4 × 2 men = 4 × 4 women = 5×4 children

\(\displaystyle \Rightarrow \) 2 men = 4 women = 5 children

Therefore, 2 men + 4 women + 10 children

= 20 children

Therefore, latex \displaystyle {{M}_{1}}{{D}_{1}}={{M}_{2}}{{D}_{2}}$

\(\displaystyle \Rightarrow \) 5 × 4 = \(\displaystyle 20\times {{D}_{2}}\) \(\displaystyle \Rightarrow {{D}_{2}}=1day\)

Alternative method

1 man can do the work in 8 days.
1 woman can do the work in 16 days.
1 child can do the work in 20 days.
2m + 4w + 10c can do the work in –
\(\displaystyle \left( {\frac{2}{8}+\frac{4}{{16}}+\frac{{10}}{{20}}} \right)\times \)d = 1
\(\displaystyle \Rightarrow \left( {\frac{1}{4}+\frac{1}{4}+\frac{1}{2}} \right)\times \)d = 1
\(\displaystyle \Rightarrow \)d = 1

Work done by 6 women in 1 day = \(\displaystyle \frac{1}{{10}}\)

Work done by 6 women in 6 days = \(\displaystyle \frac{6}{{10}}=\frac{3}{5}\)

Therefore, Remaining work = \(\displaystyle (1-\frac{3}{5})=\frac{2}{5}\) which is completed by 10 children in 6 days

Therefore, Work done by 10 children in 1 day = \(\displaystyle \frac{2}{{5\times 6}}=\frac{1}{{15}}\)

Therfore, Time taken in completing the work = 15 days.

Alternate Method:

Number of days required = \(\displaystyle \frac{{6\times 10}}{{10-6}}=\frac{{6\times 10}}{4}=15days\)

Solution for MCQ on Time and Work is option (b)

8 women can do a work in 6 days.

Therefore, 2 women can do same work in = \(\displaystyle \frac{{8\times 6}}{2}=24days\)

2 women can do \(\displaystyle \frac{1}{{24}}\) work in 1 day.

(2 women + 10 children) can do a work in 8 days.

Therefore, (2 women + 10 children)’s 1 days work \(\displaystyle \frac{1}{8}\).

10 Children 1 day work = \(\displaystyle \frac{1}{8}-\frac{1}{{24}}=\frac{1}{{12}}\) work

Hence 10 children can do same work in 12 days.

Alternate method

Consider the speed of work of women as ‘X’ and that of children be ‘Y’ and the Work as ‘W’, then

2X+10Y=W/8 and

8X=W/6

\(\displaystyle \Rightarrow \) X=W/48

Y substituting the value of X in first equation, we get

2(W/48)+10Y=W/8

Y=W/120

\(\displaystyle \Rightarrow \) Speed of 1 Child = W/120

\(\displaystyle \Rightarrow \) Speed of 10 Child = W/12

Hence 10 children can do same work in 12 days.

Solution for MCQ on Time and Work is option (c)

Let the A can do the task in x days

\(\displaystyle \frac{1}{x}+\frac{1}{{10}}=\frac{1}{8}\)

\(\displaystyle \frac{1}{x}=\frac{1}{8}-\frac{1}{{10}}=\frac{{10-8}}{{80}}=\frac{2}{{80}}=\frac{1}{{40}}\) 

 X = 40

Alternate Method

If total work = 80 units

Then in 1 day (A+B) will do 80/10 = 10 unit of work

Therefore, A does 2 unit of work each day

Hence, A requires 80/2 = 40 days to complete work

Solution for MCQ on Time and Work is option (b)

10 hr A pipe\(\displaystyle \to \)1

16 hr B pipe\(\displaystyle \to \)1

32 hr C pipe\(\displaystyle \to \)1

\(\displaystyle \frac{1}{{10}}+\frac{1}{{16}}-\frac{1}{{32}}=\frac{{21}}{{160}}\)

\(\displaystyle \frac{{160}}{{21}}=7\frac{{13}}{{21}}hr\)

Alternate Method without pen and paper

LCM of 10, 16 and 32 is 160. Therefore,

Units filled by A pipe is 16

Units filled by B pipe is 10

Units filled by B pipe is – 5

If they work together for 1 hour, they will fill (16+10-5) units= 21 units

Time required to fill the tank=\(\displaystyle \frac{{160}}{{21}}=7\frac{{13}}{{21}}hr\)

Solution for MCQ on Time and Work is option (e)

A’s 1 day’s work = \(\displaystyle \frac{1}{6}\)\(\displaystyle \frac{1}{6}\)

B’s 1 day’s work = \(\displaystyle \frac{1}{{12}}\)

Therefore, (A + B)’s 1 day’s work = \(\displaystyle \frac{1}{6}+\frac{1}{{12}}=\frac{{2+1}}{{12}}=\frac{1}{4}\)

Hence, A and B together will make 100 baskets in 4 days.

Alternate method

A alone can make 100 baskets in 6 days and B alone can make in 12 days.

Therefore, Rate of efficiency of A=100/6 baskets per day

And rate of efficiency of B=100/12 baskets per day

⇒ Baskets made by A and B together in 1 day = \(\displaystyle \frac{{100}}{6}+\frac{{100}}{{12}}=\frac{{300}}{{12}}=25\) baskets per day

Therefore, Time taken by A and B together to make 100 baskets= \(\displaystyle \frac{{100}}{{25}}\)=4 days

Solution for MCQ on Time and Work is option (c)

In 1 day (6 × 10) women can complete the piece of work and in 1 day (10 × 15) children can complete the same piece of work.

Therefore, 6 × 10 women º 10 × 15 children

\(\displaystyle \Rightarrow \) 2 women = 5 children

Hence, 6 women + 10 children = (15 + 10) children = 25 children

 \(\displaystyle {{M}_{1}}{{D}_{1}}={{M}_{2}}{{D}_{2}}\) Þ \(\displaystyle 10\times 15=25\times {{D}_{2}}\)

\(\displaystyle \Rightarrow \) \(\displaystyle {{D}_{2}}=\frac{{10\times 15}}{{25}}=6days\)