Problem 1 :
Worker A takes 8 hours to do a work. Worker B takes 10 hours to do the same work. How long will it take both A and B, working together but independently, to do the same work?
Problem 2 :
A and B together can complete a piece of work in 4 days. If A alone can complete the same work in 12 days, in how many days can B alone complete that work?
Problem 3 :
A can do a piece of work in 7 days of 9 hours each day and B can do it in 6 days of 7 hours each day. How long will it take for them to complete the work, working together 8⅖ hours a day?
Problem 4 :
A and B can do a piece of work in 18 days; B and C can do the same work in 24 days; A and C can do it 36 days. In how many days will A, B and C finish it, working together and separately?
Problem 5 :
A is twice as good as B and together they finish a piece of work in 18 days. In how many days will A alone finish the work?
Problem 6 :
A can do a certain work in 12 days. B is 60% more efficient than A. How many days does B alone take to do the same work?
Problem 7 :
A can do a piece of work in 80 days. He works at it for 10 days and then B alone finishes the remaining work in 42 days. In how much time will A and B, working together, finish the work?
Problem 8 :
A and B undertake to do a piece of work for $600. A alone can do it 6 days while B alone can do it in 8 days. With the help of C, they finish it ion 3 days. Find the share of each.
Problem 9 :
A and B working separately can do a piece of work in 9 and 12 days respectively. If they work for a day alternately, A begining, in how many days will the work be completed?
Problem 10 :
45 men can complete a work in 16 days. Six days afetr they started working, 30 more men joined them. How many days will they now take to complete the remaning work?
Problem 11 :
2 men and 3 boys can do a piece of work in 10 days, while 3 men and 2 boys can do the same work in 8 days. In how many days can 2 men and 1 boy do the same work?
1. Answer :
A's 1 hour work = ⅛
B's 1 hour work = ⅒
(A + B)'s 1 hour work = ⅛ + ⅒ = ⁹⁄₄₀
Time taken by both A and B to finish the work :
= ⁴⁰⁄₉
= 4⁴⁄₉ hours
2. Answer :
(A + B)'s 1 day work = ¼
A's 1 day work = ¹⁄₁₂
B's 1 day work = ¼ - ¹⁄₁₂ = ⅙
Therefore, B alone can finsih the work in
= ⁶⁄₁
= 6 days
3. Answer :
Time taken by A to complete the work :
= 7 ⋅ 9
= 63 hours
A's 1 hour work = ¹⁄₆₃
Time taken by B to complete the work :
= 6 ⋅ 7
= 42 hours
B's 1 hour work = ¹⁄₄₂
(A + B)'s 1 hour work = ¹⁄₆₃ + ¹⁄₄₂ = ⁵⁄₁₂₆
Time taken by both A and B to finsih the work :
= ¹²⁶⁄₅ hours
Number of days taken by both A and B to finsih the work, working 8⅖ hours a day :
= ¹²⁶⁄₅ ÷ 8⅖
= ¹²⁶⁄₅ ÷ ⁴²⁄₅
= ¹²⁶⁄₅ ⋅ ⁵⁄₄₂
= 3 days
4. Answer :
(A + B)'s 1 day work = ¹⁄₁₈ ----(1)
(B + C)'s 1 day work = ¹⁄₂₄ ----(2)
(A + C)'s 1 day work = ¹⁄₃₆ ----(3)
(1) + (2) + (3) :
(A + B) + (B + C) + (A + C) = ¹⁄₁₈ + ¹⁄₂₄ + ¹⁄₃₆
A + B + B + C + A + C = ⁹⁄₇₂
2A + 2B + 2C = ⅛
2(A + B + C) = ⅛
(A + B + C)'s 1 day work = ¹⁄₁₆
Number of days taken by A, B and C together to finish the work :
= ¹⁶⁄₁
= 16 days
A's 1 day work :
= (A + B + C)'s 1 day work - (B + C)'s 1 day work
= ¹⁄₁₆ - ¹⁄₂₄
= ¹⁄₄₈
Number of days taken by A alone to finish the work :
= ⁴⁸⁄₁
= 48 days
B's 1 day work :
= (A + B + C)'s 1 day work - (A + C)'s 1 day work
= ¹⁄₁₆ - ¹⁄₃₆
= ⁵⁄₁₄₄
Number of days taken by B alone to finish the work :
= ¹⁴⁴⁄₅
= 28⅘ days
B's 1 day work :
= (A + B + C)'s 1 day work - (A + C)'s 1 day work
= ¹⁄₁₆ - ¹⁄₃₆
= ⁵⁄₁₄₄
Number of days taken by B alone to finish the work :
= ¹⁴⁴⁄₅
= 28⅘ days
C's 1 day work :
= (A + B + C)'s 1 day work - (A + B)'s 1 day work
= ¹⁄₁₆ - ¹⁄₁₈
= ¹⁄₁₄₄
Number of days taken by C alone to finish the work :
= ¹⁴⁴⁄₁
= 144 days
5. Answer :
(A's 1 days work ) : (B's 1 day work) = 2 : 1
(A + B)'s 1 day work = ¹⁄₁₈
Divide ¹⁄₁₈ in the ratio 2 : 1.
A's 1 day work = ¹⁄₁₈ ⋅ ⅔ = ¹⁄₂₇
Number of days taken by A alone to finish the work :
= ²⁷⁄₁
= 27 days
6. Answer :
Ratio of times taken by A and B :
= 160 : 100
= 8 : 5
Number of days taken by A = 8x
Number of days taken by B = 5x
Given : A takes 12 days to complete the work.
8x = 12
x = 1.5
Number of days taken by B to complete the work :
= 5(1.5)
= 7.5 days
7. Answer :
A's 1 day work = ¹⁄₈₀
Work done by A in 10 days :
= 10 ⋅ ¹⁄₈₀
= ¹⁄₈
Remaining work :
= 1 - ¹⁄₈
= ⅞
⅞ of the work done by B in 42 days.
Number of days taken by to complete the whole work :
= 42 ÷ ⅞
= 42 ⋅ ⁸⁄₇
= 48 days
B's 1 day work = ¹⁄₄₈
(A + B)'s 1 day work = ¹⁄₈₀ + ¹⁄₄₈ = ¹⁄₃₀
Number of days taken by A and B, working together :
= ³⁰⁄₁
= 30 days
8. Answer :
A's 1 day work = ⅙
B's 1 day work = ⅛
(A + B + C)'s 1 day work = ⅓
C's 1 day work = ⅓ - (⅙ + ⅛) = ¹⁄₂₄
Ratio of 1 day work :
A : B : C = ⅙ : ⅛ : ¹⁄₂₄
A : B : C = 4 : 3 : 1
A's share = 4x
B's share = 3x
C's share = x
Given : The work is undertaken for $600.
4x + 3x + x = 600
8x = 600
x = 75
Therefore,
A's share = 4(75) = $300
B's share = 3(75) = $225
C's share = $75
9. Answer :
A's 1 day work = ⅑
B's 1 day work = ¹⁄₁₂
A and B work for a day alternately and A starts on the first day. So, B works on the second day, A works on the third day and so on.
(A + B)'s 2 days work = ⅑ + ¹⁄₁₂ = ⁷⁄₃₆
Work completed in the first ten days (5 pairs of days) :
= 5 ⋅ ⁷⁄₃₆ = ³⁵⁄₃₆
Remaning work = 1 - ³⁵⁄₃₆ = ¹⁄₃₆
On 11^{th} day, its A's turn. He can complete ⅑ of the work in one day.
Time taken by A to complete ¹⁄₃₆ of the work on 11^{th} day :
= ¹⁄₃₆ ÷ ⅑
= ¹⁄₃₆ ⋅ ⁹⁄₁
= ¼ day
Total time taken to complete the work :
= (10 + ¼) days
= 10¼ days
10. Answer :
45 men can complete the work in 16 days.
45 men's 1 day work = ¹⁄₁₆
1 man's 1 day work = ¹⁄₍₁₆ₓ₄₅₎ = ¹⁄₇₂₀
45 men's 6 days work = 6 ⋅ ¹⁄₁₆ = ⅜
Remaning Work = ⅝
This remaning ⅝ of the work is completed by 75 men.
We already know that 1 man's 1 day work is ¹⁄₇₂₀.
75 men's 1 day work = 75 ⋅ ¹⁄₇₂₀ = ⁵⁄₄₈
Time taken by 75 men to complete the remanining work :
= ⅝ ÷ ⁵⁄₄₈
= ⅝ ⋅ ⁴⁸⁄₅
= 6 days
11. Answer :
Let us assume x and y as follows.
1 man's 1 day work = x
1 boy's 1 day work = y
Given : 2 men and 3 boys can do the work in 10 days.
(2 men + 3 boys)'s 1 day work = ⅒
2x + 3y = ⅒ ----(1)
Given : 3 men and 2 boys can do the work in 8 days.
(3 men + 2 boys)'s 1 day work = ⅛
3x + 2y = ⅛ ----(2)
Solving (1) and (2),
x = ⁷⁄₂₀₀ and y = ¹⁄₁₀₀
(2 men + 1 boy)'s 1 day work = 2 ⋅ ⁷⁄₂₀₀ + ¹⁄₁₀₀
= ⁷⁄₁₀₀ + ¹⁄₁₀₀
= ⁸⁄₁₀₀
= ²⁄₂₅
Number of days taken by 2 men and 1 boy to complete the work :
= ²⁵⁄₂
= 12½ days
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