Solving a value mixture problem.

Question

The Rogers Family and the Gray Family each used their sprinklers last summer. The Roger’s family sprinkler was used for 20 hours. The Gray’s family sprinkler was used for 15 hours. There was a combined total output of 800 L of water. What was the water output rate for each sprinkler if the sum of the two rates was 45 L per hour?

Answer 1

You have information about the total water used and other information about the two parts of the water usage. You can write two independent equations from this.

The first objective in solving a system of equations is to use both equations to eliminate one variable so you have a new equation that only has that one variable in it. Solve for that variable's value, then use that in either equation to find the other one. (You get to pick which equation you think will be easier to work with, but either one will get you there.)

Let R be the Rogers' rate and G the Gray's rate.

Eqn 1: 20R + 15G = 800 total consumption

Eqn 2: R+G = 45 sum of the rates

Substitution is the easier way to solve this one.

Solve equation 2 for R or G and substitute that into equation 1.

G=45-R

20R + 15(45-R) = 800

20R + 675-15R = 800

5R = 800 - 675 = 125

R = 25

G = 45-25 = 20

Just for illustration of the elimination method (the other way to solve this):

Eqn 1: 20R + 15G = 800

Eqn 2: R+G = 45

Multiply both sides of Eqn2 by 15

Eqn 2: 15R+15G = 45*15 = 675

Subtract this equation from Eqn 1:

20R + 15G = 800

-15R - 15G = -675

5R = 125

R = 25

Substitue for R in Eqn 2:

R+G = 45

25+G = 45

G = 20

You can see that the same numbers crop up since we're actually doing the same arithmetic, just in a different sequence.

I hope this helps!