1. $A$ can do a piece of work in $10$ days while $B$ can do it in $15$ days. In how many days can $A$ and $B$ working together do it ?
a. 25 days
b. 12.5 days
c. 9 days
d. 6 days
Answer: D
Explanation:
Part of the work done by $A$ per day $ = \dfrac{1}{{10}}$
($\because$ We are assuming that total work = $1$)
Part of the work done by $B$ per day $ = \dfrac{1}{{15}}$
Part of the work done by $(A + B)$ per day = $\left( {\displaystyle\frac{1}{{10}} + \displaystyle\frac{1}{{15}}} \right) = \displaystyle\frac{1}{{6}}$
Days required for both persons to finish the work = $\dfrac{{Total\; work}}{{Part\; of\; work\; per\; day}} $ $= \dfrac{1}{{\left( {1/6} \right)}}$ $ = 6$ days.
Alternative method 1:
Let us assume that the total work $= \text{LCM}(10,\; 15) = 30$ units.
Capacity of $A$ = 30/10 = 3
Capacity of $B$ = 30/15 = 2
$\biggl(\because Capacity = \dfrac{{Total\; work}}{{Days}}\biggr)$
$
\begin{array}{c|c} \hdashline
Days \downarrow & Total work = 30 \\ &Capacities \downarrow \\[4px]
\hline
A = 10& 3 \\[4px]
B = 15& 2 \\[4px] \hdashline
\end{array}
$
$A$ and $B$ combined capacity $= 3 + 2 = 5$ units.
Days required by A and B to complete the work = $\dfrac{{30}}{5} = 6$ days.
Alternative method 2:
To save time, the following formula is useful. $\dfrac{{xy}}{{x + y}}$. Here, x and y are the times taken by A and B.
So Time taken by both to finish the work = $\dfrac{{10 \times 15}}{{10 + 15}}$ = $\dfrac{{150}}{{25}} = 6$
2. A and B can do a piece of work in 12 days; B and C in 15 days; C and A in 20 days. A alone can do the work in :
a. $15\displaystyle\frac{2}{3}$ days
b. 24 days
c. 30 days
d. 40 days
Answer: C
Explanation:
A's 1 day's work $= (A + B + C)$'s one day's work $- (B + C)$'s one day's work
(A + B)'s one day work = $\dfrac{1}{{12}}$
(B + C)'s one day work = $\dfrac{1}{{15}}$
(C + A)'s one day work = $\dfrac{1}{{20}}$
$\bigl((A + B ) + (B + C) + ( C + A )\bigr)$'s one day work = $2(A + B + C)$'s one day work = $\left( {\displaystyle\frac{1}{{12}} + \displaystyle\frac{1}{{15}} + \displaystyle\frac{1}{{20}}} \right)$$= \dfrac{{5 + 4 + 3}}{{60}}$ = $\displaystyle\frac{1}{5}$
$2(A + B + C)$'s one day work = $\displaystyle\frac{1}{{5}}$
$(A + B + C)$'s one day work = $\displaystyle\frac{1}{{10}}$
A's one day work $= (A + B + C)$'s one day's work $- (B + C)$'s one day's work = $\left({\displaystyle\frac{1}{{10}} - \displaystyle\frac{1}{{15}}} \right) = \displaystyle\frac{1}{{30}}$
A alone can finish it in 30 days.
Alternative method:
$
\begin{array}{c|c} \hdashline
Days \downarrow & Total work = 60 \\ &Capacities \downarrow \\[4px]
\hline
A+B = 12& 5 \\[4px]
B + C = 15& 4 \\[4px]
C + A = 20& 3 \\[4px]\hdashline
2(A+B+C) & 12
\end{array}
$
$A + B + C$ $\frac{{12}}{2} = 6$ units.
A's Capacity $= (A + B + C) - (B + C)$ $= 6 - 4 = 2$
Days required by A to complete the work = \(\dfrac{{60}}{2} = 30\) days.
3. A can do $\left( {\displaystyle\frac{1}{3}} \right)$ of a work in 5 days and B can do $\left( {\displaystyle\frac{2}{5}} \right)$ of the work in 10 days. In how many days both A and B together can do the work ?
a. $7\displaystyle\frac{3}{4}$
b. $8\displaystyle\frac{4}{5}$
c. $9\displaystyle\frac{3}{8}$
d. 10
Answer: C
Explanation:
$\displaystyle\frac{1}{3}$ work is done by A in 5 days.
Whole work will be done by A in $ 3 \times 5 = 15$ days.
$\displaystyle\frac{2}{5}$ of work is done by B in 10 days.
Whole work will be done by B in $\left({10 \times \displaystyle\frac{5}{2}} \right)$ i.e. 25 days
(A+B)'s 1 day's work = $\left( {\displaystyle\frac{1}{{15}} + \displaystyle\frac{1}{{25}}} \right) = \displaystyle\frac{8}{{75}}$
So, both together can finish it in $\displaystyle\frac{{75}}{8}$ days i.e. $9\displaystyle\frac{3}{8}$ days.
4. A and B can together do a piece of work in 15 days. B alone can do it in 20 days. In how many days can A alone do it ?
a. 30 days
b. 40 days
c. 45 days
d. 60 days
Answer: D
Explanation:
$A$'s one day work = one day work of $(A+B) -$ one day work of $A$ = $\left( {\displaystyle\frac{1}{{15}} - \displaystyle\frac{1}{{20}}} \right) = \displaystyle\frac{1}{{60}}$
A alone can finish it in 60 days
5. A alone can finish a work in 10 days and B alone can do it in 15 days. If they work together and finish, then out of a total wages of Rs.75. A will get :
a. Rs.30
b. Rs.37.50
c. Rs.45
d. Rs.50
Answer: C
Explanation:
Ratio of time taken by A and B= 10 : 15 = 2 : 3
Ratio of work done in the same time = 3:2, So, the money is to be divided among A and B in the ratio 3 : 2
A's share = Rs. ${\left( {75 \times \displaystyle\frac{3}{5}} \right)}$ = Rs.45.
6. A can do a certain job in 12 days. B is 60% more efficient than A. The number of days it takes for B to do the same piece of work, is :
a. 6
b. $6\displaystyle\frac{1}{4}$
c. $7\displaystyle\frac{1}{2}$
d. 8
Answer: C
Explanation:
Let us assume that A's capacity = 100
B's capacity = 160%(100) = 160
$
\begin{array}{l|c} \hdashline
Days \downarrow & Capacities \downarrow \\[4px]
\hline
A = 12& 100 \\[4px]
B = x& 160 \\[4px] \hdashline
\end{array}
$
Total work = Days $\times$ Capacity
$\therefore$ $12\times 100 = x \times 160$
$ \Rightarrow x = \dfrac{{1200}}{{160}} = 7\dfrac{1}{2}$
