P and Q together can complete a project in 12 days, and Q and R together can complete it in 10 days. R works one and a half times as fast as P. How many days would Q alone take to complete the project?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 20
Write each person’s one-day work (rate) and use the link between R and P.
Let P do \(p\) of the project in a day; then R does \(1.5p\)
P and Q: \(\displaystyle p+q=\frac{1}{12}\)
Q and R: \(\displaystyle q+1.5p=\frac{1}{10}\)
Subtracting: \(\displaystyle 0.5p=\frac{1}{10}-\frac{1}{12}=\frac{6-5}{60}=\frac{1}{60}\), so \(\displaystyle p=\frac{1}{30}\)
\(\displaystyle q=\frac{1}{12}-\frac{1}{30}=\frac{5-2}{60}=\frac{1}{20}\)
Q alone would take 20 days.
Why the other options are incorrect
Option (1) 15 — Incorrect. Then P\(=\frac{1}{12}-\frac{1}{15}=\frac{1}{60}\) and R\(=\frac{1}{10}-\frac{1}{15}=\frac{1}{30}\): R would be twice as fast as P, not 1.5 times.
Option (3) 24 — Incorrect. Then P\(=\frac{1}{24}\) and R\(=\frac{1}{10}-\frac{1}{24}=\frac{7}{120}\), which is 1.4 times P.
Option (4) 30 — Incorrect. Then P\(=\frac{1}{12}-\frac{1}{30}=\frac{1}{20}\) and R\(=\frac{1}{10}-\frac{1}{30}=\frac{1}{15}\), so R would be only \(\frac{4}{3}\) times as fast as P, not 1.5 times.