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A and B together complete a piece of job in 10 days, B and C together

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A and B together complete a piece of job in 10 days, B and C together complete the same piece of job in 15 days, whereas A and C together complete it in 8 days. How many days will it take to complete the job if all of them work together?


A. \(3 \frac{3}{7}\)

B. \(6 \frac{6}{7}\)

C. \(7\)

D. \(7 \frac{23}{35}\)

E. \(7 \frac{6}{7}\)
[Reveal] Spoiler: OA

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A and B together complete a piece of job in 10 days, B and C together [#permalink]

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Bunuel wrote:
A and B together complete a piece of job in 10 days, B and C together complete the same piece of job in 15 days, whereas A and C together complete it in 8 days. How many days will it take to complete the job if all of them work together?


A. \(3 \frac{3}{7}\)

B. \(6 \frac{6}{7}\)

C. \(7\)

D. \(7 \frac{23}{35}\)

E. \(7 \frac{6}{7}\)


(A+B)'s 1 day job = \(\frac{1}{10}\)
(B+C)'s 1 day job = \(\frac{1}{15}\)
(A+C)'s 1 day job = \(\frac{1}{8}\)

Adding the three equations we get -
\(2\)(A+B+C)'s 1 day job = \(\frac{1}{10}+\frac{1}{15}+\frac{1}{8} = \frac{35}{120}\)
or (A+B+C)'s 1 day job = \(\frac{35}{120}*\frac{1}{2}\) \(=\frac{7}{48}\)
Hence time taken by All to complete the job = \(\frac{48}{7}\) = \(6 \frac{6}{7}\)

Option B

Last edited by niks18 on 14 Aug 2017, 22:05, edited 1 time in total.

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A and B together complete a piece of job in 10 days, B and C together [#permalink]

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If A and B can together do the piece of work in \(10(2*5)\) days, B and C do the work in \(15(3*5)\) days
, and A and C do the work in \(8(2^3)\) days

Let us assume the work to be \(120(2^3 * 3 * 5)\) units.

Therefore A & B's combined rate is 12 units, B & C's combined rate is 8 units, and A & C's combined rate is 15 units.

Together they would do half of 35(12+8+15) units in a day.
So their combined rate is \(\frac{35}{2}\) units per day.

In order to finish 120 units at that rate, it would require \(120 * (\frac{2}{35}) = \frac{120*2}{35} = \frac{24*2}{7} = \frac{48}{7} = 6\frac{6}{7}\) days(Option B)
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A and B together complete a piece of job in 10 days, B and C together [#permalink]

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When we are given AB, BC and CA rates, there is a quicker method. Eventually we have to add the rates, hence, the shortcut is

2*(combined rate) = Sum of rates of AB + BC + CA

\(2*x= \frac{1}{8}+\frac{1}{10}+\frac{1}{15}\)

\(x=\frac{7}{48}\)

Flip the rate to get the time.

Answer is \(\frac{48}{7}\)
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Re: A and B together complete a piece of job in 10 days, B and C together [#permalink]

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Bunuel wrote:
A and B together complete a piece of job in 10 days, B and C together complete the same piece of job in 15 days, whereas A and C together complete it in 8 days. How many days will it take to complete the job if all of them work together?


A. \(3 \frac{3}{7}\)

B. \(6 \frac{6}{7}\)

C. \(7\)

D. \(7 \frac{23}{35}\)

E. \(7 \frac{6}{7}\)


We can let a = the number of days A will complete the job alone, b = the number of days B will complete the job alone and c = the number of days C will complete the job alone. We can now create the following combined rate equations:

1/a + 1/b = 1/10

and

1/b + 1/c = 1/15

and

1/a + 1/c = 1/8

Adding the equations together, we have:

2/a + 2/b + 2/c = 1/10 + 1/15 + 1/8

2/a + 2/b + 2/c = 12/120 + 8/120 + 15/120

2/a + 2/b + 2/c = 35/120= 7/24

Thus, 1/a + 1/b + 1/c = 7/24 x 1/2 = 7/48.

So, the time it would take the machines to work together is 1/(7/48) = 48/7 = 6 6/7.

Answer: B
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Re: A and B together complete a piece of job in 10 days, B and C together [#permalink]

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New post 14 Nov 2017, 07:55
Bunuel wrote:
A and B together complete a piece of job in 10 days, B and C together complete the same piece of job in 15 days, whereas A and C together complete it in 8 days. How many days will it take to complete the job if all of them work together?


A. \(3 \frac{3}{7}\)

B. \(6 \frac{6}{7}\)

C. \(7\)

D. \(7 \frac{23}{35}\)

E. \(7 \frac{6}{7}\)

Let the total work be 120 units

So, Efficiency of A and B together is 12
Efficiency of B and C together is 8
Efficiency of A and C together is 15

Combined efficiency is \(\frac{12+8+15}{2} = \frac{35}{2}\)

Time required to complete the work is \(120*\frac{2}{35}\) = \(6 \frac{6}{7}\)

Answer will be (B) \(6 \frac{6}{7}\)
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A and B together complete a piece of job in 10 days, B and C together [#permalink]

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New post 19 Nov 2017, 07:20
Lets solve this using LCM method
A+B = 10 days
B+C = 15 days
C+D = 8 days
LCM of 10,15 and 8 is 120.
so 120 units of work done (Assume)
therefore rate of work
A+B = 120/10 = 12 units per day
B+C = 120/15 = 8 units per day
C+D = 120/8 = 15 units per day.
So if all of them work together i.e A+B+C = 35/ 2units ((12+8+15)/2)(we are dividing by 2 because each rate gets repeated
twice)
hence number of days = 120/ (35/2) = 48/7 (answer B)

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Re: A and B together complete a piece of job in 10 days, B and C together [#permalink]

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New post 19 Nov 2017, 09:35
sasyaharry wrote:
When we are given AB, BC and CA rates, there is a quicker method. Eventually we have to add the rates, hence, the shortcut is

2*(combined rate) = Sum of rates of AB + BC + CA

\(2*x= \frac{1}{8}+\frac{1}{10}+\frac{1}{15}\)

\(x=\frac{7}{48}\)

Flip the rate to get the time.

Answer is \(\frac{48}{7}\)


what is combined rate?

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Re: A and B together complete a piece of job in 10 days, B and C together [#permalink]

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New post 19 Nov 2017, 19:57
Combined rate here will refer to the rate at which the three of them(A,B, and C) do the work.

I have used a different method to solve this problem. Check if that method is clear
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Re: A and B together complete a piece of job in 10 days, B and C together   [#permalink] 19 Nov 2017, 19:57
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