Solved Questions

How to Use These Questions

The 35 questions here jump between types on purpose. A removal-and-replacement question can come right after one on mixing rice for profit, so first decide what kind it is: does one part stay fixed, is a mean price given, or is the same amount taken out and put back? Once you know the type, the method is usually one or two lines.

Work each one on paper against a clock before you look at the answer; 36 to 45 seconds is Tier 1 speed. If a question runs past a minute, find the quicker route in Shortcuts & Speed Methods and redo it the next day.

About a third of the questions are mixture basics: adding or removing one part, mixing two mixtures, transfers between containers, alloys and the average price of a mix. Another third use alligation: two prices, two strengths or two ratios, three varieties, averages that aren't mixtures, and mixing for profit. The rest, close to 30%, are removal and replacement, a type that comes up often in recent CGL papers.

A [Tier 2] question needs one more step than usual, such as a third variety or a second pour. Tier 1 papers ask simpler forms of the same ideas, so do them too.

Solved Examples

Question 1

Two kinds of rice, at ₹33 and ₹51 per kg, are mixed to make 84 kg of rice worth ₹39 per kg. How many kg of the ₹51 rice are in it?

Answer. I use the cross. Cheaper : dearer = (51 − 39) : (39 − 33) = 12 : 6 = 2 : 1. So the ₹51 rice is 1 part out of 3, which is $\frac{1}{3}$ of 84 = 28 kg. It makes sense, because ₹39 is much nearer ₹33, so most of the rice should be the cheap one.

Question 2

A tank holds 160 litres of pure milk. 32 litres are taken out and replaced with water, and this is done three times in all. How much milk is left in the tank?

Answer. Each time $\frac{32}{160} = \frac{1}{5}$ of whatever is there goes out, so $\frac{4}{5}$ of the milk stays. After three rounds: 160 × $\left(\frac{4}{5}\right)^3$ = 160 × $\frac{64}{125}$ = 81.92 litres. Don't do 160 − 96 = 64, because from the second round on, part of what comes out is water.

Question 3

In a mixture, milk and water are in the ratio 6 : 1. When 16 litres of water are added, the ratio becomes 6 : 5. How much milk is in the mixture?

Answer. Milk doesn't change and it's 6 in both ratios, so I can compare the water straight away. Water went from 1 unit to 5 units, so 4 units = 16 litres and 1 unit = 4 litres. Milk is 6 units = 24 litres.

Question 4

Two kinds of sugar, at ₹37 and ₹58 per kg, are mixed, and the mixture is sold at ₹56.10 per kg for a 10% profit. In what ratio were they mixed?

Answer. First I take out the profit: cost of the mix = 56.10 × $\frac{100}{110}$ = ₹51. Now the cross: (58 − 51) : (51 − 37) = 7 : 14 = 1 : 2 (cheaper : dearer). Check: 37 + 2 × 58 = 153, and 153 ÷ 3 = 51.

Question 5

A 60-litre mixture has milk and water in the ratio 4 : 1. 15 litres of the mixture are taken out and replaced with water, and this is done once more. What is the ratio of milk to water now?

Answer. Milk is 48 litres at the start, and only water goes back in, so the replacement formula works on the milk. Each round keeps $\frac{45}{60} = \frac{3}{4}$ of it: 48 × $\frac{9}{16}$ = 27 litres. The vessel still holds 60 litres, so water is 33 litres and milk : water = 9 : 11.

Question 6

28 litres of a mixture with alcohol and water in the ratio 5 : 2 are mixed with 12 litres of a mixture with alcohol and water in the ratio 1 : 5. What percentage of the new mixture is alcohol?

Answer. Alcohol from the first: 28 × $\frac{5}{7}$ = 20 litres. From the second: 12 × $\frac{1}{6}$ = 2 litres. That's 22 litres of alcohol in 40 litres, which is 55%.

Question 7

How many litres of 15% acid must be added to 60 litres of 35% acid to get 23% acid?

Answer. Cross 15 and 35 around 23: 15% acid : 35% acid = (35 − 23) : (23 − 15) = 12 : 8 = 3 : 2. The 60 litres are the 2 parts, so 1 part is 30 litres and I need 90 litres of the 15% acid. Check: 13.5 + 21 = 34.5 litres of acid in 150 litres is 23%.

Question 8

A jar holds 84 litres of milk and water in the ratio 9 : 5. 28 litres of the mixture are taken out and 7 litres of water are poured in. What is the ratio of milk to water now?

Answer. Taking out mixture doesn't change the ratio, so the 56 litres left are still 9 : 5, which is 36 litres of milk and 20 of water. With 7 more litres of water it becomes 36 : 27 = 4 : 3.

Question 9

A container has 75 litres of pure acid. A certain amount is drawn out and replaced with water, and this is done once more with the same amount. Now there are 48 litres of acid in it. How much was drawn out each time?

Answer. The acid left is $\frac{48}{75} = \frac{16}{25}$ of the start, which is $\left(\frac{4}{5}\right)^2$. So each round kept $\frac{4}{5}$ and took out $\frac{1}{5}$ of 75 = 15 litres. Check: 75 → 60 → 48.

Question 10

Milk costs ₹85 a litre. How much water must be added to 36 litres of milk so that selling the mixture at ₹81.60 a litre gives a profit of 20%?

Answer. Cost of the mixture = 81.60 ÷ 1.2 = ₹68 a litre. Water is a free variety at ₹0, so crossing ₹85 and ₹0 around ₹68 gives milk : water = 68 : 17 = 4 : 1. For 36 litres of milk that's 9 litres of water. Check: cost ₹3,060, sale 45 × 81.60 = ₹3,672, and the ₹612 profit is 20%.

Question 11

Vessel P has milk and water with 40% water, and vessel Q has milk and water in the ratio 5 : 4. 10 litres from P and 18 litres from Q are poured into an empty vessel R. Find the ratio of milk to water in R.

Answer. I follow the milk. From P: 60% of 10 = 6 litres. From Q: $\frac{5}{9}$ of 18 = 10 litres. That's 16 litres of milk in 28, so water is 12 litres and the ratio is 4 : 3.

Question 12

A mixture has milk and water in the ratio 11 : 4. What part of it must be drawn off and replaced with water so that the ratio becomes 11 : 7?

Answer. Milk only goes down here. It is $\frac{11}{15}$ of the mixture now and has to become $\frac{11}{18}$. The part kept is $\frac{11}{18} \div \frac{11}{15} = \frac{15}{18} = \frac{5}{6}$, so $\frac{1}{6}$ of the mixture has to be replaced. Check with 15 litres: taking out 2.5 litres leaves $\frac{55}{6}$ of milk and $\frac{20}{6}$ of water, and adding 2.5 litres of water makes it $\frac{35}{6}$, so milk : water = 55 : 35 = 11 : 7.

Question 13

Tea at ₹220 per kg is mixed with another tea in the ratio 3 : 2, and the mixture is worth ₹244 per kg. What is the price of the second tea?

Answer. The mix is ₹24 above ₹220. The second tea is $\frac{2}{5}$ of the mix, so the full gap between the two teas is 24 ÷ $\frac{2}{5}$ = ₹60. The second tea costs 220 + 60 = ₹280 per kg. Check: 3 × 220 + 2 × 280 = 1,220, and 1,220 ÷ 5 = 244.

Question 14

A cask holds three syrups A, B and C in the ratio 2 : 5 : 3. 40 litres of the mixture are drawn off, and then 8 litres of A and 4 litres of C are poured in. Now there is 28 litres more of B than of A. How much syrup was in the cask at first?

Answer. Before the 8 litres of A went in, B was 28 + 8 = 36 litres ahead of A (the C doesn't touch this gap). In the leftover mixture B − A = 5 − 2 = 3 units, so one unit is 12 litres. The leftover was 10 units = 120 litres, and the cask held 120 + 40 = 160 litres at first.

Question 15

From a 90-litre cask of pure apple juice, 9 litres are drawn out and replaced with water. This is done once more. Find the ratio of juice to water.

Answer. Each round keeps $\frac{9}{10}$ of the juice, so after two rounds juice is $\frac{81}{100}$ of the cask and water is $\frac{19}{100}$. Juice : water = 81 : 19.

Question 16

Wheat at ₹26, ₹30 and ₹38 per kg is mixed to get wheat worth ₹33 per kg. Which of these can be the mixing ratio: 1 : 2 : 3, 1 : 1 : 2 or 2 : 3 : 4?

Answer. I measure from ₹33: the three wheats are −7, −3 and +5. A ratio works only if the pluses and minuses cancel.

• 1 : 2 : 3 gives −7 − 6 + 15 = 2. No.
• 1 : 1 : 2 gives −7 − 3 + 10 = 0. Yes.
• 2 : 3 : 4 gives −14 − 9 + 20 = −3. No.

So 1 : 1 : 2 is the one. Check: 26 + 30 + 76 = 132, and 132 ÷ 4 = 33.

Question 17

A 48-litre mixture of milk and water contains 12.5% water. How much water must be added so that water makes up 30% of the new mixture?

Answer. The milk stays at 87.5% of 48 = 42 litres. In the new mixture milk is 70%, so the new total is 42 ÷ 0.7 = 60 litres, and the water added is 12 litres.

Question 18

A jar holds 30 litres of pure juice. 6 litres are taken out and replaced with water. Then 9 litres of the mixture are taken out and replaced with water. How much juice is left?

Answer. The two rounds take out different amounts, so I multiply each round's own "kept" fraction. The first keeps $\frac{24}{30}$ and the second keeps $\frac{21}{30}$. Juice = 30 × $\frac{4}{5}$ × $\frac{7}{10}$ = 16.8 litres.

Question 19

₹36,000 is lent in two parts, one at 8% and the other at 13% simple interest a year. The total interest after 2 years is ₹6,840. How much was lent at 8%?

Answer. The interest for one year is ₹3,420, which is 9.5% of ₹36,000. Crossing 8 and 13 around 9.5: 8% part : 13% part = 3.5 : 1.5 = 7 : 3. So the 8% part is $\frac{7}{10}$ of 36,000 = ₹25,200. Check: 2,016 + 1,404 = 3,420 a year.

Question 20

30 kg of an alloy with copper and tin in the ratio 7 : 3 is melted with 15 kg of another alloy with copper and tin in the ratio 2 : 1. Find the ratio of copper to tin in the new alloy.

Answer. Copper: 30 × $\frac{7}{10}$ + 15 × $\frac{2}{3}$ = 21 + 10 = 31 kg. Tin is the rest, 45 − 31 = 14 kg. Copper : tin = 31 : 14.

Question 21

A 140-litre mixture has milk and water in the ratio 3 : 4. How much of it must be taken out and replaced with pure milk to make the ratio 3 : 2?

Answer. Milk goes back in, so I track the water. It's $\frac{4}{7}$ now and has to become $\frac{2}{5}$. The part kept is $\frac{2}{5} \div \frac{4}{7} = \frac{7}{10}$, so $\frac{3}{10}$ of 140 = 42 litres must be replaced. Check: 60 and 80 become 42 and 56 after 42 litres come out, and adding 42 litres of milk gives 84 : 56 = 3 : 2.

Question 22

Milk and water are in the ratio 5 : 2 in vessel A and 2 : 3 in vessel B. In what ratio must A and B be mixed to get a mixture that is half milk and half water?

Answer. Milk is $\frac{5}{7}$ of A, $\frac{2}{5}$ of B and $\frac{1}{2}$ of the target. Over 70ths these are 50, 28 and 35. Cross: A gets 35 − 28 = 7 and B gets 50 − 35 = 15, so A : B = 7 : 15. Check: 7 litres of A have 5 litres of milk and 15 litres of B have 6, so 11 litres of milk in 22.

Question 23

A vessel holds milk and water in the ratio 9 : 2. 15 litres of milk are taken out of it and 3 litres of water are added, and the ratio becomes 3 : 1. How much mixture was in the vessel at first?

Answer. Say milk is 9x and water is 2x. After the change, (9x − 15) : (2x + 3) = 3 : 1, so 9x − 15 = 6x + 9. That gives 3x = 24 and x = 8, so the vessel held 11 × 8 = 88 litres. Check: 72 − 15 = 57 and 16 + 3 = 19, and 57 : 19 = 3 : 1.

Question 24

A dairy tank holds 128 litres of pure milk. Each time, 32 litres are drawn out and replaced with water. After how many such operations does the milk first fall below 50 litres?

Answer. Each round keeps $\frac{3}{4}$: 128 → 96 → 72 → 54 → 40.5. It's still 54 litres after three rounds, so it first drops below 50 after the 4th operation. Multiplying by $\frac{3}{4}$ step by step is quicker than any formula here.

Question 25

A bus covers 312 km in 6 hours, partly at 40 km/h and partly at 60 km/h. How many km does it travel at 60 km/h?

Answer. Average speed = 312 ÷ 6 = 52 km/h. Speed is per hour, so the cross gives a ratio of times: time at 40 : time at 60 = (60 − 52) : (52 − 40) = 8 : 12 = 2 : 3. That's 2.4 hours and 3.6 hours, so the bus goes 3.6 × 60 = 216 km at 60 km/h (and 96 km at 40 km/h).

Question 26

A grocer mixes 18 kg of rice at ₹62 per kg, 12 kg at ₹68 per kg and 30 kg at ₹74 per kg. At what price per kg must he sell the mix to gain 25%?

Answer. Cost of the mix with ₹68 as the base: 18 × (−6) + 30 × (+6) = −108 + 180 = 72. Over 60 kg that's +1.20, so the mix costs ₹69.20 a kg. For a 25% gain he sells at 69.20 × 1.25 = ₹86.50 per kg.

Question 27 [Tier 2]

A 30-litre mixture has milk and water in the ratio 2 : 1. 6 litres of it are taken out and replaced with 6 litres of another mixture in which milk and water are in the ratio 1 : 2. Find the new ratio of milk to water.

Answer. Taking out 6 of the 30 litres removes $\frac{1}{5}$ of each part, so 20 litres of milk and 10 of water become 16 and 8. The 6 litres poured in bring 2 of milk and 4 of water. That's 18 : 12 = 3 : 2. The power formula doesn't work here, because what goes back in isn't pure water.

Question 28

Two kinds of dal, at ₹64 and ₹80 per kg, are mixed, and the mix is sold at ₹66.50 per kg at a loss of 5%. In what ratio were they mixed?

Answer. For a loss, cost = selling price × $\frac{100}{95}$ = 66.50 ÷ 0.95 = ₹70. Cross: (80 − 70) : (70 − 64) = 10 : 6 = 5 : 3 (cheaper : dearer). Check: 320 + 240 = 560, and 560 ÷ 8 = 70.

Question 29

45 kg of an alloy has copper and zinc in the ratio 7 : 2. How much zinc must be melted in to make the ratio 5 : 3?

Answer. Copper stays at 35 kg. For 5 : 3, zinc has to be $\frac{3}{5}$ of 35 = 21 kg. It's 10 kg now, so 11 kg of zinc must be added.

Question 30 [Tier 2]

A vessel is full of pure milk. 8 litres are drawn out and replaced with water, and this is done once more. Now milk and water are in the ratio 16 : 9. What is the capacity of the vessel?

Answer. Milk is $\frac{16}{25}$ of the vessel, which is $\left(\frac{4}{5}\right)^2$. So each round kept $\frac{4}{5}$ and took out $\frac{1}{5}$ of the vessel. That $\frac{1}{5}$ is 8 litres, so the vessel holds 40 litres. Check: milk goes 40 → 32 → 25.6, and 25.6 : 14.4 = 16 : 9.

Question 31

How many kg of tea at ₹320 per kg must be added to 30 kg of tea at ₹260 per kg to get a mix worth ₹284 per kg?

Answer. Cross: ₹260 tea : ₹320 tea = (320 − 284) : (284 − 260) = 36 : 24 = 3 : 2. The 30 kg are the 3 parts, so 1 part is 10 kg and 20 kg of the ₹320 tea are needed.

Question 32

A milkman buys 45 litres of milk at ₹48 a litre, adds 9 litres of water, and sells the whole mixture at ₹46 a litre. What is his profit percentage?

Answer. Cost = 45 × 48 = ₹2,160. He sells 54 litres at ₹46, which is ₹2,484. Profit = ₹324, and 324 ÷ 2,160 = 15%. He sells below his buying price per litre and still gains, because the water is free.

Question 33

A radiator holds 16 litres of coolant that is 40% antifreeze. How much of it must be drained and replaced with pure antifreeze to make it 60% antifreeze?

Answer. Pure antifreeze goes back in, so I track the water. Water is 60% now and has to become 40%, so the part kept is $\frac{40}{60} = \frac{2}{3}$ and the part replaced is $\frac{1}{3}$ of 16 = $5\frac{1}{3}$ litres. Check: water goes from 9.6 litres to 6.4 litres, which is 40% of 16.

Question 34 [Tier 2]

Three teas are mixed in the ratio 2 : 3 : 5. The first two cost ₹150 and ₹170 per kg, and the mixture is worth ₹200 per kg. What does the third tea cost per kg?

Answer. Think of 10 kg of mix worth ₹2,000. The first two teas give 2 × 150 + 3 × 170 = ₹810, so the 5 kg of the third tea are worth ₹1,190, which is ₹238 per kg.

Question 35 [Tier 2]

Jug A has 20 litres of pure milk and jug B has 20 litres of water. 5 litres are poured from A into B and stirred, then 5 litres are poured from B back into A, and finally 5 litres are poured from A into B again. How much milk is in B at the end?

Answer. I keep a small table.

• After A to B: A has 15 litres of milk, and B has 5 of milk and 20 of water.
• 5 litres of B is $\frac{1}{5}$ of it, so 1 of milk and 4 of water go back. A has 16 milk and 4 water, and B has 4 milk and 16 water.
• 5 litres of A is $\frac{1}{4}$ of it, so 4 of milk and 1 of water go to B.

B ends with 4 + 4 = 8 litres of milk (and 17 of water).