Official Solution:

Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of \(x\)?

A. \(1\)
B. \(1.25\)
C. \(2\)
D. \(2.5\)
E. \(4\)


After 2 hours \(2*\frac{1}{10}=\frac{1}{5}\) of the taks will be done (as only A works);

After 4 hours \(\frac{1}{5}+2*(\frac{1}{10}+\frac{1}{5})=\frac{4}{5}\) of the task will be done and \(\frac{1}{5}\) will be left to be done;

We are told that 1/5th of the task is done in 15 minutes (1/4th of an hour) by all three workers: \(\frac{1}{4}*(\frac{1}{10}+\frac{1}{5}+\frac{1}{x})=\frac{1}{5}\). From which we can find that \(x=2\) hours.


Answer: C
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A works alone for 2 hours,
0.2x(work) completed

joins B
0.2x completed by worker A
1/5 => 2/5 completed or 0.4x completed by worker B

after 4 hours we have 0.8x completed
joins C and everything is finished in 15 minutes
or 0.2x (work) is completed in 0.25 hours

now we can set an equation
1/10 + 1/5+ 1/x = 2/10 * 4/1 simplify 1/5 or 4/5
multiply everyting by 10
1+2+10/x = 8
10/x = 5
x = 2.

not clear why 1/4 has been multiplied at end

1 hour is 1.
15 minutes is 1/4

The last calculation is not clear. How we are multiplying with with 1/4 and equalizing it to 1/5. I want to know the mathematical formula. Thanks

\(\frac{1}{4}*(\frac{1}{10}+\frac{1}{5}+\frac{1}{x})=\frac{1}{5}\).

(time)(combined rate) = (job done).
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Consider the total work as 10, and A can do 1 work in 1hr, B can do 2 work in 1 hr. A starts work for 2 hrs. So 2 work completed(1×2=2), then B joins for 2 hrs, so B does 4 work (2×2=4), now total work = A's 4+B's 4 work = 8. Balance work is 2. C joins and complete the work in 15 mins. That is A do .25 work and B do .5 work in15 mins., total .75 work(.25+.5=.75).Balace 1.25 work is done by C in 15 mins. So C can do 5 work in 1 hr. It means C takes 2hrs to complete the total work.
Ans : opt.c, 2hrs

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10 A = 5 B =Cx= k = since work remain same that each does

10A = since work remains the same

10 A = A*2 + (A+B)2 + (A+B+C) *1/4

=> K = K/10 *2 + (K/10+K/5)2 + (K/10 + K/5 +k/x) *1/4

solving for above equation

=> 1/2 = 1/x

x =2 hours for C

A - 10 hours to complete the work
B- 5 hours to complete the work
C- X hours to complete the work
Take the LCM of 10,5,x = 10x (total work)
Hence A rate of work = 10x/10 = X units/hour
B rate of work = 10x/5 = 2x units/hour
C rate of work = 10x/x = 10 units/hour
therefore
X*2 +(2x+x)*2+(2x+x+10)*1/4 =10x
this gives X =2 (quick and easy)

Hi Bunuel: Question M28-19 and D01-28 are the same. Just in case if it was not known earlier.

I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. Please show how to solve for "X" once the last equation is setup. Thank you

Why is the question of poor quality? That's not right. Both the question and the solution are correct and detailed enough.


After you get \(\frac{1}{4}*(\frac{1}{10}+\frac{1}{5}+\frac{1}{x})=\frac{1}{5}\), only simple algebraic manipulations are left to be done:

\(\frac{1}{10}+\frac{1}{5}+\frac{1}{x}=\frac{4}{5}\)

\(\frac{1}{x}=\frac{4}{5}-(\frac{1}{10}+\frac{1}{5})\)

\(\frac{1}{x}=\frac{5}{10}\)

\(x = 2\).

Hope it's clear.
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I apologize. I was stuck on something very dumb last night. Is there a way to change my feedback?

I think this is a high-quality question and I agree with explanation. I feel we take too long to answer this kind of question (in general), despite being simple math.

Hi Bunuel
Please identify the mistake in my approach.

A = 1/10 * 2 * 2 * 1/4 = 1/10
B = 1/5 * 2 * 1/4 = 1/10
C = 1/x * 1/4 = 1/x

1/10 + 1/10 + 1/x = 1

Of course, am not getting the answer, but what is my equation missing?

Thanks in advance.

Seeing as no one got back to you, I'll offer to clarify the error I see. For each worker you are multiplying by the time intervals worked one after the other rather than adding the amount of work done in each of these time intervals. You should add up all the times and then multiply their rates by that. It would look like this:

A = 1/10 * (2+2+1/4=17/4) = 17/40
B = 1/5 * (2+1/4=9/4) = 9/20 = 18/40
C = 1/x * 1/4 = 1/4x

A+B+C=1
17/40 + 18/40 + 1/4x = 1
35/40 + 1/4x = 1
1/4x = 5/40
Cross Multiply
20x=40
x=2

Hope it's clear now.

Assumes total work= 10 Unit

A Can complete the work in = 10 Hours or Work Completed 1 Unit/Hour
B Can complete the work in = 5 Hours or Work Completed 2 Unit/Hour
C Can complete the work in = X Hours or Work Completed 10/X Unit/Hour

As per the question- A work 2 Hours independently then B joins after 2 hours and then c joins after 2 hours.
Work Completed- 1 Unit*2+ 2 Unit*3= 8 Unit
Remaining Unit= 2 Hours

Then A+B+C work for 15 min and completed the task in 15 Min or 1/4 Hours
1 Unit/Hour+2 Unit/hour+10/x Unit/Hour= 2/1/4
Solving this, you ll get X= 2 Hours

This is an excelent question to plug in some numbers , and solve it less then a minute !

FIrst lets think about good number that the work can take, 50 ! (as a multiple of both 5, and 10).

Now if the total Job is 50 (imagine it is a big wall that need 50 litter of paint), lets find out the work speed of each worker:

A: 50/10 = 5
B: 50/5 = 10.
C: i will leave it as it is .

Now:
worker A works 4+1/4 hours *5(his work speed) = 21.25
worker B works 2+1/4 Hours * 10(his work speed) = 22.5
the work that left for worker C is 50-43.75=6.25 (because he worked only 1/4 hours lets multiple that by 4 to get is speed in a hour) *4=25 (we got now his work speed )

To caculate the time for X = 50/25= 2

I think this is a high-quality question and I agree with explanation.