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
