Arithmetic - CAT Quant LaTeX Source

IntermediateHardTITA Source: LaTeX

Arithmetic/Time & Work

Question

[Arithmetic -- Time & Work, Hard, TITA]

Working alone, $A$ and $B$ can complete a certain piece of work in 20 days and 30 days respectively. They are assigned to work together every day, except that $A$ takes leave on every 5th day (i.e.\ on days $5, 10, 15, \dots$ ); on such days only $B$ works. Starting on Day 1, the smallest positive integer $N$ such that the work is completely finished by the end of day $N$ is ____.

Answer

$N = 14$ .

Detailed solution

[Work with leave schedule]

Daily rates: $A = \tfrac{1}{20},\ B = \tfrac{1}{30}$ . Joint rate (no leave) $= \tfrac{1}{12}$ . On leave days only $B$ : rate $\tfrac{1}{30}$ .

Cumulative work at end of each day (in units of $\tfrac{1}{60}$ ):

\begin{array}{c|cccccccccccccc} \text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \text{Cum.} & 5 & 10 & 15 & 20 & 22 & 27 & 32 & 37 & 42 & 44 & 49 & 54 & 59 & 64 \end{array}

(Each non-leave day adds $5$ ; leave days $5, 10$ add $2$ .) The total work corresponds to $60$ . The cumulative crosses $60$ during Day 14. Hence the work is completely finished by the end of Day 14. Answer: $N = 14$ .

Why CAT-level: The trap is to compute average daily rate over a 5-day block, miss the residual, and undercount.

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