ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics
Class 8: Reference Books               Class 8: NTSE Preparation
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Q1. A can do a piece of work in 15 days while B can do it in 10 days. How long will they take together to do it?

Answer:

A’s 1 Day =   \frac{1}{15}

B’s 1 Day work = \frac{1}{10}

A’s + B’s  1 Day work = (\frac{1}{15}+\frac{1}{10})=\ \frac{5}{30}=\ \frac{1}{6}

Therefore both can finish the work in 6 Days.

 

Q2. A, B and C can do a piece of work in 12 days, 15 days and 10 days respectively. In what time will they all together finish it?

Answer:

A’s 1 Day = \frac{1}{12}

B’s 1 Day work = \frac{1}{15}

C’s 1 Day work = \frac{1}{10}

(A’s + B’s + C’s) 1 Day work = (\frac{1}{12} +\frac{1}{15} + \frac{1}{10})  = \frac{15}{60} = \frac{1}{4}

Therefore all three can finish the work in 4 Days.

 

Q3. A and B together can do a piece of work in 35 days, while A alone can do it in 60 days. How long would B alone take to do it?

Answer:

A’s 1 Day = \frac{1}{60}

B’s 1 Day work = \frac{1}{x}

(A’s + B’s) 1 Day work = (\frac{1}{60} + \frac{1}{x} ) = \frac{1}{35}

Solving for x  = 84 Days

 

Q4. A can do a piece of work in 20 days while B can do it in 15 days. With the help of C, they finish the work in 5 days. In what time would C alone do it?

Answer:

A’s 1 Day = \frac{1}{20}

B’s 1 Day work = \frac{1}{15}

C’s 1 Day work = \frac{1}{x}

(A’s + B’s + C’s) 1 Day work = (\frac{1}{20} +\frac{1}{15} + \frac{1}{x} ) =  \frac{1}{5} 

Solving for x   = 12 Days

 

Q5. A can do a piece of work in 12 days and B alone can do it in 16 days. They worked together on it for 3 days and then A left. How long did B take to finish the remaining work?

Answer:

A’s 1 Day = \frac{1}{12}

B’s 1 Day work = \frac{1}{16}

(A’s + B’s) 1 Day work = (\frac{1}{12} +\frac{1}{16} ) =  \frac{7}{48} 

The amount of work that is completed in 3 days = \frac{3\times7}{48} =  \frac{7}{16}

Amount of work left for B to complete = 1 - \frac{7}{16} =  \frac{9}{16}

Therefore the number of days that B will take to finish the work = \frac {\frac{9}{16}} {\frac{1}{16} }  = 9 days

 

Q6. A can do  \frac{1}{4}  of a work in 5 days, while B can do \frac{1}{5}   of the work in 6 days. In how many days can both do it together?

Answer:

If A can do  \frac{1}{4}  of a work in 5 days, then A can do the entire work in 20 days.

Therefore A’s 1 Day Work =  \frac{1}{20}

If B can do  \frac{1}{5}  of a work in 6 days, then B can do the entire work in 30 days.

Therefore B’s 1 Day Work = \frac{1}{30}

(A’s + B’s) 1 Day work = (\frac{1}{20} +\frac{1}{30} ) =  \frac{1}{12} 

Therefore both can do the work in 12 days.

 

Q7. A can dig a trench in 6 days while B can dig it in 8 days. They dug the trench working together and received 1120 for it. Find the share of each in it.

Answer:

A’s 1 Day = \frac{1}{6}

B’s 1 Day work = \frac{1}{8}

Therefore the ratio of work = \frac {\frac{1}{6}} {\frac{1}{8} } = \frac{8}{6}

Therefore A’s share = \frac{8}{14} \times1120 = 640

Therefore B’s share = \frac{6}{14} \times1120 = 480

 

Q8. A can mow a field in 9 days; B can mow it in 12 days while C can mow it in 8 days. They all together mowed the field and received 1610 for it. How will the money be shared by them?

Answer:

A’s 1 Day = \frac{1}{9}

B’s 1 Day work = \frac{1}{12}

C’s 1 Day work = \frac{1}{18}

Therefore the ratio of their one day’s work = \frac{1}{9} \colon  \frac{1}{12} \colon  \frac{1}{18}  = 8 \colon 6 \colon 9 

Hence A’s share = \frac{8}{23} \times1120 = 1610

Hence A’s share = \frac{6}{23} \times1120 = 1610

Hence A’s share = \frac{9}{23} \times1120 = 1610

 

Q9. A and B can do a piece of work in 30 days; B and C in 24 days; C and A in 40 days. How long will it take them to do the work together? In what time can each finish it, working alone?

Answer:

(A’s + B’s) 1 Day work = \frac{1}{30}

(B’s + C’s) 1 Day work = \frac{1}{24}

(C’s + D’s) 1 Day work = \frac{1}{40}

Adding the above three 2\times(A + B + C)   day work = (\frac{1}{30} +\frac{1}{24} + \frac{1}{40} ) =  \frac{1}{10} 

Therefore if they all work together, they will take 5 days to finish the work.

 

Q10. A can do a piece of work in 80 days. He works at it for 10 days and then B alone finishes the remaining work in 42 days. In how many days could both do it?

Answer:

A’s 1 Day = \frac{1}{80}

Work finished by A in 10 days = \frac{1}{80} \times 10 = \frac{1}{8}

B finished the remainder of work (1 - \frac{1}{8}  = \frac{7}{8})  in 42 days

Therefore 1 Days work for B = \frac{\frac{7}{8}}{42}  = \frac{1}{48}

Hence B can do the work in 48 days

(A’s + B’s) 1 Day work = (\frac{1}{80} + \frac{1}{48} )= \frac{1}{30}

Therefore both can do the work in 30 days.

 

Q11. A and B can together finish a work in 30 days. They worked at it for 20 days and then B left. The remaining work was done by A alone in 20 more days. In how many days can A alone do it?

Answer:

(A’s + B’s) 1 Day work = \frac{1}{30}

Amount of work finished by both in 20 days = 20 \times \frac{1}{30}  = \frac{2}{3}

Work left to be finished =  1 - \frac{2}{3}  = \frac{1}{3}  

Work done by A in 1 Day = \frac{\frac{1}{3}}{20} = \frac{1}{60}

Therefore A can do the work alone in 60 days.

 

Q12. A can do a certain job in 25 days which B alone can do in 20 days. A started the work and was joined by B after 10 days. In how many days was the whole work completed?

Answer:

A’s 1 Day work = \frac{1}{25}

B’s 1 Day work = \frac{1}{20}

(A’s + B’s) 1 Day work = (\frac{1}{25} +\frac{1}{20}) = \frac{9}{100}

Amount of work finished by A in 10 days = 10 \times \frac{1}{25}

Work left to be finished = (1-\frac{2}{5}) = \frac{3}{5} 

Days taken by both A and B working together = \frac{\frac{3}{5}}{\frac{9}{100}} = 6\frac{2}{3}  

The work got completed in  10 + 6\frac{2}{3} = 16\frac{2}{3}

 

Q13. A can do a piece of work in 14 days, while B can do in 21 days. They begin together. But, 3 days before the completion of the work, A leaves off. Find the total number of days taken to complete the work.

Answer:

A’s 1 Day work = \frac{1}{14}

B’s 1 Day work = \frac{1}{21}

(A’s + B’s) 1 Day work = (\frac{1}{14} +\frac{1}{21}) = \frac{5}{42}

Amount of work finished by B in 3 days = (3 \times \frac{1}{21}) =\frac{1}{7}   

Work left to be finished by A and B together = (1- \frac{1}{7}) =  \frac{6}{7}

Days taken by A + B working together = \frac{\frac{6}{7}}{\frac{5}{42}} = 7\frac{1}{5}  

The work got completed in  3+7\frac{1}{5} = 10\frac{1}{5}

 

Q14.  A is thrice as good a workman as B and B is twice as good a workman as C. All the three took up a job and received 1800 as remuneration. Find the share of each.

Answer:

If C takes x   days to complete the job

Then B will complete the job in \frac{x}{2}   days

And A will complete the job in \frac{x}{3}   days days

Therefore the ratio of 1 days’ work of  A \colon B \colon C = \frac{x}{3} \colon \frac{x}{2} \colon \frac{x}{1}   = 3 \colon 2 \colon 1  

Therefore the share of A = 1800 \times \frac{3}{6}  = 900  Rs.

Therefore the share of B = 1800 \times \frac{2}{6} =  600    Rs.

Therefore the share of C = 1800 \times \frac{1}{6}  = 300   Rs.

 

Q15.  A can do a certain job in 12 days. B is 60% more efficient than A. Find the number of days taken by B to finish the job.

Answer:

Time taken to finish the job = 12 days

A’s 1 Day’s work = \frac{1}{12}

B is 60% more efficient

B’s 1 Day’s work = 1.6 \times\frac{1}{12}

Therefore the number of days B will take to finish the job =  \frac{1}{\frac{1.6}{12}} = 7\frac{1}{2}  days

 

Q16.  A is twice as good a workman as B and together they finish a piece of work in 14 days. In how many days can A alone do it?

Answer:

Let B take x   days to finish the work.

B’s 1 day’s work = \frac{1}{x}

The A will take \frac{x}{2}  days to finish the work.

A’s 1 day’s work = \frac{2}{x}

(A+B) one day’s work =  (\frac{1}{x} +\frac{2}{x}) = \frac{1}{14}

Solving for x=42

Therefore A will take 21 days to finish the job.

 

Q17. Two pipes A and B can separately fill a tank in 36 minutes and 45 minutes respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

Answer:

A’s 1 minute fill rate = \frac{1}{36}

B’s 1 minute fill rate = \frac{1}{45}

A’s and B’s fill rate together = (\frac{1}{36} + \frac{1}{45}) = \frac{1}{20}

Therefore if A and B are opened simultaneously, the tank will take 20 minutes to fill up.

Q18.  One tap can fill a cistern in 3 hours and the waste pipe can empty the full tank in 5 hours. In what time will the empty cistern be full, if the tap and the waste pipe are kept open together?

Answer:

Tap’s 1 minute fill rate = \frac{1}{3}

Waste Pipe’s 1 minute empty rate = \frac{1}{5}

Therefore the net fill rate = (\frac{1}{3} + \frac{1}{5}) = \frac{2}{15}

Therefore if both the tap and the waste pipe are opened simultaneously then it will take 7\frac{1}{2}  hours to fill up.

 

Q19.  Two pipes A and B can separately fill a cistern in 20 minutes and 30 minutes respectively , while a third pipe C can empty the full cistern in 15 minutes. If all the pipes are opened together, in what time the empty cistern is filled?

Answer:

A’s 1 minute fill rate = \frac{1}{20}

B’s 1 minute fill rate = \frac{1}{30}

C’s 1 minute empty rate = \frac{1}{15}

Therefore the net fill rate = (\frac{1}{20} + \frac{1}{30} - \frac{1}{15}) = \frac{1}{60}

Therefore if all the tap are opened simultaneously then it will take 60 minutes or one hour to fill up.

 

Q20.  A pipe can fill a tank in 16 hours. Due to a leak in the bottom, it is filled in 24 hours. If the tank is full, how much time will the leak take to empty it?

Answer:

Pipe’s fill rate = \frac{1}{16}

Let the leak is at a rate of = \frac{1}{x}

Therefore the net fill rate =  (\frac{1}{16} - \frac{1}{x}) = \frac{1}{24}

Solving for x = 48  hours.

 

ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics
Class 8: Reference Books               Class 8: NTSE Preparation
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