GMAT 700 800 Level Quant Question 6 | Rates Word Problem

This one is a word problem in Rates - Work and Time. Word problems test your ability to translate information given in words into mathematical expressions and equations and solve them. The gist of what is tested is the ability to translate effectively.

Question 6: Working alone at their respective constant rates, A can complete a task in ‘a’ days and B in ‘b’ days. They take turns in doing the task with each working 2 days at a time. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more. How long does it take to complete the task if they both work together?

1. \\frac{\text{46}}{\text{9}})
2. \\frac{\text{50}}{\text{9}})
3. \\frac{\text{50}}{\text{11}})
4. \\frac{\text{36}}{\text{7}})
5. \\frac{\text{210}}{\text{41}})

Video Explanation

Explanatory Answer | GMAT Rates - Work and Time

Working alone at a constant rate if A takes 'a' days to complete a task, A will complete \\frac{\text{1}}{\text{a}}) of the task in a day.

Step 1: Translate words into mathematical expressions

A will complete \\frac{\text{1}}{\text{a}}) of the task in a day.
Therefore, in 2 days A will complete \\frac{\text{2}}{\text{a}}) of the task in a day.
Similarly, A will complete \\frac{\text{5}}{\text{a}}) of the task in 5 days.

If A starts, they finish the task in exactly 10 days.

A starts and works for 2 days. So, A will work on day 1 and day 2.
Then B will work for the next 2 days. B will work on day 3 and day 4.
A will continue for the next 2 days. i.e., on day 5 and day 6.
B will work on day 7 and day 8.
A will for the last 2 days i.e., day 9 and day 10.
Therefore, A will work on day 1, day 2, day 5, day 6, day 9, and day 10. i.e., for 6 days.
And B will work on day 3, day 4, day 7, and day 8. i.e., for 4 days.

In 6 days A will complete \\frac{\text{6}}{\text{a}}) of the task.
In 4 days B will complete \\frac{\text{4}}{\text{b}}) of the task.

With A working 6 days and B working 4 days, the task is completed.
i.e., \\frac{\text{6}}{\text{a}}) + \\frac{\text{4}}{\text{b}}) = 1 .... eqn (1).

If B starts, they finish the task they take half a day more. i.e., 10.5 days.

Therefore, B will work on day 1, day 2, day 5, day 6, day 9, and day 10. i.e., for 6 days.
And A will work on day 3, day 4, day 7, day 8 and half a day on day 11. i.e., for 4.5 days.

In 6 days B will complete \\frac{\text{6}}{\text{b}}) of the task.
In 4 days A will complete \\frac{\text{4.5}}{\text{a}}) of the task.

With B working 6 days and A working 4.5 days, the task is completed.
i.e., \\frac{\text{4.5}}{\text{a}}) + \\frac{\text{6}}{\text{b}}) = 1 .... eqn (2).

Step 2: Solve the two equations

\\frac{\text{6}}{\text{a}}) + \\frac{\text{4}}{\text{b}}) = 1 .... eqn (1)
\\frac{\text{4.5}}{\text{a}}) + \\frac{\text{6}}{\text{b}}) = 1 .... eqn (2)
Solving the two equations we get a = 9 days and b = 12 days.

The question: How long does it take to complete the task if they both work together?

Working together A and B will complete \\frac{\text{1}}{\text{9}}) + \\frac{\text{1}}{\text{12}})= \\frac{\text{4 + 3}}{\text{36}}) = \\frac{\text{7}}{\text{36}}) th of the task in a day.
Hence, they will complete the task in \\frac{\text{36}}{\text{7}}) days.

Choice D is the correct answer.