5k : (5k + 4,000) = 10 : 11, which gives 55k = 50k + 40,000, so k = 8,000. B's old salary = 5 × 8,000 = ₹40,000. Check: A goes from 32,000 to 40,000, B from 40,000 to 44,000, and 40,000 : 44,000 = 10 : 11.

Question 25

A starts a business with ₹48,000. B joins 4 months later. Out of a profit of ₹72,000 at the end of the year, B gets ₹40,000. How much did B invest?

Answer. A gets 72,000 − 40,000 = 32,000, so A : B = 32 : 40 = 4 : 5. A's capital-months are 48 × 12 = 576 (in thousands), so B's are 576 × 54\frac{5}{4} = 720. B was in for 8 months, so B put in 720 ÷ 8 = 90 thousand, that is ₹90,000.

Question 26 [Tier 2]

Solve x+6+x−6x+6−x−6=3\frac{\sqrt{x + 6} + \sqrt{x - 6}}{\sqrt{x + 6} - \sqrt{x - 6}} = 3.

Answer. The top and bottom are the sum and difference of the same two roots, so componendo and dividendo works: x+6x−6=3+13−1\frac{\sqrt{x + 6}}{\sqrt{x - 6}} = \frac{3 + 1}{3 - 1} = 2. Squaring both sides, x + 6 = 4(x − 6), so 3x = 30 and x = 10. Check: 16\sqrt{16} = 4 and 4\sqrt{4} = 2, and 4+24−2\frac{4 + 2}{4 - 2} = 3.

Question 27

A cash box has ₹10, ₹20 and ₹50 notes. The values of the three kinds of notes are in the ratio 3 : 4 : 5, and there are 180 notes in all. What is the total amount?

Answer. To get the number ratio, I divide each value by its note: 310:420:550\frac{3}{10} : \frac{4}{20} : \frac{5}{50} = 0.3 : 0.2 : 0.1 = 3 : 2 : 1. So the 180 notes are 90, 60 and 30. The amount is 900 + 1,200 + 1,500 = ₹3,600. Check: 900 : 1,200 : 1,500 = 3 : 4 : 5.

Question 28 [Tier 2]

A, B and C start a business. A puts in ₹60,000 for the whole year. B puts in ₹20,000 at the start and adds ₹20,000 more at the end of every 3 months. C joins after 4 months with ₹75,000 and takes out ₹25,000 at the end of the 8th month. The profit is ₹1,82,000. Find each share.

Answer. In thousands, partner by partner:

  • A: 60 × 12 = 720.
  • B: 20 for months 0 to 3, 40 for months 3 to 6, 60 for months 6 to 9 and 80 for months 9 to 12. That's (20 + 40 + 60 + 80) × 3 = 600.
  • C: 75 for months 4 to 8 and 50 for months 8 to 12, so 300 + 200 = 500.

A : B : C = 720 : 600 : 500 = 36 : 30 : 25, which is 91 units. 1 unit = 1,82,000 ÷ 91 = 2,000. So A gets ₹72,000, B gets ₹60,000 and C gets ₹50,000.

Question 29

Three numbers are in the ratio 3 : 4 : 6, and the sum of their squares is 1,525. Find the largest number.

Answer. I take them as 3k, 4k and 6k. Their squares add up to 9k2+16k2+36k2=61k29k^2 + 16k^2 + 36k^2 = 61k^2, so 61k261k^2 = 1,525 and k2k^2 = 25. That makes k = 5, and the largest number is 6 × 5 = 30. The three numbers are 15, 20 and 30.

Question 30

A box has red and green balls in the ratio 9 : 7. After 48 red balls are taken out, the ratio becomes 3 : 5. How many green balls are there?

Answer. The green balls don't change, so I make the green terms equal. The LCM of 7 and 5 is 35: 9 : 7 becomes 45 : 35 and 3 : 5 becomes 21 : 35. Red went from 45 units to 21 units, so 24 units = 48 balls and 1 unit = 2. Green = 35 × 2 = 70. Check: red goes from 90 to 42, and 42 : 70 = 3 : 5.

Question 31

If (x + 2), (x + 5) and (x + 11) are in continued proportion, find x.

Answer. Continued proportion means the middle one squared equals the product of the other two: (x+5)2=(x+2)(x+11)(x + 5)^2 = (x + 2)(x + 11). Expanding, x2+10x+25=x2+13x+22x^2 + 10x + 25 = x^2 + 13x + 22, so 3x = 3 and x = 1. Check: the numbers are 3, 6 and 12, and 3 : 6 = 6 : 12.

Question 32 [Tier 2]

A and B invest ₹1,80,000 and ₹2,70,000. A manages the business and gets ₹3,500 a month for it. The rest of the profit is shared in the ratio of capitals. A's total for the year is ₹74,000. What was the total profit?

Answer. A's salary for the year is 3,500 × 12 = 42,000. So his share of the rest is 74,000 − 42,000 = 32,000. The capitals are 180 : 270 = 2 : 3, so A gets 25\frac{2}{5} of the rest. That means the rest is 32,000 × 52\frac{5}{2} = 80,000, and the total profit is 80,000 + 42,000 = ₹1,22,000.

Question 33

A father is now 3 times as old as his son. Eight years ago, he was 5 times as old as his son. How old is the father now?

Answer. Now it's 3 : 1, a gap of 2 units, and then it was 5 : 1, a gap of 4 units. I make the gaps equal by writing now as 6 : 2. The son went from 2 units to 1 unit in 8 years, so 1 unit = 8 years. The father is 6 × 8 = 48 and the son is 16. Check: eight years ago they were 40 and 8, and 40 is 5 times 8.

Question 34 [Tier 2]

The monthly cost of running a hostel is partly fixed and partly varies with the number of students. It is ₹2,64,000 with 120 students and ₹3,15,000 with 150 students. What will it be with 180 students?

Answer. The extra 30 students add 3,15,000 − 2,64,000 = 51,000, so each student costs ₹1,700. The fixed part is 2,64,000 − 120 × 1,700 = 2,64,000 − 2,04,000 = ₹60,000. With 180 students the cost is 60,000 + 180 × 1,700 = 60,000 + 3,06,000 = ₹3,66,000. Even quicker: going from 150 to 180 is another 30 students, so I just add another ₹51,000 to ₹3,15,000.

Question 35 [Tier 2]

A puts in 13\frac{1}{3} of the capital for 14\frac{1}{4} of the time, B puts in 14\frac{1}{4} of the capital for 12\frac{1}{2} of the time, and C puts in the rest of the capital for the whole time. The business makes a loss of ₹22,500. How much of the loss does C bear?

Answer. C's part of the capital is 1 − 13\frac{1}{3} − 14\frac{1}{4} = 512\frac{5}{12}. Capital × time: A = 112\frac{1}{12}, B = 18\frac{1}{8} and C = 512\frac{5}{12}. Multiplying by 24 gives 2 : 3 : 10, which is 15 units. A loss is shared the same way as a profit, so 1 unit = 22,500 ÷ 15 = 1,500 and C bears 10 × 1,500 = ₹15,000.

Question 36

Two numbers are in the ratio 5 : 8. When 6 is subtracted from each, the ratio becomes 2 : 5. Find the numbers.

Answer. Taking the same number from both keeps the gap fixed. Both ratios have a gap of 3 units, so they already match. The first number went from 5 units to 2 units, so 3 units = 6 and 1 unit = 2. The numbers are 10 and 16. Check: 4 : 10 = 2 : 5.