Free GMAT Problem Solving Practice Question

If the two looms together weave one bolt in 6 hours, their combined rate is 1/6 of a bolt per hour.

Let Loom Q take t hours alone, so Loom P takes t + 5 hours alone. Rates add: 1/(t + 5) + 1/t = 1/6. Multiplying both sides by 6t(t + 5) gives 6t + 6(t + 5) = t(t + 5), so 12t + 30 = t² + 5t, which rearranges to t² − 7t − 30 = 0, or (t − 10)(t + 3) = 0. The root t = −3 is rejected because a time cannot be negative, so t = 10. Loom Q takes 10 hours alone, giving it a rate of 1/10, and Loom P takes 15 hours alone, giving it a rate of 1/15.

Now handle the staggered start described in the problem. Loom P runs alone for the first 5 hours at its rate of 1/15, weaving 5 × (1/15) = 1/3 of the bolt. The work still remaining is 1 − 1/3 = 2/3. Once Loom Q joins, the two work together at the combined rate of 1/6 per hour, so the remaining 2/3 takes (2/3) ÷ (1/6) = 4 hours. Adding the two phases gives a total time of 5 + 4 = 9 hours. So the answer is E. Plugging answer choices back in is no shortcut, since checking any total still requires solving the quadratic and redoing the staggered-completion logic.

As an independent check, confirm the solo rates against the given joint time: 1/15 + 1/10 = 2/30 + 3/30 = 5/30 = 1/6, which matches the stated 6-hour together-time, so the rates 1/15 and 1/10 are correct. You can also verify by tracking the work done rather than the time: in the first 5 hours Loom P lays down 1/3 of the bolt, and over the final 4 hours the pair together lay down 4 × (1/6) = 2/3; the two pieces sum to 1/3 + 2/3, which is the whole bolt, so the 9-hour total is consistent.