A and B together can do a piece of work in 16 days and B and C can do

Question

A and B together can do a piece of work in 16 days and B and C can do the same work in 24 days. From starting A and B worked for 4 days and 7 days respectively and remaining work is completed by C in 23 days, then find in how many days will C complete the work alone?

(a) 32 days
(b) 16 days
(c) 8 days
(d) 24 days
(e) 36 days

(a) 32 days
(b) 16 days
(c) 8 days
(d) 24 days
(e) 36 days

Miyo311 · Accepted Answer

A and B together can do a piece of work in 16 days and B and C can do the same work in 24 days. From starting A and B worked for 4 days and 7 days respectively and remaining work is completed by C in 23 days, then find in how many days will C complete the work alone?

(a) 32 days
(b) 16 days
(c) 8 days
(d) 24 days
(e) 36 days

(a) 32 days
(b) 16 days
(c) 8 days
(d) 24 days
(e) 36 days

I found a simple way!

w=rt
4A+7B+23C=1 can be rewritten as 4(A+B) + 3(B+C) + 20C = 1

A+B has rate of 1/16
B+C has rate of 1/24
so...

4(1/16) + 3(1/24)+20C = 1
C=1/32

since the rate of C is 1/32, the time it takes C to finish 1 work is  32 days

yashikaaggarwal · Answer

Total work done by A B and C
=> 4/A+7/B+23/C

16/A+16/B = 4/A+7/B+23/C
12/A + 9/B = 23/C
12/A = 23/C - 9/B -----------(1)

24/B+24/C = 4/A+7/B+23/C
17/B + 1/C = 4/A
Multiply both side by 3
51/B + 3/C = 12/A -----------(2)

23/C - 9/B = 51/B + 3/C 
20/C = 60/B
1/C = 3/B
B = 3C

24/3C+24/C = 1
72+24/3c = 1
96/3C = 1
C = 32

So C can do work in 32 days.

Answer is A

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CrackverbalGMAT · Answer

The sum of fractions of work done by each person to complete the work is = 1

Since A and B together finish the work together in 16 days,  \frac{16}{A}  +  \frac{16}{B}  = 1 or  \frac{1}{A}  +  \frac{1}{B}  =  \frac{1}{16}  - Equation (1)

Since B and C finish the work together in 24 days,  \frac{24}{B}  +  \frac{24}{C}  = 1 or  \frac{1}{B}  +  \frac{1}{C}  =  \frac{1}{24}  - Equation (2)

Also  \frac{4}{A}  +  \frac{7}{B}  +  \frac{23}{C}  = 1 - Equation (3)

Since we need to find the number of days in which C finishes the work, we need to find A and B in terms of C.
 
From Equation (2)  \frac{1}{B}  =  \frac{1}{24}  -  \frac{1}{C}  - Equation (4)

Subtracting Equation (1) from (2)

\frac{1}{A}  +  \frac{1}{B}  -  \frac{1}{B}  +  \frac{1}{C} ] =  \frac{1}{16}  -  \frac{1}{24}

\frac{1}{A}  -  \frac{1}{C}  =  \frac{1}{16}  -  \frac{1}{24}  =  \frac{1}{48}

\frac{1}{A}  =  \frac{1}{48}  +  \frac{1}{C}  - (Equation 5)

Putting 4 and 5 in Equation (3)

4 * ( \frac{1}{48}  +  \frac{1}{C} ) + 7 * ( \frac{1}{24}  -  \frac{1}{C} ) +  \frac{23}{C}  = 1

\frac{4}{48}  +  \frac{4}{C}  +  \frac{7}{24}  -  \frac{7}{C}  +  \frac{23}{C}  = 1

\frac{20}{C}  = 1 -  \frac{4}{48}  -  \frac{7}{24}  =  \frac{5}{8}

Therefore C = 32 days

Option A

Arun Kumar

Fdambro294 · Answer

If A started off doing 4 days of work and B started off doing 7 days of working ———>

Same as if:

(A and B) worked together simultaneously at their respective rates for the first 4 days

And

B worked another 3 days on top of that.

C will then finish the remaining work.

(1st) A and B working together for 4 days

We are told that A and B working together take 16 days to complete the work.

Rate = (1 / 16) job/day

working for 4 days ———-> (4 / 16) = (1 / 4) of job completed.

3/4 of job remains

(2nd) B does 3 more days on his own

We are told that C and B can finish the job in 24 days. Therefore, we can put B’s rate of work in terms of C’s rate of work

Rate of B + Rate of C = (1 / 24)

(1 / B) + (1 / C) = (1 / 24)

Rate of B = (1 / B) = (1 / 24) - (1 / C) =

(C - 24) / (24C)

———-> rate of B worked for 3 days ——> multiples by the 3 extra days B works (the 3 cancels and simplifies with the 24C in the DEN)

Work finished in 3 days by B = (C - 24) / (8C)

Work remaining after A and B are done =

(3 / 4) - (C - 24)/(8C) =

(6C / 8C) - (C - 24)/(8C) =

(5C + 24) / (8C) ———-> work left for C to complete in terms of “C”, where C = the number of days it takes C to finish 1 job

This remaining work is done in 23 days ——-> rate of C = (work done) / (23 days)

(1 / C) = (5C + 24) / (8C * 23)

Cross multiply

(8C * 23) = (C) * (5C + 24)

Since C = a positive number of hours, we can divide by C on both sides of the equation

(8 * 23) = (5C + 24)

184 - 24 = 5C

5C = 160

C = 32 days

Answer

It takes C 32 days to complete the job on his/her own

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Crytiocanalyst · Answer

w = 24*(a+b).........1

w= 16*(b+c).............2

w= 4*a + 7*b + 23*c .........3

Solving for C in terms of W

we get W/32 =C

THerefore IMO A

Brujisimo2 · Answer

Awesome explanation!

Dawoodgmat · Answer

how could you rewrite this as 4(A+B) + 3(B+C) + 20C =1

Bunuel · Answer

4A + 7B + 23C =
 
= 4A + (4B + 3B) + (3C + 20C) =

= (4A + 4B) + (3B + 3C) + 20C =
 
 = 4(A+B) + 3(B+C) + 20C

datnguyen123 · Answer

because B and C together cost 24 days to make it so any answer <=24 is eliminated. 
Only 32 and 36 remains, just try simple maths.
and the answer is 32

A and B together can do a piece of work in 16 days and B and C can do

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