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Loom P, working alone, takes 5 hours longer than Loom Q takes to weave one bolt of cloth, and the two looms running together weave one bolt in 6 hours. On a certain day, Loom P weaves a bolt alone for the first 5 hours, after which Loom Q joins it and the two finish the bolt together. How many hours in total does it take to weave that bolt?
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Correct answer
E
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. Multiply through by 6t(t + 5): 6t + 6(t + 5) = t(t + 5), so 12t + 30 = t² + 5t, giving t² − 7t − 30 = 0, or (t − 10)(t + 3) = 0. The root t = −3 is rejected, so t = 10. Loom Q takes 10 hours alone (rate 1/10) and Loom P takes 15 hours alone (rate 1/15).
Now handle the staggered start. Loom P runs alone for the first 5 hours at rate 1/15, weaving 5 × (1/15) = 1/3 of the bolt. The remaining work is 1 − 1/3 = 2/3. Once Loom Q joins, the two work at 1/6 per hour, so the remaining 2/3 takes (2/3) ÷ (1/6) = 4 hours. Total time is 5 + 4 = 9 hours. The answer is E.
The wrong choices each mishandle one step. 4 (A) is only the finishing phase, reported without adding the 5-hour head start. 6 (B) reports the plain together-time and ignores the head start. 7 (C) treats the remaining work as 1/3 instead of 2/3, confusing the fraction done with the fraction left. 8 (D) swaps which loom runs solo first, using the faster loom for the opening 5 hours. 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. Then verify by the work done rather than by re-deriving the time: in the first 5 hours Loom P lays down 1/3 of the bolt, and over the final 4 hours the pair lay down 4 × (1/6) = 2/3; the two pieces sum to 1/3 + 2/3 = 1 whole bolt, so the 9-hour total is consistent.
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