M05-21

Question

Otto takes 6 days to complete a research project. If Otto and Hannah, working together, can complete the same project in 4 days, how many days would it take Hannah to finish the project on her own?

A. 6
B. 8
C. 10
D. 12
E. 14

Bunuel · Accepted Answer

Official Solution:

Otto takes 6 days to complete a research project. If Otto and Hannah, working together, can complete the same project in 4 days, how many days would it take Hannah to finish the project on her own?

A. 6
B. 8
C. 10
D. 12
E. 14

Otto takes 6 days to finish the project. Hence, Otto's rate is  \frac{1}{6}  of the project per day.
 
Let's assume Hannah takes  x  days to finish the project. Therefore, her rate is  \frac{1}{x}  of the project per day.
 
When they work together, they complete the project in 4 days, giving a combined rate of  \frac{1}{4}  of the project per day.
 
Thus,  \frac{1}{6}+\frac{1}{x}=\frac{1}{4} ;
 
 \frac{1}{x}=\frac{1}{12} ;
 
 x=12  days.

Answer: D

kimnguyen · Answer

Without any equation:
In one day Otto can do 1/6 Project, then in 4 days Otto can do 4/6 or 2/3 Project.
The remain 1/3 Project is made by Hannah.
If Hannah can do 1/3 Project in 4 days, then she can do the whole project in 12 days.

AndrewN · Answer

An alternative approach  - With these sorts of work/rate problems, you can also supply an imaginary target to qualify "a research project." In this case, the two numbers we have are 6 and 4, which cross paths at 12 (and 24, 36, 48, and so on). We can just as easily think of the project as a 12-page report. Then, we can reinterpret the first bit of information in the following manner:

It takes Otto 6 days to finish a 12-page report

and the second as saying

Otto and Hannah  (does someone have an affinity for palindromes here?)  complete a 12-page report in 4 days .

In the former case, 6 days for 12 pages would reduce to 2 pages per day (by dividing by 6), so we have Otto figured out, in terms of productivity. In the latter case, Otto and Hannah together complete 3 pages per day (by dividing by 4). Putting the two pieces of information together, we can see quite clearly that Hannah is adding 1 page per day to what Otto completes on his own. Thus, to "finish the project by herself" and complete the 12-page report at a rate of 1 page per day, it would take Hannah 12 days to do so. Any way you look at it, choice (D) will be the answer.

If an algebraic solution eludes you, try a more intuitive approach with numbers. If you can get the answer just the same in about as much time, who cares? (You do not have to tell anyone your secret when you get a higher Quant score.)

Good luck with your studies.

- Andrew

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M05-21

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