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Time And Work Word Problems Worksheets

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1. Rafiq can do a work in 10 days and Shafiq can do that in 15 days. In how many days do they together finish the work ?

2. A can do a work in p days and B can do it in 2p days. They started to do the work together and after some days A left the work unfinished. B completed the rest of the work in r days. In how many days was the work finished ?

3. 10 persons can do a work in 7 days by working 6 hours. Working how many hours per day can 14 persons finish the work in 6 days ?

1. Answer :

Let «d» be the number days taken by together to finish the work.

Rafiq is taking 10 days to complete the work and Shafiq is taking 15 days.

Work done by Rafiq in 1 day  =  1/10

Work done by Shafiq in 1 day  =  1/15

Part of work done by Rafiq in d days  =  d/10

Part of work done by Shafiq in d days  =  d/15

Part of work finished by both in d day  =  d/10 + d/15

In d days they are finishing the work :

d/10 + d/15  =  1

5d/30  =  1

5d  =  30

d  =  6

So, they will take 6 days to complete the work.

2. Answer :

Time taken by A   =  p days

A’s one day work  =  1/p

Time taken by B   =  2p days

B’s one day work  =  1/2p

Let x be the number of days they work together.

Work done in x days  =  x(1/p + 1/2p)

=  x(3/2p)

Unfinished work  =  1 — (3x/2p)

To finish the remaining work, B is taking r days.

work finished in r days  =  r (1/1-(3x/2p))

=  r(2p/2p-3x)

r(2p/2p-3x)  =  2p

r  =  2p-3x

3x  =  2p-r

x  =  (2p-r)/3

Number of days taken to complete the work  =  x + r

= (2p-r)/3 + r

=  (2p+2r)/3

=  (2/3)(p+r)

3. Answer :

By comparing persons and hours,

if number of persons increased then hours taken will reduce. So, it comes under inverse proportion.

By comparing days and hours,

if the days increased then hours taken will reduce. So, it also comes under inverse proportion.

10 ⋅ 6  =  14 ⋅ x     and        7 ⋅ 6  =  6 ⋅ x

So, 

10 ⋅ 6 ⋅ 7  =  14 ⋅ x ⋅ 6

x  =  (10 ⋅ 6 ⋅ 7) / (14 ⋅ 6)

x  =  (10 ⋅ 6 ⋅ 7) / (14 ⋅ 6)

x  =  5 h

Therefore, the required number of hours is 5 hours.

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Twelve machines work twelve hours a day to complete a task .

1. How many machines will be required to finish double the task in the same time?
2. How many machines will be required to finish the task in same time if four of the machines broke down after 3^rd hour?

Solution:

Using Logic :

Let number of machines be y.

a) 12 machines complete the task in 12 hrs. Double the machines are required to finish double the task in the same time. Therefore, number of machines required to complete double the task = 24 .

b) 4 machines worked for 3 hrs = 12 machine hrs – work of one machine in 12 hrs

To finish the task 4 machines should have worked for 12 hrs = 12 X 4 = 48 machine hrs. Therefore, 12 machines + 3 extra machines = 15 machines.

Using Formula:

Let number of machines be Y.

a) Task = 12 machines x 12 hrs = 144 machine hrs

Double task = 2 x 144 hrs = 288  machine hrs

288 machine hrs = Y machines X 12hrs

Number of machines Y = 288 machine hrs/ 12 hrs = 24 machines

b) Task = 12 machines x 12 hrs = 144 machine-hrs

     144 = (4 machines x 3 hrs ) + (Y-4) machines  x 12 hrs

           = 12 machine hrs + 12 Y machine hrs  – 12 x 4 machine hrs

     144 = 12 y machine hrs – 36 machine hrs

     180 machine hrs = 12 Y machine hrs

     Y= 180 machine hrs/12 hrs  = 15 machines  

Example:

A road can be constructed by twelve employees working eight hours a day for thirty days. To speed completion of the project , a contractor deploys 30 employees to complete the project in sixteen days. How many hours a day must the employees work , in order to finish the construction work?

Solution:

Using logic:

Number of men in 1^st case= 12

Number of men in 2^nd case=30

Number of men in 2^nd case is 2.5 times more than that in 1^st case.

Therefore, number of hrs in 1^st case will be = 2.5 x number of hrs in 2^nd case

i.e. 30 days x 8 hrs = 2.5 (16days x h)

h = 30×8 / 2.5×16 hrs = 6hrs

Using Formula:

Construction is done by 12 men in 30 days working 8 hrs a day

Construction of road  is Work W = man x hrs = man x (days x hrs)

                                            = 12 men x 30 days x 8 hrs

                                            = 2880man hrs

Construction of same road by 30 men takes h hrs per day for 16 days

 W = 2880 man-hrs = 30 men x 16 days x h hrs

 h = 2800 man-hrs / (30 men x 16) = 6 hrs

Chapter 9: Radicals

9.10 Rate Word Problems: Work and Time

If it takes Felicia 4 hours to paint a room and her daughter Katy 12 hours to paint the same room, then working together, they could paint the room in 3 hours. The equation used to solve problems of this type is one of reciprocals. It is derived as follows:

[latex]text{rate}times text{time}=text{work done}[/latex]

For this problem:

[latex]begin{array}{rrrl} text{Felicia’s rate: }&F_{text{rate}}times 4 text{ h}&=&1text{ room} \ \ text{Katy’s rate: }&K_{text{rate}}times 12 text{ h}&=&1text{ room} \ \ text{Isolating for their rates: }&F&=&dfrac{1}{4}text{ h and }K = dfrac{1}{12}text{ h} end{array}[/latex]

To make this into a solvable equation, find the total time [latex](T)[/latex] needed for Felicia and Katy to paint the room. This time is the sum of the rates of Felicia and Katy, or:

[latex]begin{array}{rcrl} text{Total time: } &T left(dfrac{1}{4}text{ h}+dfrac{1}{12}text{ h}right)&=&1text{ room} \ \ text{This can also be written as: }&dfrac{1}{4}text{ h}+dfrac{1}{12}text{ h}&=&dfrac{1 text{ room}}{T} \ \ text{Solving this yields:}&0.25+0.083&=&dfrac{1 text{ room}}{T} \ \ &0.333&=&dfrac{1 text{ room}}{T} \ \ &t&=&dfrac{1}{0.333}text{ or }dfrac{3text{ h}}{text{room}} end{array}[/latex]

Karl can clean a room in 3 hours. If his little sister Kyra helps, they can clean it in 2.4 hours. How long would it take Kyra to do the job alone?

The equation to solve is:

[latex]begin{array}{rrrrl} dfrac{1}{3}text{ h}&+&dfrac{1}{K}&=&dfrac{1}{2.4}text{ h} \ \ &&dfrac{1}{K}&=&dfrac{1}{2.4}text{ h}-dfrac{1}{3}text{ h}\ \ &&dfrac{1}{K}&=&0.0833text{ or }K=12text{ h} end{array}[/latex]

Doug takes twice as long as Becky to complete a project. Together they can complete the project in 10 hours. How long will it take each of them to complete the project alone?

The equation to solve is:

[latex]begin{array}{rrl} dfrac{1}{R}+dfrac{1}{2R}&=&dfrac{1}{10}text{ h,} \ text{where Doug’s rate (} dfrac{1}{D}text{)}& =& dfrac{1}{2}times text{ Becky’s (}dfrac{1}{R}text{) rate.} \ \ text{Sum the rates: }dfrac{1}{R}+dfrac{1}{2R}&=&dfrac{2}{2R} + dfrac{1}{2R} = dfrac{3}{2R} \ \ text{Solve for R: }dfrac{3}{2R}&=&dfrac{1}{10}text{ h} \ text{which means }dfrac{1}{R}&=&dfrac{1}{10}timesdfrac{2}{3}text{ h} \ text{so }dfrac{1}{R}& =& dfrac{2}{30} \ text{ or }R &= &dfrac{30}{2} end{array}[/latex]

This means that the time it takes Becky to complete the project alone is [latex]15text{ h}[/latex].

Since it takes Doug twice as long as Becky, the time for Doug is [latex]30text{ h}[/latex].

Joey can build a large shed in 10 days less than Cosmo can. If they built it together, it would take them 12 days. How long would it take each of them working alone?

[latex]begin{array}{rl} text{The equation to solve:}& dfrac{1}{(C-10)}+dfrac{1}{C}=dfrac{1}{12}, text{ where }J=C-10 \ \ text{Multiply each term by the LCD:}&(C-10)(C)(12) \ \ text{This leaves}&12C+12(C-10)=C(C-10) \ \ text{Multiplying this out:}&12C+12C-120=C^2-10C \ \ text{Which simplifies to}&C^2-34C+120=0 \ \ text{Which will factor to}& (C-30)(C-4) = 0 end{array}[/latex]

Cosmo can build the large shed in either 30 days or 4 days. Joey, therefore, can build the shed in 20 days or −6 days (rejected).

The solution is Cosmo takes 30 days to build and Joey takes 20 days.

Clark can complete a job in one hour less than his apprentice. Together, they do the job in 1 hour and 12 minutes. How long would it take each of them working alone?

[latex]begin{array}{rl} text{Convert everything to hours:} & 1text{ h }12text{ min}=dfrac{72}{60} text{ h}=dfrac{6}{5}text{ h}\ \ text{The equation to solve is} & dfrac{1}{A}+dfrac{1}{A-1}=dfrac{1}{dfrac{6}{5}}=dfrac{5}{6}\ \ text{Therefore the equation is} & dfrac{1}{A}+dfrac{1}{A-1}=dfrac{5}{6} \ \ begin{array}{r} text{To remove the fractions, } \ text{multiply each term by the LCD} end{array} & (A)(A-1)(6)\ \ text{This leaves} & 6(A)+6(A-1)=5(A)(A-1) \ \ text{Multiplying this out gives} & 6A-6+6A=5A^2-5A \ \ text{Which simplifies to} & 5A^2-17A +6=0 \ \ text{This will factor to} & (5A-2)(A-3)=0 end{array}[/latex]

The apprentice can do the job in either [latex]dfrac{2}{5}[/latex] h (reject) or 3 h. Clark takes 2 h.

A sink can be filled by a pipe in 5 minutes, but it takes 7 minutes to drain a full sink. If both the pipe and the drain are open, how long will it take to fill the sink?

The 7 minutes to drain will be subtracted.

[latex]begin{array}{rl} text{The equation to solve is} & dfrac{1}{5}-dfrac{1}{7}=dfrac{1}{X} \ \ begin{array}{r} text{To remove the fractions,} \ text{multiply each term by the LCD}end{array} & (5)(7)(X)\ \ text{This leaves } & (7)(X)-(5)(X)=(5)(7)\ \ text{Multiplying this out gives} & 7X-5X=35\ \ text{Which simplifies to} & 2X=35text{ or }X=dfrac{35}{2}text{ or }17.5 end{array}[/latex]

17.5 min or 17 min 30 sec is the solution

Questions

For Questions 1 to 8, write the formula defining the relation. Do Not Solve!!

1. Bill’s father can paint a room in 2 hours less than it would take Bill to paint it. Working together, they can complete the job in 2 hours and 24 minutes. How much time would each require working alone?
2. Of two inlet pipes, the smaller pipe takes four hours longer than the larger pipe to fill a pool. When both pipes are open, the pool is filled in three hours and forty-five minutes. If only the larger pipe is open, how many hours are required to fill the pool?
3. Jack can wash and wax the family car in one hour less than it would take Bob. The two working together can complete the job in 1.2 hours. How much time would each require if they worked alone?
4. If Yousef can do a piece of work alone in 6 days, and Bridgit can do it alone in 4 days, how long will it take the two to complete the job working together?
5. Working alone, it takes John 8 hours longer than Carlos to do a job. Working together, they can do the job in 3 hours. How long would it take each to do the job working alone?
6. Working alone, Maryam can do a piece of work in 3 days that Noor can do in 4 days and Elana can do in 5 days. How long will it take them to do it working together?
7. Raj can do a piece of work in 4 days and Rubi can do it in half the time. How long would it take them to do the work together?
8. A cistern can be filled by one pipe in 20 minutes and by another in 30 minutes. How long would it take both pipes together to fill the tank?

For Questions 9 to 20, find and solve the equation describing the relationship.

1. If an apprentice can do a piece of work in 24 days, and apprentice and instructor together can do it in 6 days, how long would it take the instructor to do the work alone?
2. A carpenter and his assistant can do a piece of work in 3.75 days. If the carpenter himself could do the work alone in 5 days, how long would the assistant take to do the work alone?
3. If Sam can do a certain job in 3 days, while it would take Fred 6 days to do the same job, how long would it take them, working together, to complete the job?
4. Tim can finish a certain job in 10 hours. It takes his wife JoAnn only 8 hours to do the same job. If they work together, how long will it take them to complete the job?
5. Two people working together can complete a job in 6 hours. If one of them works twice as fast as the other, how long would it take the slower person, working alone, to do the job?
6. If two people working together can do a job in 3 hours, how long would it take the faster person to do the same job if one of them is 3 times as fast as the other?
7. A water tank can be filled by an inlet pipe in 8 hours. It takes twice that long for the outlet pipe to empty the tank. How long would it take to fill the tank if both pipes were open?
8. A sink can be filled from the faucet in 5 minutes. It takes only 3 minutes to empty the sink when the drain is open. If the sink is full and both the faucet and the drain are open, how long will it take to empty the sink?
9. It takes 10 hours to fill a pool with the inlet pipe. It can be emptied in 15 hours with the outlet pipe. If the pool is half full to begin with, how long will it take to fill it from there if both pipes are open?
10. A sink is ¼ full when both the faucet and the drain are opened. The faucet alone can fill the sink in 6 minutes, while it takes 8 minutes to empty it with the drain. How long will it take to fill the remaining ¾ of the sink?
11. A sink has two faucets: one for hot water and one for cold water. The sink can be filled by a cold-water faucet in 3.5 minutes. If both faucets are open, the sink is filled in 2.1 minutes. How long does it take to fill the sink with just the hot-water faucet open?
12. A water tank is being filled by two inlet pipes. Pipe A can fill the tank in 4.5 hours, while both pipes together can fill the tank in 2 hours. How long does it take to fill the tank using only pipe B?

Answer Key 9.10

General Rules

Problems on Time and Work :

1. Aaron alone can finish a piece of work in 12 days and Brandon alone can do it in 15 days. If both of them work at it together, how much time will they take to finish it?

2. A and B together can do a piece of work in 15 days, while B alone can finish it 20 days. In how many days can A alone finish the work?

3. A can do a piece of work in 25 days and B can finish it in 20 days. They work together for 5 days and then A leaves. In how many days will B finish the remaining work?

4. A and B can do a piece of work in 18 days; B and C can do it in 24 days while C and A can finish it in 36 days. If A, B, C works together, in how many days will they finish the work?

5. A and B can do a piece of work in 12 days; B and C can do it in 15 days while C and A can finish it in 20 days. If A, B, C works together, in how many days will they finish the work? In how many days will each one of them finish it, working alone?

● Time and Work

Time and Work

Pipes and Cistern

Practice Test on Time and Work

● Time and Work — Worksheets

Worksheet on Time and Work