Suppose A and B can complete a work together in 10 days. If B alone can complete the work in 15 days, then in how many days can A alone finish the work?
- (a)20 days
- (b)24 days
- (c)25 days
- (d)30 days
Correct — D, 30 days. Work in rates rather than days. Together A and B finish the job in 10 days, so their combined rate is 1/10 of the work per day. B alone takes 15 days, a rate of 1/15 per day. A's rate is the difference: 1/10 − 1/15 = 3/30 − 2/30 = 1/30 of the work per day. A working alone therefore needs 30 days.
- (a)20 days — A rate of 1/20 plus B's 1/15 gives 7/60 per day, which finishes the job in about 8.6 days, not the 10 the question states.
- (b)24 days — 1/24 + 1/15 = 13/120, a joint time of about 9.2 days. Still faster than the given 10.
- (c)25 days — 1/25 + 1/15 = 8/75, a joint time of about 9.4 days. The only value that makes the pair take exactly 10 days is 30.
Days do not add; rates do. Convert every 'takes n days' into 'does 1/n of the work per day', combine by addition or subtraction as the situation requires, then invert the result to get back to days.
The wrong instinct is to subtract the times — 15 minus 10 gives 5, and no option is 5, which is a useful sign that the shortcut is nonsense. An alternative that avoids fractions is the LCM method: take the total work as 30 units (the LCM of 10 and 15), so the pair does 3 units a day and B does 2, leaving 1 unit a day for A and 30 days for the job.
- If a worker takes n days, the daily rate is 1/n of the job.
- Combined rate of two workers is the sum of their individual rates; one worker's rate is the combined rate minus the other's.
- Time together for two workers taking a and b days is ab/(a + b) — here 10 = (30 × 15)/45, which confirms the answer.
- Taking total work as the LCM of the given times turns every fraction into a whole number of units.
Or take the work as 30 units: pair 3 a day, B 2 a day, A 1 a day.
- Subtracting the days (15 − 10) instead of the rates.
- Adding rates when the question calls for a difference, or the reverse.
- Forgetting to invert at the end and reporting the rate as the number of days.
Asked as a pair working together with one member's solo time given and the other's asked for, sometimes extended by a third worker or an emptying pipe.
A and B can complete work together in 5 days. If A works at twice his speed and B at half of his speed, this work can be finished in 4 days. How many days would it take for A alone to complete the job?
- (a) 10
- (b) 12
- (c) 15
- (d) 18
Answer(a) 10
The harder cousin — two rate equations instead of one, with the speeds scaled in the second. The opening move is identical: write each worker's contribution as a fraction of the job per day.
- practice — not a real PYQ
A can do a piece of work in 12 days and B in 18 days. Working together they will finish it in
- (a)6.6 days
- (b)7.2 days
- (c)8.4 days
- (d)9 days
Answer(b) 7.2 days — combined rate 1/12 + 1/18 = 5/36 per day, so the time is 36/5 = 7.2 days.
- practice — not a real PYQ
A pipe fills a tank in 6 hours and an outlet empties it in 9 hours. With both open, the tank fills in
- (a)15 hours
- (b)18 hours
- (c)12 hours
- (d)3.6 hours
Answer(b) 18 hours — the net rate is 1/6 − 1/9 = 1/18 of the tank per hour, so 18 hours are needed.