Time and Work
- A and B can complete a piece of work in 45 and 40 days respectively. Both started to work together, but after some days A left and B alone completed the rest work in 23 days. For how many days did A work?
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Using Rule 2,
Let A worked for x days.A’s 1 day’s work = 1 45 ∴ A’s x day’s work= x 45 B’s 1 day’s work = 1 40
∴ B’s x day’s work= 1 × x = x 40 40
(A + B) together worked for x days.
∴ (A + B)’s x day’s work= x + x 45 40 = 8x + 9x = 17x 360 360
∴ Remaining work= 1 - 17x = 360 - 17x 360 360
This part of work, i.e.,360 - 17x 360
is completed by B alone in 23 days.∴ 360 - 17x = 23 × B’s 1 day’s work 360 360 - 17x 360 = 23 × 1 = 23 40 40
∴ 360 – 17x= 23 × 360 = 207 40
∴ 17x = 360 – 207 = 153⇒ x = 153 = 9 days 17
Hence, A worked for 9 days.Correct Option: D
Using Rule 2,
Let A worked for x days.A’s 1 day’s work = 1 45 ∴ A’s x day’s work= x 45 B’s 1 day’s work = 1 40
∴ B’s x day’s work= 1 × x = x 40 40
(A + B) together worked for x days.
∴ (A + B)’s x day’s work= x + x 45 40 = 8x + 9x = 17x 360 360
∴ Remaining work= 1 - 17x = 360 - 17x 360 360
This part of work, i.e.,360 - 17x 360
is completed by B alone in 23 days.∴ 360 - 17x = 23 × B’s 1 day’s work 360 360 - 17x 360 = 23 × 1 = 23 40 40
∴ 360 – 17x= 23 × 360 = 207 40
∴ 17x = 360 – 207 = 153⇒ x = 153 = 9 days 17
Hence, A worked for 9 days.
- Ram can do a piece of work in 20 days and Shyam in 30 days. They work together for 10 days. After that Shyam leaves and rest of the work is completed by Ram alone. How long does it take Ram to finish the remaining work?
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Ram’s 1 day’s work = 1 20
Shyam’s 1 day’s work = 1 30
∴ (Ram + Shyam)’s 1 day’s work= 1 + 1 = 3 + 2 20 30 60 = 5 = 1 60 12
∴ (Ram + Shyam)’s 10 days’ work= 10 × 1 = 5 12 6
⇒ Remaining work= 1 - 5 = 1 6 6
Now, 1/6 work is completed by Ram alone.
To finish this part Ram will take= = 1 Remaining work 6 Ram’s1 day’s work 1 20 = 1 × 20 = 10 = 3 1 days. 6 3 3 Correct Option: C
Ram’s 1 day’s work = 1 20
Shyam’s 1 day’s work = 1 30
∴ (Ram + Shyam)’s 1 day’s work= 1 + 1 = 3 + 2 20 30 60 = 5 = 1 60 12
∴ (Ram + Shyam)’s 10 days’ work= 10 × 1 = 5 12 6
⇒ Remaining work= 1 - 5 = 1 6 6
Now, 1/6 work is completed by Ram alone.
To finish this part Ram will take= = 1 Remaining work 6 Ram’s1 day’s work 1 20 = 1 × 20 = 10 = 3 1 days. 6 3 3
- A can do a piece of work in 40 days. He works on it for 5 days and then B completes it in 21 days. How long will A and B together take to complete the work?
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Using Rule 2,
A’s 1 day’s work = 1 40
∴ A’s 5 days’ work= 5 = 1 40 8
Remaining work= 1 - 1 = 7 8 8
This part of work is done by B in 21 days.
∴ B’s 1 day’s work= 7 = 1 8 × 21 24
∴ (A + B)’s 1 day’s work= 1 + 1 = 3 + 5 40 24 120 = 8 = 1 120 15
Hence, A and B together will
complete the work in 15 days.Correct Option: B
Using Rule 2,
A’s 1 day’s work = 1 40
∴ A’s 5 days’ work= 5 = 1 40 8
Remaining work= 1 - 1 = 7 8 8
This part of work is done by B in 21 days.
∴ B’s 1 day’s work= 7 = 1 8 × 21 24
∴ (A + B)’s 1 day’s work= 1 + 1 = 3 + 5 40 24 120 = 8 = 1 120 15
Hence, A and B together will
complete the work in 15 days.
- A, B and C can complete a work in 8 days. B alone can do it in 18 days and C alone can do it in 24 days. In how many days can A alone do the same work?
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(A + B + C)’s 1 day’s work
= 1 8 B’s 1 day’s work = 1 18 C’s 1 day’s work = 1 24
∴ A’s 1 day’s work
= (A + B + C)’s 1 day’s work – B’s
1 day’s work – C’s 1 day’s work= 1 - 1 - 1 8 18 24 = 9 - 4 - 3 = 2 = 1 72 72 36 ⇒ A’s 1 day’s work = 1 36
∴ A alone can do the same work in 36 days.
Aliter : Using Rule 18,
Here, x = 8, y = 18, z = 24
Required time= xyz zy - x(y + z) = 8 × 18 × 24 24 × 18 - 8(18 + 24) = 8 × 18 × 24 432 - 336 = 8 × 18 × 24 = 36 days 96 Correct Option: A
(A + B + C)’s 1 day’s work
= 1 8 B’s 1 day’s work = 1 18 C’s 1 day’s work = 1 24
∴ A’s 1 day’s work
= (A + B + C)’s 1 day’s work – B’s
1 day’s work – C’s 1 day’s work= 1 - 1 - 1 8 18 24 = 9 - 4 - 3 = 2 = 1 72 72 36 ⇒ A’s 1 day’s work = 1 36
∴ A alone can do the same work in 36 days.
Aliter : Using Rule 18,
Here, x = 8, y = 18, z = 24
Required time= xyz zy - x(y + z) = 8 × 18 × 24 24 × 18 - 8(18 + 24) = 8 × 18 × 24 432 - 336 = 8 × 18 × 24 = 36 days 96
- A and B can do a piece of work in 12 days, B and C in 15 days; C and A in 20 days. In how many days will they finish it working together? In what time can A do it separately?
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(A + B)’s 1 day’s work
= 1 12
(B + C)’s 1 day’s work= 1 15
(C + A)’s 1 day’s work= 1 20
Adding all,
2 (A + B + C)’s 1 day’s work= 1 + 1 + 1 12 15 20 = 5 + 4 + 3 = 12 = 1 60 60 5
∴ (A + B + C)’s 1 day’s work= 1 = 1 5 × 2 10
∴ (A + B + C) together can complete the work in 10 days.
Now, A’s 1 day’s work
= (A + B + C)’s 1 day’s work –
(B + C)’s 1 day’s work= 1 - 1 = 3 - 2 = 1 10 15 30 30
∴ A alone can finish the work in 30 days.
Aliter : Using Rule 19,
Here, x = 12, y = 15, z = 20
A alone can do in= 2xyz xy + yz - zx = 2 × 12 × 15 × 20 12 × 15 + 15 × 20 - 20 × 12 = 24 × 300 180 + 300 - 240 = 24 × 300 = 30 days 240 Correct Option: D
(A + B)’s 1 day’s work
= 1 12
(B + C)’s 1 day’s work= 1 15
(C + A)’s 1 day’s work= 1 20
Adding all,
2 (A + B + C)’s 1 day’s work= 1 + 1 + 1 12 15 20 = 5 + 4 + 3 = 12 = 1 60 60 5
∴ (A + B + C)’s 1 day’s work= 1 = 1 5 × 2 10
∴ (A + B + C) together can complete the work in 10 days.
Now, A’s 1 day’s work
= (A + B + C)’s 1 day’s work –
(B + C)’s 1 day’s work= 1 - 1 = 3 - 2 = 1 10 15 30 30
∴ A alone can finish the work in 30 days.
Aliter : Using Rule 19,
Here, x = 12, y = 15, z = 20
A alone can do in= 2xyz xy + yz - zx = 2 × 12 × 15 × 20 12 × 15 + 15 × 20 - 20 × 12 = 24 × 300 180 + 300 - 240 = 24 × 300 = 30 days 240
