Time and Work 1/2

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7.  A is thrice as good a work man as B and takes 10 days less to do a piece of work than B takes. B can do the work in :
a. 12 days
b. 15 days
c. 20 days
d. 30 days

Answer: B

Explanation:
If the capacity of B is one, then Capacity of A is three.
we know that days are inversely proportional to capacities.
So days are in the ratio 1 : 3
Let us assume that the days are $x$ and $3x$ respectively for A and B.
$
\begin{array}{c|c} \hdashline
Days \downarrow & Capacities \downarrow \\[4px]
\hline
A = x& 3 \\[4px]
B = 3x& 1 \\[4px] \hdashline
\end{array}
$
Given, $3x - x = 10$
$ \Rightarrow x = 5$
B's days to complete the work = $3x$ = $3 \times 5$ = $15$

8.  A can complete a job in 9 days B in 10 days and C in 15 days. B and C start the work and are forced to leave after 2 days. The time taken to complete the remaining work is :
a. 6 days
b. 9 days
c. 10 days
d. 13 days

Answer: A

Explanation:
(B + C)'s 2 days work = ${2\left( {\displaystyle\frac{1}{{10}} + \displaystyle\frac{1}{{15}}} \right) = \displaystyle\frac{1}{3}}$
Remaining work = ${\left( {1 - \displaystyle\frac{1}{3}} \right) = \displaystyle\frac{2}{3}}$
${\displaystyle\dfrac{1}{9}}$ work is done by A in 1 day
Time required by A to complete ${\displaystyle\frac{2}{3}}$ work = $\dfrac{{2/3}}{{1/9}}$ = $6$ days

9.  A completes a work in 6 days, B works $1\displaystyle\frac{1}{2}$ times as fast as A. How many days it will take for A and B together to complete the work ?
a. $4\displaystyle\frac{7}{{12}}$
b. $3\displaystyle\frac{5}{{12}}$
c. $4\displaystyle\frac{4}{5}$
d. None of these

Answer: C

Explanation:
B's capacity $1\displaystyle\frac{1}{2}$ or $\dfrac{3}{2}$ times of A.
B will take $\dfrac{2}{3}$ time taken by A.
B's time = $\dfrac{2}{3} \times 6 = 4$ days.
Days required for A and B together to complete the work = $\dfrac{{2xy}}{{x + y}}$ = $\dfrac{{2 \times 6 \times 4}}{{6 + 4}}$ = $4\dfrac{4}{5}$

10.  Twelve men can complete a work in 8 days. Three days after they started the work, 3 more men joined them. In how many days will all of them together complete the remaining work ?
a. 2
b. 4
c. 5
d. 6

Answer: B

Explanation:
Let us assume 1 man's one day's work = $1$ unit.
Work completed by 12 men in 8 days = $12 \times 1 \times 8$ = $96$ units.
Work completed by 12 men in 3 days = $12 \times 1 \times 3$ = $36$ units.
Remaining work = $96 - 36$ = $60$ units.
Total men after 3 days = 12 + 3 = 15.
Capacity of 15 men = $15 \times 1 = 15$ units.
Days required for 15 men to complete 60 units = $\dfrac{{60}}{{15}} = 4$ days.

Alternative method:
12 men's 3 day's work = $\left({3 \times \displaystyle\frac{1}{8}} \right) = \displaystyle\frac{3}{8}$
Remaining work = $\left( {1 - \displaystyle\frac{3}{8}} \right) = \displaystyle\frac{5}{8}$
15 men's 1 day's work = $\displaystyle\frac{{15}}{{96}}$ Now, $\displaystyle\frac{{15}}{{96}}$ work is done by them in 1 day,
Days required to complete ${\displaystyle\frac{5}{8}}$ work = $\dfrac{{5/8}}{{15/96}}$ = $\left( {\displaystyle\frac{{96}}{{15}} \times \displaystyle\frac{5}{8}} \right)$ = $4$ days

11.  A and B can complete a work in 10 days and 15 days respectively. B starts the work and after 5 days A also joins him. In all, the work would be completed in :
a. 7 days
b. 9 days
c. 11days
d. None of these

Answer: B

Explanation:
B's 5 day's work = $5 \times \displaystyle\frac{1}{{15}} = \displaystyle\frac{1}{3}$
Remaining work = $\left( {1 - \displaystyle\frac{1}{3}} \right) = \displaystyle\frac{2}{3}$
A and B combined work in 1 day = $\left( {\displaystyle\frac{1}{{10}} + \displaystyle\frac{1}{{15}}} \right)$ = $\dfrac{1}{6}$
Days required to complete $\displaystyle\frac{2}{3}$ work = $\dfrac{{2/3}}{{1/6}} = \dfrac{2}{3} \times \dfrac{6}{1} = 4$ days
Hence the work was completed in 9 days.

12.  A can do a piece of work in 80 days. He works at it for 10 days and then B alone finishes the work in 42 days. The two together could complete the work in :
a. 24 days
b. 25 days
c. 30 days
d. 35 days

Answer: C

Explanation:
A's 10 day's work = $\left( {10 \times \displaystyle\frac{1}{{80}}} \right) = \displaystyle\frac{1}{8}$
Remaining work = $\left( {1 - \displaystyle\frac{1}{8}} \right) = \displaystyle\frac{7}{8}$
$\displaystyle\frac{7}{8}$ work is done by A in 42 days
Days required by A to complete the whole work = $\dfrac{{42}}{{7/8}}$ = $\left( {42 \times \displaystyle\frac{8}{7}} \right)$ i.e.48 days
(A + B)'s 1 day's work = $\left({\displaystyle\frac{1}{{80}} + \displaystyle\frac{1}{{48}}} \right) = \displaystyle\frac{8}{{240}} = \displaystyle\frac{1}{{30}}$
Hence A and B together can finish it in 30 days.

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