In algebra, problems dealing with two (or more) people working together are encountered quite often. (These are quite similar to "pipes filling a pool" problems and are solved similarly.)
c) Use a formula.
Bill can paint a room in 6 hours and Dave can paint a room in 4 hours. If they both work togeher, how much time will it take?
There are several ways in which this can be solved:
a) Set up rates based on time per room making sure the same time is used for each person.
In 6 hours Bill paints one room.
Dave's rate of 4 hours per room can be converted to 6 hours per 1½ rooms.
So, 2½ rooms could be painted in 6 hours, and this converts to:
6 ÷ 2½ = 2.4 hours.
b) Set up rates based on rooms per time.
In one hour, Bill paints ⅙ of a room and Dave paints ¼ of a room.
So together they paint (⅙ + ¼) = 10/24 = 5/12 of a room in one hour.
5/12 rooms per hour = 12/5 hours per room = 2.4 hours.
Formulas are usually the best way for solving problems but memorize them exactly especially if you will be taking a test.
It might seem unusual to have a special formula to use for two people, but you have to admit it is much easier than using formula 2.
Using Formula 1 with the data from example "1"
Time = (6 * 4) ÷ (6 + 4)
Time = (24) ÷ (10)
Time = 2.4 hours.
Using Formula 2
Time = 1 ÷ [(1/6) + (1/4)]
Time = 1 ÷ [(4/24) + (6/24)]
Time = 1 ÷ (10/24)
Time = 24/10
Time = 2.4 hours
See? Formula 1 is much better isn't it?
d) Use a calculator
This calculator is designed for solving parallel resistances but can also be used for these type of problems.
Just for a quick example, 4 pipes, working separately, can each fill a pool in 7, 8, 9 and 12 hours. If they all work together, how quickly will the pool be filled?
Entering these numbers into the calculator shows that the pool will be filled in 2.1631 hours.
Example 2A) (F × S) ÷ (F + S) = 72
Two people working together can mow a lawn in 72 minutes.
When working separately, one person can mow the lawn one hour faster than the other.
How much time would it take for each person to mow the lawn when working separately?
Looking at Formula 1, let's set up equation "A" where "F" is the faster speed and "S" is the slower speed.
We know that one person is 1 hour (60 minutes) faster than the other, giving us equation "B"
B) F + 60 = S
Inserting equation B into equation A, we get:
(F × F + 60) ÷ (F + F + 60) = 72
(F² + 60F) ÷ (2F + 60) = 72
(F² + 60F) = 72 • (2F + 60)
(F² + 60F) = 144F + 4,320
F² - 84F - 4,320 = 0
Using the quadratic equation calculator, we find F = 120 and inserting this into
equation B) F + 60 = S B) 120 + 60 = S we get S =180.
Two people can paint a room in 4 hours 48 minutes (or 4.8 hours).
If 1 person can paint the room in 12 hours, how long would it take the other person working alone?
Basically, you could do this by solving Formula 1 for either Worker 1 Time or Worker 2 Time.
But that's okay because we've done it for you.
We input this information into Formula 3.
Worker 2 Time = (12 × 4.8) ÷ (12 - 4.8)
Worker 2 Time = 57.6 ÷ 7.2
Worker 2 Time = 8 hours.