Time and Work 1/3

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13.  A, B and C can complete a piece of work in 15, 12, 10 days respectively. A and B started the work together and left after 4 days. If the remaining work can be done by C, then in how many days the total work get completed?
a. 12 days
b. 10 days
c. 6 days
d. 8 days

Answer: D

Explanation:
Assume total work = LCM (15,12,10) = 60
The capacities of A, B, C are 4, 5, 6 respectively. (Capacity = Total work/ Days)
$
\begin{array}{c|c} \hdashline
Days \downarrow & Total work = 60 \\ &Capacities \downarrow \\[4px]
\hline
A = 15& 4 \\[4px]
B = 12& 5 \\[4px]
C = 10& 6 \\[4px] \hdashline
\end{array}
$
A and B per day work = 4 + 5 = 9.
Work completed in 4 days= $4 \times 9 = 36$.
Remaining work = $60 - 36= 24$
This work has to be done by C alone.
We know that C's efficiency is 6 units per day.
So he can complete the remaining work in $\displaystyle\frac{{24}}{6}$ days = 4 days.
Total work can be completed in $4 + 4 = 8$ days.

14.  A does half as much work as B in three-fourth of the time. If together they take 18 days to complete a work, how much time shall B take to do it ?
a. 30 days
b. 35 days
c. 40 days
d. None of these

Answer: A

Explanation:
Let us assume that $B$ did 4 units in 4 days.
Then $A$ do 2 units in 3 days.
B's capacity = $\dfrac{{Work}}{{days}} = \dfrac{4}{4} = 1$
A's capacity = $\dfrac{{Work}}{{days}} = \dfrac{2}{3}$
Total work = $\left( {1 + \dfrac{2}{3}} \right) \times 18 = \dfrac{5}{3} \times 18 = 30$
Days taken by $B$ to complete the work = $\dfrac{{Work}}{{Capacity}}$ = $\dfrac{{30}}{1} = 30$ days.

15.  A and B working separately can do a piece of work in 9 and 12 days respectively. If they work for a day alternately. If A begins first, in how many days the work will be completed ?
a. $10\displaystyle\frac{1}{2}$ days
b. $10\displaystyle\frac{1}{4}$ days
c. $10\displaystyle\frac{2}{3}$ days
d. $10\displaystyle\frac{1}{3}$ days

Answer: B

Explanation:
$
\begin{array}{c|c} \hdashline
Days \downarrow & Total work = 36 \\ &Capacities \downarrow \\[4px]
\hline
A = 9& 4 \\[4px]
B = 12& 3 \\[4px] \hdashline
\end{array}
$
Day 1, A will complete = 4 units
Day 2, B will complete = 3 units
2 days work = 7 units
10 days work = 35 units ($\because $ multipled 7 by 5 so that it is close to total work 36).
11th day, A will come for work.
Part of the day taken by him to complete the remaining 1 unit = $\dfrac{1}{4}$
Total work gets completed in $10\dfrac{1}{4}$ days.

Alternative method:
(A + B)'s 2 day's work = ${\left({\displaystyle\frac{1}{9} + \displaystyle\frac{1}{{12}}} \right) = \displaystyle\frac{7}{{36}}}$
Evidently, the work done by A and B during 5 pairs of days = ${\left({5 \times \displaystyle\frac{7}{{36}}} \right) = \displaystyle\frac{{35}}{{36}}}$
Remaining work = ${\left( {1 - \displaystyle\frac{{35}}{{36}}} \right) = \displaystyle\frac{1}{{36}}}$
Now, on 11th day it is A's turn.
Now ${\displaystyle\frac{1}{9}}$ work is done by A in 1 day.
Days required for A to complete the remaining work = $\dfrac{\text{Remaining work}}{{Capacity}}$ = $\dfrac{{1/36}}{{1/9}}$ = $\dfrac{9}{{36}}$ = $\dfrac{1}{{4}}$.
So, total time taken = ${10\displaystyle\frac{1}{4}}$ days.

16.  A, B and C together earn Rs.150 per day while A and C together earn Rs.94 and B and C together earn Rs.76. The daily earning of C is :
a. Rs.75
b. Rs.56
c. Rs.34
d. Rs.20

Answer: D

Explanation:
B's daily earning = Rs.(150 - 94) = Rs.56
A's daily earning = Rs.(150 - 76) = Rs.74
C's daily earning = Rs.(150 - (56 + 74)) = Rs.20

17.  A, B and C contract a work for Rs.550. Together A and B are to do $\displaystyle\frac{7}{{11}}$ of the work. The share of C should be :
a. Rs.$183\displaystyle\frac{1}{3}$
b. Rs.200
c. Rs.300
d. Rs.400

Answer: B

Explanation:
Work to be done by C = $\left( {1 - \displaystyle\frac{7}{{11}}} \right) = \displaystyle\frac{4}{{11}}$
(A + B) : C = $\displaystyle\frac{7}{{11}}:\displaystyle\frac{4}{{11}} = 7:4$
C's share = Rs.$\left( {550 \times \displaystyle\frac{4}{{7 + 4}}} \right)$ = Rs. 200

18.  Two men undertake to do a piece of work for Rs.400. One alone can do it in 6 days, the other in 8 days. With the help of a boy, they finish it in 3 days. The boy's share is
a. Rs.40
b. Rs.50
c. Rs.60
d. Rs.80

Answer: B

Explanation:
One man's 1 day's work = $\displaystyle\frac{1}{6}$
Another man's 1 day's work = $\displaystyle\frac{1}{8}$
Boy's 1 day's work = $\displaystyle\frac{1}{3} - \left( {\displaystyle\frac{1}{6} + \displaystyle\frac{1}{8}} \right) = \displaystyle\frac{1}{{24}}$
Ratio of their shares = ${\displaystyle\frac{1}{6}:\displaystyle\frac{1}{8}:\displaystyle\frac{1}{{24}}}$ = $4:3:1$
Boy's share = Rs. $\left( {400 \times \displaystyle\frac{1}{(4 + 3 + 1)}} \right) = Rs.50$

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