UPSC Prelims 2025 CSAT, question 59

Topic: Arithmetic

Q59

$X$ can complete one-third of a certain work in 6 days, $Y$ can complete one-third of the same work in 8 days and $Z$ can complete three-fourth of the same work in 12 days. All of them work together for $n$ days and then $X$ and $Z$ quit and $Y$ alone finishes the remaining work in $8\frac{2}{3}$ days. What is $n$ equal to ?

(a) 3
(b) 4Correct answer
(c) 5
(d) 6

Solution.

Answer: (b) 4

$X$ completes the whole work in $6 \times 3 = 18$ days, $Y$ in $8 \times 3 = 24$ days, and $Z$ in $12 \times \frac{4}{3} = 16$ days.
In $8\frac{2}{3} = \frac{26}{3}$ days alone, $Y$ completes $\frac{26/3}{24} = \frac{26}{72} = \frac{13}{36}$ of the work, so all three working together for $n$ days completed $1 - \frac{13}{36} = \frac{23}{36}$ of the work.
Their combined daily rate is $\frac{1}{18} + \frac{1}{24} + \frac{1}{16} = \frac{8 + 6 + 9}{144} = \frac{23}{144}$ of the work per day, so $n = \frac{23/36}{23/144} = 4$ days.