Solved Questions

How to Use These Questions

These 36 questions jump between the three topics with no warning, so a pipes question can come right after one on men and women. First decide what kind it is: a rate question, a men-days question or a tank. After that the working in units is usually two or three lines.

Do each one on paper before you read the answer, at about 36 to 45 seconds a question. If one takes you over a minute, find the matching method in Shortcuts & Speed Methods and redo the question the next day.

Roughly a third are on rates: efficiency, three people, people joining or leaving partway, and work given in fractions. Another third are men-days, mixed groups of men, women and boys, alternate days, negative work and wages. The rest are pipes, from outlets and leaks to pipes in turns, part-full tanks and capacity in litres.

The ones marked [Tier 2] add one more stage, such as three people leaving at different times or pipes closed one after another. A Tier 1 student should still try them, since the set-up is the same, only longer.

Solved Examples

Question 1

Pipes A and B can fill a tank in 15 minutes and 18 minutes, and pipe C can empty it in 45 minutes. If all three are opened together on an empty tank, how long will it take to fill?

Answer. I take the tank as 90 units. A puts in 6 a minute, B puts in 5, and C takes out 2. So the tank gains 9 units a minute and fills in 90 ÷ 9 = 10 minutes.

Question 2

A can do $\frac{3}{5}$ of a job in 12 days, and B can do $\frac{2}{3}$ of it in 20 days. How long will they take to do the whole job together?

Answer. First the full times. A does $\frac{3}{5}$ in 12 days, so the whole job takes him 12 × $\frac{5}{3}$ = 20 days. B does $\frac{2}{3}$ in 20 days, so he needs 20 × $\frac{3}{2}$ = 30 days. Together: $\frac{20 \times 30}{50}$ = 12 days.

Question 3

8 carpenters working 9 hours a day make 360 chairs in 15 days. How many carpenters working 6 hours a day are needed to make 480 chairs in 12 days?

Answer. The first group puts in 8 × 9 × 15 = 1,080 man-hours for 360 chairs, so one chair takes 3 man-hours. 480 chairs need 1,440 man-hours. Each carpenter now gives 6 × 12 = 72 hours, so we need 1,440 ÷ 72 = 20 carpenters.

Question 4

A takes 12 days more than B to finish a job. Working together, they finish it in 8 days. How long does A take alone?

Answer. If B takes x days, A takes x + 12, and $\frac{x(x + 12)}{2x + 12} = 8$. That gives $x^2 + 12x = 16x + 96$, so $x^2 - 4x - 96 = 0$, which is $(x - 12)(x + 8) = 0$. So B takes 12 days and A takes 24 days. Check: $\frac{24 \times 12}{36}$ = 8. In the exam, trying the options in the product-over-sum check is even quicker.

Question 5

A pipe can fill a tank in 6 hours. Because of a leak, it takes 7 hours 30 minutes. How long would the leak alone take to empty the full tank?

Answer. 7 hours 30 minutes is 7.5 hours. Leak formula: $\frac{6 \times 7.5}{7.5 - 6} = \frac{45}{1.5}$ = 30 hours. Check: $\frac{1}{6} - \frac{1}{30} = \frac{4}{30} = \frac{2}{15}$, and $\frac{15}{2}$ is 7.5 hours.

Question 6

A can do a job in 9 days and B in 15 days. They work on alternate days, and B works on the first day. In how many days will the job be done?

Answer. I take 45 units: A does 5 a day and B does 3, so every 2 days they do 8. 5 rounds (10 days) do 40 units, and 5 are left. Day 11 is B's turn: he does 3, so 2 are left. Day 12 is A's, and he needs only 2 of his 5 units, which is $\frac{2}{5}$ of a day. So the job takes $11\frac{2}{5}$ days.

Question 7

A is 50% more efficient than B, and B is 20% more efficient than C. C alone can finish a job in 36 days. How long will A, B and C take together?

Answer. B is 20% better than C, so B takes 36 × $\frac{100}{120}$ = 30 days. A is 50% better than B, so A takes 30 × $\frac{100}{150}$ = 20 days. With 180 units, A does 9 a day, B 6 and C 5, which is 20 a day. Together they take 9 days.

Question 8

Pipe A can fill a tank in 21 hours and pipe B in 42 hours. A is opened at 6 a.m. and B at 9 a.m. At what time will the tank be full?

Answer. Tank = 42 units: A puts in 2 an hour and B puts in 1. From 6 to 9 a.m. A alone puts in 6 units, so 36 are left. Together they put in 3 an hour, so that takes 12 hours. 9 a.m. + 12 hours = 9 p.m.

Question 9

3 men and 2 boys can do a job in 18 days, while 2 men and 5 boys can do it in 16 days. How long will 4 men and 4 boys take?

Answer. The job is the same, so 18(3m + 2b) = 16(2m + 5b). That is 54m + 36b = 32m + 80b, so 22m = 44b and 1 man = 2 boys. The job is 18 × (6 + 2) = 144 boy-days. 4 men and 4 boys are worth 8 + 4 = 12 boys, so they take 144 ÷ 12 = 12 days.

Question 10 [Tier 2]

A, B and C can do a job in 15, 25 and 75 days. They start together, but A leaves after 3 days, and B leaves 4 days before the job is finished. How many days does the job take in all?

Answer. I take 75 units: A does 5 a day, B 3 and C 1. In the first 3 days all three work: 27 units. In the last 4 days only C works: 4 units. That leaves 75 − 27 − 4 = 44 units for B and C in the middle, at 4 a day, which is 11 days. In all, 3 + 11 + 4 = 18 days.

Question 11

A tank is one-third full. Pipe A can fill the whole tank in 9 hours and pipe B can empty it in 18 hours. They are opened alternately for an hour each, starting with A. After how many hours will the tank be full?

Answer. Tank = 18 units, and it already has 6. A puts in 2 an hour and B takes out 1, so each 2-hour round adds 1. The tank fills in an A hour once 6 + k + 2 reaches 18, so after k = 10 rounds (20 hours) it has 16 units, and in the 21st hour A adds the last 2. 21 hours. Doing 12 ÷ 1 × 2 = 24 hours misses that the tank fills before B's turn.

Question 12

A, B and C can do a job in 9, 18 and 36 days. All three start together, but A leaves after 3 days, and B and C finish the job. They are paid ₹4,500 for it. What is C's share?

Answer. I take 36 units: A does 4 a day, B 2 and C 1. In 3 days all three do 21 units. B and C do the other 15 in 15 ÷ 3 = 5 days, so the job lasts 8 days. A did 12 units, B did 16 and C did 8. C's share is $\frac{8}{36}$ of 4,500 = ₹1,000.

Question 13

A and B together can do a job in 20 days. A alone can do it in 36 days. How long will B take alone?

Answer. B = $\frac{36 \times 20}{36 - 20} = \frac{720}{16}$ = 45 days. Check with 180 units: the pair does 9 a day and A does 5, so B does 4 and takes 45 days.

Question 14

Pipe A fills a tank three times as fast as pipe B, and B fills it twice as fast as pipe C. All three together fill it in 12 minutes. How long would C take alone?

Answer. If C puts in 1 unit a minute, B puts in 2 and A puts in 6, so together 9 a minute. The tank is 9 × 12 = 108 units, and C alone takes 108 minutes.

Question 15

60 workers can build a road in 45 days. After 15 days, 20 workers leave. In how many days in all will the road be finished?

Answer. The road needs 60 × 45 = 2,700 worker-days. In 15 days, 900 are done, so 1,800 are left. 40 workers take 1,800 ÷ 40 = 45 more days. In all, 15 + 45 = 60 days.

Question 16

A and B can do a job in 12 and 30 days. They start together, but after some days A leaves, and B finishes the rest in 9 days. After how many days did A leave?

Answer. I take 60 units: A does 5 a day and B 2. B's last 9 days give 18 units, so the two of them together did 42 units at 7 a day. That is 6 days, so A left after 6 days.

Question 17

A pipe fills a tank in 6 hours. If it let in 10 litres a minute more, it would fill the tank in 5 hours. What is the capacity of the tank?

Answer. I take the tank as 30 units. The pipe gives 5 units an hour, and the faster one would give 6. So 10 litres a minute, which is 600 litres an hour, is worth 1 unit. The tank holds 30 × 600 = 18,000 litres.

Question 18

A can build a wall in 18 days, and B can knock it down in 27 days. A works alone for 9 days, and then B also starts working against him. How many days does the wall take in all?

Answer. I take 54 units. A's 9 days give 27 units. After that A adds 3 a day and B removes 2, so the wall grows by 1 a day, and the other 27 units take 27 days. In all, 9 + 27 = 36 days.

Question 19

A does half as much work as B in three-fourths of the time. Together they finish a job in 18 days. How long would B take alone?

Answer. In the same time, A does $\frac{1}{2} \div \frac{3}{4} = \frac{2}{3}$ of B's work, so A : B = 2 : 3. The job is 5 × 18 = 90 units, and B does 3 a day, so B alone takes 30 days.

Question 20

Pipes A and B can fill a tank in 32 and 48 minutes. Both are opened together, and after 9 minutes B is closed. How much longer will A take to fill the tank?

Answer. Tank = 96 units: A puts in 3 a minute and B 2. In 9 minutes they put in 45. A fills the other 51 in 17 minutes.

Question 21 [Tier 2]

A can finish a job in 20 days. He works at full speed for 8 days, and after that his efficiency falls by 25%. In how many days in all is the job finished?

Answer. In 8 days A does $\frac{8}{20} = \frac{2}{5}$ of the job, so $\frac{3}{5}$ is left. At 75% speed he does $\frac{3}{4} \times \frac{1}{20} = \frac{3}{80}$ of the job a day, so the rest takes $\frac{3}{5} \div \frac{3}{80}$ = 16 days. In all, 8 + 16 = 24 days.

Question 22 [Tier 2]

Pipes A, B and C can fill a tank in 12, 15 and 60 hours. All three are opened together. After 2 hours C is closed, and after 1 more hour B is closed. How long does the tank take to fill?

Answer. Tank = 60 units: A puts in 5 an hour, B 4 and C 1. 2 hours of all three give 20 units. In the next hour A and B give 9, so 29 are in. A fills the other 31 in 31 ÷ 5 = 6.2 hours. In all, 3 + 6.2 = 9.2 hours, which is 9 hours 12 minutes.

Question 23

Ramesh can weed a 3-acre field in 9 days, and Suresh can weed a 2-acre field in 4 days. Working together, how long will they take to weed a 10-acre field?

Answer. Ramesh weeds $\frac{3}{9} = \frac{1}{3}$ acre a day and Suresh $\frac{2}{4} = \frac{1}{2}$ acre a day. Together that is $\frac{5}{6}$ acre a day, so 10 acres take 10 ÷ $\frac{5}{6}$ = 12 days.

Question 24

12 men can finish a job in 10 days, and 15 women can finish it in 16 days. If 4 men are put on the job, how many women must join them to finish it in 8 days?

Answer. The job is 120 man-days or 240 woman-days, so 1 man = 2 women. To finish in 8 days we need 120 ÷ 8 = 15 men's worth of work every day. 4 men give 4, so we need 11 men's worth more, which is 22 women.

Question 25

A tank is $\frac{3}{5}$ full. Pipe A can fill the tank in 28 minutes and pipe B can empty it in 20 minutes. If both are opened together, how long will the tank take to become empty?

Answer. Tank = 140 units, and it has 84. A puts in 5 a minute and B takes out 7, so the tank loses 2 units a minute. 84 units go in 42 minutes.

Question 26

A can do a job in 40 days. He works alone for 8 days, then B joins him, and together they finish the rest in 12 days. How long would B take alone?

Answer. A's 8 days are $\frac{8}{40} = \frac{1}{5}$ of the job, so $\frac{4}{5}$ is left. Together they do $\frac{4}{5}$ in 12 days, so they would need 15 days for the whole job. Then B = $\frac{40 \times 15}{40 - 15} = \frac{600}{25}$ = 24 days.

Question 27

A can do a job in 12 days and B in 24 days. A works every day, and B helps him on every third day. In how many days is the job finished?

Answer. I take 24 units: A does 2 a day and B 1. In each 3-day round A works 3 days and B works 1, so a round does 7 units. 3 rounds (9 days) do 21, and 3 are left. Day 10: A alone does 2, so 1 is left. Day 11: A needs half a day. $10\frac{1}{2}$ days.

Question 28

Pipes A and B can fill a tank in 12 hours and 20 hours. They are opened alternately for one hour each, starting with B. How long will the tank take to fill?

Answer. Tank = 60 units: A puts in 5 an hour and B 3, so every 2 hours give 8. 7 rounds (14 hours) give 56, and 4 units are left. The 15th hour is B's turn: 3 units, so 1 is left. The 16th hour is A's, and 1 of his 5 units takes $\frac{1}{5}$ hour. $15\frac{1}{5}$ hours, which is 15 hours 12 minutes.

Question 29

The efficiencies of A and B are in the ratio 5 : 3, and A takes 12 days less than B to finish a job. How long will they take working together?

Answer. Times go the other way round, 3 : 5. The gap is 2 parts = 12 days, so 1 part is 6 days: A takes 18 days and B takes 30. Together: $\frac{18 \times 30}{48}$ = $11\frac{1}{4}$ days.

Question 30

Pipe A can fill a tank in 33 hours. When pipe B is opened along with it, the tank fills 11 hours sooner. How long would B take alone?

Answer. Together they take 33 − 11 = 22 hours. Then B = $\frac{33 \times 22}{33 - 22}$ = 66 hours.

Question 31

60 men were hired to finish a job in 50 days. After 30 days, only half the job was done. How many more men must be hired to finish the job on time?

Answer. 60 × 30 = 1,800 man-days did half the job, so the other half also needs 1,800 man-days. Only 20 days are left, so we need 1,800 ÷ 20 = 90 men. That means 30 more men.

Question 32 [Tier 2]

A and B can do a job in 15 days, and B and C in 20 days. A works for 6 days, then B works for 8 days, and then C finishes the job in 17 days. How long would C take alone?

Answer. I pair things up. A's 6 days with 6 of B's days make 6 days of A + B. B's other 2 days with 2 of C's days make 2 days of B + C. That leaves 15 days of C alone. So $\frac{6}{15} + \frac{2}{20} + 15c = 1$, which gives $15c = 1 - \frac{2}{5} - \frac{1}{10} = \frac{1}{2}$. C does $\frac{1}{30}$ of the job a day and takes 30 days.

Question 33

A pipe can fill a tank in 10 hours. When the tank is half full, a leak starts that could empty the full tank in 40 hours. How long does the tank take to fill in all?

Answer. The first half takes 5 hours. After that the tank gains $\frac{1}{10} - \frac{1}{40} = \frac{3}{40}$ of itself an hour, so the other half takes $\frac{1}{2} \div \frac{3}{40} = \frac{20}{3}$ = $6\frac{2}{3}$ hours. In all, 5 + $6\frac{2}{3}$ = $11\frac{2}{3}$ hours, which is 11 hours 40 minutes.

Question 34

A man's daily wage is twice a woman's. 4 men and 6 women together earn ₹21,000 in 5 days. How much will 6 men and 8 women earn in 7 days?

Answer. Counting a man as 2 women, the first group is worth 14 women, so 70 woman-days = ₹21,000 and one woman-day = ₹300. The second group is worth 12 + 8 = 20 women for 7 days, which is 140 woman-days, so they earn 140 × 300 = ₹42,000.

Question 35 [Tier 2]

Pipe A alone takes 25 minutes more to fill a tank than pipes A and B together, and B alone takes 4 minutes more than the two together. How long does A take alone?

Answer. If together they take T minutes, then $T^2$ = 25 × 4, so T = 10. (This comes from $\frac{1}{T + 25} + \frac{1}{T + 4} = \frac{1}{T}$, which simplifies to $T^2 = 25 \times 4$.) So A alone takes 10 + 25 = 35 minutes, and B takes 14. Check: $\frac{35 \times 14}{49}$ = 10.

Question 36

A does $\frac{2}{3}$ of a job in 16 days, and then B finishes the rest in 6 days. How long would A and B take to do the whole job together?

Answer. A's full time is 16 × $\frac{3}{2}$ = 24 days. B did the last $\frac{1}{3}$ in 6 days, so his full time is 18 days. Together: $\frac{24 \times 18}{42} = \frac{72}{7}$ = $10\frac{2}{7}$ days.