We will solve “working together word problems” with proportion. It may easier to tackle than algebra.
Problem #1:
Maria can clean all bathrooms in her house in 5 minutes. Her husband can do the same in 20 minutes. How long will it take them if they work together?
One important observation is that an entire task or job can be represented with the number 1.
If it takes Maria 5 minutes to do the entire job, then she can do
15
of the job in 1 minute
By the same fashion, if it takes the husband 20 minutes to do the entire job, then he can do
120
of the job in 1 minute.
Together, the amount of the job they can do in 1 minute is
| 15 | + 120 |
| 15 | + 120 | = 420 | + 120 | = 520 |
Together, they can do
520
of the job in 1 minute
Now set up a proportion using the following logic.
If
520
of the job can be done in 1 minute, then the entire job or 1 can be done in x minutes
520_____________
1 minute
1 _____________
x minutes
Cross multiply
520×
x = 1 × 1
520×
x = 1
Multiply both sides of the equation by
205
The left side gives the expression below:
| 520 | × 205 × x | = 100100 = 1x = x |
Therefore, x = 1 ×
205
= 4
Together, they can clean all bathrooms in 4 minutes.
Problem #2
Say it is a big house project and Maria could finish the project in 5 days and the husband could do it in 20 days.
How many days will it take husband and wife to finish the work?
Problem #1 and problem #2 are similar working together word problems. The only difference is that instead of using hours, problem #2 is using days.
Therefore, it would take them 4 days to finish.
Some working together word problems can be complicated. Problem #3 is a little challenging!
Problem #3:
It can take Peter 25 days to build a house garage all by himself. If he works with his friend, together they can build the house garage in 20 days.
How fast can the friend work alone?
Set up a proportion to find out how much of the work can be done by both of them in 1 day.
let x represent the amount of work they can do in 1 day. Notice again how we use 1 to represent the entire job
1 _____________ 20 days
x _____________ 1 day
Cross multiply
1 × 1 = x × 20
1 = 20x
Divide both sides by 20
| 120 | = 2020 x |
120
= x
The amount of the job Peter can do in 1 day is
125
The amount of the job the friend can do in 1 day is
| 120 | – 125 |
The amount of the job the friend can do in 1 day is
| 5100 | – 4100 | or 1100 |
This means that it will take the friend 100 days to finish the house garage by himself.
Working together in sales
Michael Scott, by himself, can close 8 sales in 4 hours. If he works with Dwight, they can close 8 sales in 2 hours. How many hours does it take Dwight to close 8 sales by himself?
For Michael, 8 sales in 4 hours means 2 sales per 1 hour or 2 / 1
Michael’s sales / hour + Dwight’s sales / hour = together’s sales per hour
Dwight’s sales / hour is x / 1
Together, 8 sales in 2 hours means 4 sales per 1 hour or 4 / 1
2 / 1 + x / 1 = 4 / 1
2 + x = 4
2 – 2 + x = 4 – 2
x = 2
So Dwight can do 2 sales per hour. It is basically the same as Michael. In this case, he can also do 8 sales in 4 hours.
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