Week 11 Review: Answers and Solutions

Below are the solutions to the exercises and about work problems.

Problem 1
Aria can do a job in 7 days. What part of the job is finished after she worked for 3 days?

Aria can do a job in 7 days. So meaning, she can do 1/7 job each day.

1st day – 1/7
2nd day – 1/7
3rd day – 1/7

So, 3(1/7) = 3/7

Answer: 3/7

Problem 2
Katya can do a job in 5 days. Marie can do the same job in 6 days. If they both worked for 1 day, what part of the job is finished?

Katya can do a job in 5 days which means each day 1/5 of the job.
Marie can do a job in 6 days which means each day 1/6 of the job.

Let x = part of the job finished for 1 day

1/5 + 1/6 = 1/x
LCD: 30x

Multiplying both sides of the equation by 30, we have
(30x)(1/5 + 1/6) = (30x)(1/x)
(30x/5) + 30x(1/6) = (30x/x)
6x + 5x = 30
11x = 30
x = 30/11 or 2 8/11
Answer: 2 and 8/11 days.

Problem 3
Ramon can paint a house in 6 days. Ralph can do the same job in 10 days. If they both worked for 2 days, what part of the job is done if they were to work together the whole time?

Ramon = 6 days (or 1/6 part each day)
Ralph = 10 days (or 1/10 part each day)

Let w – part of the job done for 2 days

w = 2(1/6) + 2(1/10)
w = 2/6 + 2/10

LCD: 30

30(2/6 + 2/10)
((30/6)*2)/30 + ((30/10)*2)/30
(5*2)/30 + (3*2)/30
10/30 + 6/30
16/30 or 8/15

8/15 – work done for 2 days.

If they were to worked together, they will finish the work in x days using the equation

1/6 + 1/10 = 1/x

We multiply both sides of the equation which is 30x giving us

30x(1/6) + 30x(1/10) = 30x(1/x)
(30x)/6 + (30x)/10 = (30x/x)
5x + 3x = 30
8x = 30
x = 30/8
x = 15/4

This means that both of them will finish the work in 15/4 days. This means that the amount of worked finished is (8/15)/(15/4) = (8/15)(4/15) = 32/225.

Problem 4
One hose can fill a pool in 3 hours and a smaller hose can fill the same pool in 4 hours. How long will it take the two hoses to fill the entire pool?

Let x – total time to fill the pool

1/3 + 1/4 = 1/x
LCD: 12x

Multiplying both sides by 12x, we have

12x(1/3 + 1/4 = 1/x)12
(12x)/3 + (12x)/4 = 12
4x + 3x = 12
7x = 12
x = 12/7 or 1 5/7 hours

Answer: 1 5/7 hours

Problem 5
Marco can dig a ditch in 5 hours and he and Jimmy can do it in 2 hours. How long would it take Jimmy to dig the same ditch alone?

Let 1/x – Jimmy’s time alone
1/5 – Marco’s time

1/x + 1/5 = 1/2
LCD: 10x
10x(1/x + 1/5) = (10x)(1/2)
10 + 2x = 5x
2x – 5x = -10
-3x = -10
x = -10/-3 or 3 1/3 hrs

Problem 6
Maria can paint a fence in 6 days and Leonora can do the same job in 7 days. They start to paint it together, but after two days, Leonora left, and Maria finishes the job alone. How many days will it take Leonora to finish the job?

Job done by Maria in 2 days = 2/6
Job done by Leonora in 2 days = 2/7
Job done by Maria in the remaining days = x/6

2/6 + 2/7 + x/6 = 1

LCD:42x

Multiplying both sides of the equation by 42, we have

42(2/6 + 2/7 + x/6) = (42)(1)
84/6 + 84/7 + (42x)/6 = 42
14 + 12 + 7x = 42
26 + 7x = 42
7x = 42 – 26
7x = 16
x = 16/7 or 2 2/7 days

Problem 7
An inlet pipe can fill a pool in 4 hours. An outlet pipe can fill the same pool in 6 hours. One day, the pool was empty. The owner opened the inlet pipe but forgot to close the outlet pipe. How long will it take to fill the pool?

Let x – hours to fill the pool
1/4 – 1/6 = 1/x
LCD: 12x

(12x)(1/4) – (12x)(1/6) = (12x)(1/x)
(12x)/4 – (12x)/6 = (12x)/x
3x – 2x = 12
x = 12 hours

The Solving Work Problems Series

Work problems involves two or more persons or machines doing a particular task. In this type of problems, the rates of the persons or machines are usually given and the amount of time needed to complete a task is usually asked. Below are the posts that discuss in details these types of problems.

How to Solve Work Problems Part 1 discusses the details of the basic concepts like how the equations are formed and why the equations are equated to 1.

How to Solve Work Problems Part 2 discusses basic examples of work problems. It discusses how two hoses can fill a pool when they are opened simultaneously. It also discusses how a person can do a task alone given the rate of two persons working together.

How to Solve Work Problems Part 3 discusses two persons who worked together and after a while, the other person stopped. It also discusses the problem about how many hours a pool is filled if both the inlet and outlet pipes are open.

The Solving Work Problems Series is one of the series of Math Word Problem Series in Algebra of Ph Civil Service Exam Reviewer.

How to Solve Work Problems Part 1

This is the first part of the Solving Working Problems Series. In this post, we are going to discuss in details the basics of work problems.

Work problems usually involve the time for two or more persons or machines to complete the same job given the rate that they can work. For discussion purposes, let us have the following example.

Work problem:

Ariel can paint a house in 5 days and Ben can do the same job in 6 days. In how many days can they complete the job if they work together?

Discussion and Scratch Work

If Ariel can finish the job in 5 days, then if he were to work one day, he would have completed 1/5 of the job. If he works for two days, then he would have completed 2/5 of the job. Similarly, if Ben can finish the same job in 6 days, if he were to work for one day, then he would have completed 1/6 of the job. If he works for 2 days, he would have completed 2/6 of the job (or 1/3 of the job if reduced to lowest terms).  » Read more

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