Question

Tiana can clear a football field of debris in 3 hours. Jacob can clear the same field in 2 hours. If they decide to clear the field together, how long will it take them?
A.50 minutes
B.1 hour 5 minutes
C.1 hour 12 minutes
D.5 hours

Answer 1

Let's begin by introducing a couple variables:
Let r1 = Tiana's rate of work
Let r2 = Jacob's rate of work
t1 = Tiana's time = 3 hours
t2 = Jacob's time = 2 hours

Answer 2

While we don't have the actual rate values for Tiana and Jacob,
since they are separately clearing the same field using the 4 variables listed above ↑
we can write the following equation:
r1t1 = r2 t2
or
r1(3) = r2 (2)

Answer 3

Let's now consider what happens when Tiana and Jacob work together.
And let's call the time they need to clear the field when working together
the time capital T.

When working together,
(r1+r2) T = r1(3) = r2(2)

Answer 4

All 3 of these expressions ↑ are equal since the same field is being cleaned in each case.
So this is the setup for the problem. We now have a bit of algebra to do to find the value of T.

Answer 5

@PoetryPrincess Do you understand thus far?

Answer 6

Yes

Answer 7

Let's now try to find the value of T ...

Answer 8

A really quick way to solve problems like this. (where you have separate hours for individuals, and are asked to get the time it takes when combined is.)

In this case: Tiana can clear a football field of debris in 3 hours. Jacob can clear the same field in 2 hours. If they decide to clear the field together, how long will it take them?

set up a fraction where

total hours multiplied/total hours added up

in this case would be 6/5, just convert to hours and minutes.

Answer 9

I've indexed the 2 equations as (1) and (2) for easier reference.
Let's solve (2) for r2 in terms of r1:
If we now take this result for r2 and substitute into (1), we get:
notice in the last line, r1 will divide out from all the terms:
and like magic we now have an equation we can solve for T!