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Arithmetic aptitude : Time & Work (MCQ Questions)

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 Question 1
P alone can do a piece of work in 8 days and Q alone in 6 days. P and Q undertook to do it for Rs. 4000. With the help of R, they required 3 days to finish the work. How much is to be paid to R?
 A 500 B 800 C 600 D 400
Question 1 Explanation:
\begin{aligned} R's 1 day's work = \frac{1}{3} - \left ( \frac{1}{8} + \frac{1}{6}\right) = \frac{1}{3} - \frac{7}{24} = \frac{1}{24} \\ \end{aligned} \begin{aligned} P's Wages : Q's Wages : R's Wages = \frac{1}{8} : \frac{1}{6} : \frac{1}{24} = 3:4:1 \\ \end{aligned} \begin{aligned} \therefore R's share for 3 days = Rs. \left[ 3 * \frac{1}{24} * 4000 \right] = 500 \\ \end{aligned}
 Question 2
P can do a work in 30 days and Q in 40 days. If they work on it together for 8 days, then the fraction of the work that is left is :
 A \begin{aligned} \frac{2}{3} \end{aligned} B \begin{aligned} \frac{9}{10} \end{aligned} C \begin{aligned} \frac{1}{3} \end{aligned} D \begin{aligned} \frac{23}{30} \end{aligned}
Question 2 Explanation:
\begin{aligned} P's 1 day's work = \frac{1}{30} \\ \end{aligned} \begin{aligned} Q's 1 day's work = \frac{1}{40} \\ \end{aligned} \begin{aligned} \left(P + Q\right)'s 1 day's work = \left(\frac{1}{30}+ \frac{1}{40}\right)=\frac{7}{120} \\ \end{aligned} \begin{aligned} \left(P + Q\right)'s 4 day's work = \left(\frac{7}{120} * 4\right)=\frac{7}{30} \\ \end{aligned} \begin{aligned} \therefore Remaining Work = \left(1 - \frac{7}{30}\right)=\frac{23}{30} \\ \end{aligned}
 Question 3
P can lay railway track between two given stations in 20 days and Q can do the same job in 15 days. With help of R, they did the job in 6 days only. Then how much time is taken by R alone to do this job?
 A \begin{aligned} 7\frac{1}{2} days \end{aligned} B 7 days C \begin{aligned} 7\frac{3}{2} days \end{aligned} D 14 days
Question 3 Explanation:
\begin{aligned} \left(P + Q + R\right)'s\; 1\; day's\; work\; = \frac{1}{6} \\ \end{aligned} \begin{aligned} P's\; 1\; day's\; work\; = \frac{1}{20} \\ \end{aligned} \begin{aligned} Q's\; 1\; day's\; work\; = \frac{1}{15} \\ \end{aligned} \begin{aligned} Q's \; 1\; day's\; work\; = \frac{1}{15} \\ \end{aligned} \begin{aligned} R's\; 1\; day's\; work\; = \frac{1}{6} - \left(\frac{1}{20} + \frac{1}{15}\right) = \left(\frac{1}{4} - \frac{7}{60}\right) = \frac{2}{15} \\ \end{aligned} \begin{aligned} \therefore\; R\; alone\; can\; do\; the\; work\; in\; \frac{15}{2} = 7\frac{1}{2} days. \end{aligned}
 Question 4
P, Q and R can do a piece of work in 40, 60 and 120 days respectively. In how many days can P do the work if he is assisted by Q and R on every fourth day?
 A \begin{aligned} 27\frac{2}{3} days \end{aligned} B \begin{aligned} 26\frac{2}{3} days \end{aligned} C \begin{aligned} \frac{3}{5} days \end{aligned} D \begin{aligned} 7\frac{20}{3} days \end{aligned}
Question 4 Explanation:
\begin{aligned} P's\; 3\; day's\; work = \left(\frac{1}{30} * 3\right) = \frac{1}{10} \\ \end{aligned} \begin{aligned} \left(P+Q+R\right)'s\; 1\; day's\; work = \left(\frac{1}{40} + \frac{1}{60} + \frac{1}{120}\right) = \frac{1}{20} \\ \end{aligned} \begin{aligned} Work\; done\; in\; 4\; day's\; = \left(\frac{1}{10} + \frac{1}{20} \right) = \frac{3}{20} \\ \end{aligned} \begin{aligned} Now,\; \frac{3}{20}\; work\; is\; done\; in\; 4\; days. \\ \end{aligned} \begin{aligned} \therefore Whole\; work\; will\; be\; done\; in\; \left(\frac{20}{3} * 4\right) = 26\frac{2}{3}\; days\\ \end{aligned}
 Question 5
P is four times as good as workman as Q and therefore is able to finish a job in 40 days less than Q. Working together, they can do it in:
 A 18 days B 16 days C 22 days D 20 days
Question 5 Explanation:
\begin{aligned} Ratio\; of\; times\; taken\; by\; P\; and\; Q = 1 : 4 \\ \end{aligned} \begin{aligned} The\; time\; difference\; is\; (4 - 1) 3\; days\; while\; Q\; take\; 4\; days\; and\; P\; takes\; 1\; day \\ \end{aligned} \begin{aligned} If\; difference\; of\; time\; is\; 3\; days,\; Q\; takes\; 4\; days. \\ \end{aligned} \begin{aligned} If\; difference\; of\; time\; is\; 40\; days,\; Q\; takes\; \left(\frac{4}{2} *40\right) = 80 days. \\ \end{aligned} \begin{aligned} So,\; P\; takes\; 20\; days\; to\; do\; the\; work. \\ \end{aligned} \begin{aligned} P's\; 1\; day's\; work\; =\; \frac{1}{20}\\ \end{aligned} \begin{aligned} Q's\; 1\; day's\; work\; =\; \frac{1}{80}\\ \end{aligned} \begin{aligned} (P + Q)'s\; 1\; day's\; work\; =\; \left(\frac{1}{20} + \frac{1}{80} \right) = \frac{1}{16} \\ \end{aligned} \begin{aligned} P\; and\; Q\; together\; can\; do\; the\; work\; in\; 16\; days. \\ \end{aligned}
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