A can build up a wall in 8 days while B can break it in 3 days. A has worked for 4 days and then B joined to work with A for another 2 days only. In how many days will A alone build up the remaining part of the wall?
- (a)7 1⁄3 days
- (b)8 2⁄5 days
- (c)9 1⁄5 days
- (d)5 1⁄2 days
Answer
Why
Correct — A. Count B's breaking as negative work.
A builds 1⁄8 of the wall a day, B breaks 1⁄3 a day
A alone, 4 days: 4 × 1⁄8 = 1⁄2 built
A and B, one day: 1⁄8 − 1⁄3 = 3⁄24 − 8⁄24 = −5⁄24
A and B, 2 days: 2 × (−5⁄24) = −5⁄12
Standing after 6 days: 1⁄2 − 5⁄12 = 1⁄12
Left to build: 1 − 1⁄12 = 11⁄12
A alone: 11⁄12 ÷ 1⁄8 = 88⁄12 = 7 1⁄3 days → option (a)
Why the others are wrong
- (b)8 2⁄5 days — 8 2⁄5 days is longer than the 8 days A needs for a whole wall, yet 1⁄12 of this wall is still standing. At 1⁄8 a day it would build 21⁄20 of a wall.
- (c)9 1⁄5 days — 9 1⁄5 days also exceeds A's 8 days for a whole wall. A would build 46⁄5 × 1⁄8 = 23⁄20 of a wall, when only 11⁄12 is left.
- (d)5 1⁄2 days — 5 1⁄2 days at 1⁄8 a day builds 11⁄16 of the wall. The part left is 11⁄12, which is larger, so the wall would still be unfinished.
Concept
A work rate is the fraction of the job done per day: whoever finishes in n days does 1⁄n a day. Rates of people working at the same time add.
Someone who undoes the work adds a negative rate. With B on site the wall changes by 1⁄8 − 1⁄3 = −5⁄24 a day, so it shrinks.
Track how much stands after each phase, then divide what is left by the finisher's rate.
Check in units: call the wall 24 units, the LCM of 8 and 3. A adds 3 a day and B removes 8.
After 4 days, 12 units stand. The 2 joint days change that by 2 × (3 − 8) = −10, leaving 2. A builds the other 22 in 22 ÷ 3 = 7 1⁄3 days.
Key facts
- A worker who finishes a job in n days does 1⁄n of it per day.
- Rates of workers on the job at the same time add.
- A worker who destroys the work, like a leak or a wrecker, enters that sum with a minus sign.
- Days needed = work remaining ÷ rate of whoever finishes it.
Study next
Common traps
- Reading 'B joined to work with A' as B helping to build: adding 1⁄3 pushes the wall past 100% during the 2 joint days
- Ignoring the 1⁄12 still standing: an answer of 8 days or more means A builds at least a whole wall
- Stopping at 11⁄12, the part left, without dividing by A's rate of 1⁄8
The wrecker plays the part of an emptying pipe: also asked 13 Sep 2024, 12:30, Quant Q.13 (one pipe fills a tank in 5 minutes, another empties it in 6) and 17 Sep 2025, 16:00, Quant Q.5 (drainpipe C against two filling pipes).
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