3 Mark Question

Question 13 Marks

A and B, working together can finish a piece of work in 6 days, while A alone can do it in 9 days. How much time will B alone take to finish it?

Answer

A and B both's one days work $=\frac{1}{6}$
As alone's one days work $=\frac{1}{9}$
$\therefore$ B's alone one days work $=\frac{1}{6}-\frac{1}{9}$
$=\frac{3-2}{18}=\frac{1}{18}$
$\therefore$ B can finish the work in 18 days.

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Question 23 Marks

Pipe A can fill an empty tank in 5 hours while pipe B can empty the full tank in 6 hours. If both are opened at the same time in the empty tank, how much time will they take to fill it up completely?

Answer

Pipe A's one hour's work for filling the tank $=\frac{1}{5}$
Pipe B's one hour's work for emptying $=\frac{1}{6}$
Both pipes are opened, then,
Both's one hour's work $=\frac{1}{5}-\frac{1}{6}$
$=\frac{6-5}{30}=\frac{1}{30}$
$\therefore$ The empty tank will be filled in 30 hours.

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Question 33 Marks

Ravi can do a piece of work in 15 hours while Raman can do it in 12 hours. How long will both take to do it, working together?

Answer

Ravi's one hours $=\frac{1}{15}$
Both's one day's work $=\frac{1}{12}$
Both can finish the work in,
$\frac{20}{3}=6\frac{2}{3}\ \text{hours}$
Or 6 hours, 40 minutes.

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Question 43 Marks

A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern fills up in 10 hours. If the cistern is full, in how much time will it be emptied by the leak?

Answer

Time taken by the pipe to fill the cistern = 9 hours.
Part of the cistern filled in one hour $=\frac{1}{9}$
Suppose the leak empties the full cistern in x hours.
Part of the cistern emptied in one hour $=-\frac{1}{\text{x}}$ (negative sign implies a leak)
Time taken by the cistern to fill completely due to the leak = 10 hours.
Part of the cistern filled in one hour due to the leak $=\frac{1}{10}$
$\therefore\frac{1}{10}=\frac{1}{9}-\frac{1}{\text{x}}$
$\Rightarrow\frac{1}{\text{x}}=\frac{1}{9}-\frac{1}{10}=\frac{1}{90}$
$\text{x}=90\ \text{hours}.$
Therefore, the leak will empty a full cistern in 90 hours.

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Question 53 Marks

A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern fills up in 10 hours. If the cistern is full, in how much time will it be emptied by the leak?

Answer

The inlet pipe's 1 hour's work $=\frac{1}{9}$
The leak and inlet's 1 hours work $=\frac{1}{10}$
Leak's 1 hour work $=\frac{1}{9}-\frac{1}{10}$
$=\frac{10-9}{90}$
$=\frac{1}{90}$
The leak can empty the cistern in = 90 hours.

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Question 63 Marks

Pipe A can fill a cistern in 6 hours and pipe B can fill it in 8 hours. Both the pipes are opened and after two hours, pipe A is closed. How much time will B take to fill the remaining part of the tank?

Answer

Inlet pipe A's one hour's work $=\frac{1}{6}$
Inlet pipe B's one hour's work $=\frac{1}{8}$
Both, inlets one hour's work $=\frac{1}{6}+\frac{1}{8}$
$=\frac{4+3}{24}=\frac{7}{24}$
Both inlets 2 hour's work $=\frac{7}{24}\times2=\frac{7}{12}$
Remaining work $=1-\frac{7}{12}=\frac{12-7}{12}=\frac{5}{12}$
Now pipe B can fill the $\frac{5}{12}$ tank in $=\frac{5}{12}\times8=\frac{10}{3}=3\frac{1}{3}$ hours.

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Question 73 Marks

A, B and C can do a piece of work in 10 days, 12 days and 15 days respectively. How long will they to finish it they work together?

Answer

A's one days work $=\frac{1}{10}$
B's one days work $=\frac{1}{12}$
$\therefore$ A, B and C's one days work $=\frac{1}{10}+\frac{1}{12}$
$+\frac{1}{15}$
$=\frac{6+5+4}{60}=\frac{15}{60}=\frac{1}{4}$
$\therefore$ A, B and C can do the same work in 4 days.

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Question 83 Marks

A cistern has two inlets A and B which can fill it in 12 minutes and 15 minutes respectively. An outlet C can empty the full cistern in 10 minutes. If all the three pipes are opened together in the empty tank, how much time will they take to fill the tank completely?

Answer

Inlet A's 1 minutes work $=\frac{1}{12}$
Inlet B's 1 minutes work $=\frac{1}{15}$
Outlet C's 1 minutes work $=\frac{1}{10}$
$\therefore$ Inlet A and B and outlet C's 1 minutes work,
$=\frac{1}{12}+\frac{1}{15}-\frac{1}{10}$
$=\frac{5+4-6}{60}(\text{LCM}\ \text{of}\ \ 12,15,10=60)$
$=\frac{3}{60}=\frac{1}{20}$
$\therefore$ The empty tank can be filled completely in = 20 minutes.

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Question 93 Marks

A can do a piece of work in 24 hours while B alone can do it in 16 hours. If A, B and C working together can finish it in 8 hours, in how many hours can C alone finish the work?

Answer

A's 1 hour work $=\frac{1}{24}$
B's 1 hours work $=\frac{1}{16}$
A, B and C's 1 hours work $=\frac{1}{8}$
$\therefore$ C's 1 hour's work $=\frac{1}{8}-\Big(\frac{1}{16}+\frac{1}{24}\Big)$
$=\frac{6-(2+3)}{48}=\frac{6-5}{48}=\frac{1}{48}$
$\therefore$ C can finish the work in = 48 hours.

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Question 103 Marks

A and B can do a piece of work in 15 days; B and C in 12 days; C and A in 20 days. How many days will be taken by A, B and C working together to finish the work?

Answer

(A + B) can do a work in 15 days.
$\therefore$ (A + B)'s 1 day work $=\frac {1}{15}$
(B + C) can do a work in 12 days.
$\therefore$ (B + C)'s 1 day work $=\frac{1}{12}$
(C + A) can do a work in 20 days.
$\therefore$ (C + A)'s 1 day work $=\frac{1}{20}$
2(A+ B + C)'s 1 day work $=\frac{1}{15}+\frac{1}{12}+\frac{1}{20}=\frac{4+5+3}{60}=\frac{12}{60}=\frac{1}{5}$
(A + B + C)'s 1 day work $=\frac{1}{10}$
A, B and C working together require 10 days to complete the work.

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Question 113 Marks

Three taps A, B and C can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively. How long would the three taps take to fill the empty tank, if all of them are opened together?

Answer

Tap A's one hour's work $=\frac{1}{6}$
Tap B's one hour's work $=\frac{1}{8}$
Tap C's one hour's work $=\frac{1}{12}$
A, B and C's together one hour's work,
$=\frac{1}{6}+\frac{1}{8}+\frac{1}{12}$
$=\frac{4+3+2}{24}(\text{LCM}\ \text{of}\ \ 6,8,12=24)$
$=\frac{9}{24}=\frac{3}{8}$
$\therefore$ These three tap will fill the empty tank in, $=\frac{8}{3}$ hour,
$=2\frac{2}{3}$ hours
= 2 hours 40 minutes.

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Question 123 Marks

Tap A can fill a cistern in 8 hours and tap B can empty it in 12 hours. How long will it take to fill the cistern if both of them are opened together?

Answer

Tap A can fill a cistern in 8 hours.
Part of cistern filled by Tap A in 1 hour $=\frac{1}{8}$
Tap B empties the cistern in 12 hours.
Part of cistern emptied by Tap B in 1 hour $=\frac{1}{12}$ (negative sign shows that tap B drains the tank)
Part of cistern filled in one hour when both taps are opened together $=\frac{1}{8}-\frac{1}{12}=\frac{3-2}{24}=\frac{1}{24}$
Therefore, it will take 24 hours to fill the cistern.

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Question 133 Marks

2 men or 3 women can do a piece of work in 16 days. In how many days can 4 men and 6 women do the same work?

Answer

Work of 2 men = Work of 3 women,
⇒ Work of 1 man $=\frac{3}{2}$ women.
Three women can do a piece of work in 16 days.
As 4 men and 6 women $=\Big(4\times\frac{3}{2}\Big)$ women + 6 women = 6 women + 6 women = 12 women,
Also, 3 women can do the work in 16 days.
So, work done by 3 women in one day $=\frac{1}{16}$
$\therefore$ Work done by 1 woman in one day $=\frac{1}{48}$
⇒ Work done by 12 women in one day $=\frac{1}{4}$
Thus, 4 men and 6 women will complete the work in 4 days.

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Question 143 Marks

A can do a piece of work in 10 days while B alone can do it in 15 days. In how many days can both finish the same work?

Answer

A can do a piece of work in 10 days.
A's 1 day work $=\frac{1}{10}$
B can do a piece of work in 15 days.
B's 1 day work $=\frac{1}{15}$
(A + B)'s 1 day work $=\frac{1}{10}+\frac{1}{15}=\frac{3+2}{300}=\frac{5}{30}=\frac{1}{6}$
A and B working together can complete the work in 6 days.

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Question 153 Marks

Two motor mechanics, Raju and Siraj working together can overhaul a scoote in 6 hours. Raju alone can do the job in 15 hours. In how many hours, can Siraj alone do it?

Answer

Raju and Siraj's 1 hours work $=\frac{1}{6}$
Raju's alone 1 hours work $=\frac{1}{15}$
$\therefore$ Siraj's alone 1 hours work $=\frac{1}{6}-\frac{1}{15}$
$=\frac{5-2}{30}=\frac{3}{30}=\frac{1}{10}$
$\therefore$ Siraj's can do the same work in 10 hours.

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Question 163 Marks

A, B and C working together can finish a piece of work in 8 hours. A alone can do it 20 hours and B alone can do it in 24 hours. In how many hours will C alone do the same work?

Answer

A, B and C's 1 hours work $=\frac{1}{8}$
A's 1 hours work $=\frac{1}{20}$
B's 1 hours work $=\frac{1}{24}$
$\therefore$ C's 1 hours work $=\frac{1}{8}-\Big(\frac{1}{20}+\frac{1}{24}\Big)$
$=\frac{15-(6+5)}{120}=\frac{15-11}{120}$
$=\frac{4}{120}=\frac{1}{30}$
$\therefore$ C alone finish the work in = 30 hours.

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Question 173 Marks

Pipes A and B can fill an empty tank in 10 hours and 15 hours respectively. If both are opened together in the empty tank, how much time will they take to fill it completely?

Answer

A's one hours work $=\frac{1}{10}$
B's one hours work $=\frac{1}{15}$
$\therefore$ A and B's one hour's work $=\frac{1}{10}+\frac{1}{15}$
$=\frac{3+2}{30}=\frac{5}{30}=\frac{1}{6}$
$\therefore$ Pipes A and B can fill the empty tank in 6 hours.

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Question 183 Marks

Rajan can do a piece of work-in 24 days while Amit can do it in 30 days. In how many days can they complete it if they work together.

Answer

Rajan's one days work $=\frac{1}{24}$
Amit's one days work $=\frac{1}{30}$
Both's one day's work $=\frac{1}{24}+\frac{1}{30}$
$=\frac{5+4}{120}=\frac{9}{120}[\text{LCM}\ \text{of}\ 24,30=120]$
$=\frac{3}{40}$
$\therefore$ Both can do the work in $=\frac{40}{3}$ days
$=13\frac{1}{3}$ days.

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Maths 3 Mark Question

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