CSAT Solved Papers/ 2025/Q59

2025 CSAT — Q59

Quant Arithmetic & numeracy 2.5 marks Hard

$X$ 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. $Y$ alone finishes the remaining work in $8\tfrac{2}{3}$ days. What is $n$ equal to?

A 3
B 4 Answer
C 5
D 6

Worked rationale

Convert each “partial work in $d$ days” into a per-day rate (total work $= 1$ ):

X = \frac{1/3}{6} = \frac{1}{18},\qquad Y = \frac{1/3}{8} = \frac{1}{24},\qquad Z = \frac{3/4}{12} = \frac{1}{16}.

Combined rate (LCM denominator $144$ ):

X+Y+Z = \frac{8}{144} + \frac{6}{144} + \frac{9}{144} = \frac{23}{144}\ \text{per day}.

After $n$ days together, work done $= \dfrac{23n}{144}$ . $Y$ alone then finishes the rest in $8\tfrac23 = \dfrac{26}{3}$ days at rate $\dfrac1{24}$ :

Y\text{'s share} = \frac{26}{3}\cdot\frac{1}{24} = \frac{26}{72} = \frac{13}{36} = \frac{52}{144}.

The two pieces complete the whole:

\frac{23n}{144} + \frac{52}{144} = 1 \;\Longrightarrow\; 23n = 144 - 52 = 92 \;\Longrightarrow\; n = 4.

Answer: (b) 4.

Why the other options miss

A

an arithmetic slip: a slip in converting $8\tfrac23$ to $\tfrac{26}{3}$ (e.g. uses $\tfrac{24}{3}$ ), making $Y$ ‘s share too small and $n$ too small.
C

a rate mishandled: takes $Z = \tfrac{3/4}{12}$ as $\tfrac34\cdot12$ inverted, inflating the combined rate.
D

solved the wrong question: reads the rates as “completes the whole work in $6/8/12$ days,” ignoring the one-third / three-fourth qualifiers.

Specialist insight

The entire trap is in the rate conversion: the data gives time for a fraction of the work, not the whole, so $X$ ‘s rate is $\tfrac{1/3}{6}$ — not $\tfrac16$ . Candidates who read ” $6/8/12$ days for the work” get a clean-looking but wrong system. Once the three rates are correct, the structure is a standard “work together, then one finishes” balance: done-together $+$ done-by- $Y$ $= 1$ . Convert the mixed number $8\tfrac23 = \tfrac{26}{3}$ carefully and keep a common denominator of $144$ , and $n=4$ falls out in one line.

The trap, in one line

The times are for fractions of the work: $X=\tfrac{1/3}{6}=\tfrac1{18}$ , etc. — then $\tfrac{23n}{144} + \tfrac{52}{144} = 1$ gives $n=4$ .

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