Lesson The Robinson family and the Sanders family each used their sprinklers last summer

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<H2>The Robinson family and the Sanders family each used their sprinklers last summer</H2>

The problems of this type came so often to the forum that I decided to devote the entire lesson to them in order for to have 
a permanent reference and a permanent link for them.

<H3>Problem 1</H3>The Robinson family and the Sanders family each used their sprinklers last summer. 
The water output rate for the Robinson family's sprinkler was &nbsp;25 L per hour. 
The water output rate for the Sanders family's sprinkler was &nbsp;20 L per hour. 
The families used their sprinklers for a combined total of &nbsp;55 hours, &nbsp;resulting in a total water output of &nbsp;1250 L. 
How long was each sprinkler used?

<B>Solution 1</B> &nbsp;(using the system of two equations in two unknowns)

<pre>
Let x be the time (in hours) of use sprinklers by the Robinson family, and

let y be the time (in hours) of use sprinklers by the Sanders family.

Then from the condition, you have these two equations in two unknowns

x +   y =   55    (1)
25x + 20y = 1250    (2)

Express  x = 55 - y  from equation (1)   and substitute it into the equation (2). You will get

25*(55-y) + 20y =1250.

Simplify and solve it for y:

1375 - 25y + 20y = 1250.

-5y = 1250 - 1375

-5y = -125.

y = {{{(-125)/(-5)}}} = 25.

Then x = 55 - y = 55 - 25 = 30.

<U>Answer</U>. 30 hours of use by the Robinson family, and 25 hours of use by the Sanders family.

<U>Check</U>.  30*25 + 25*20 = 1250.   Correct !
</pre>

<B>Solution 2</B> &nbsp;(using single equation)

<pre>
Let x be the time (in hours) of use sprinklers by the Robinson family.

Then the time of use sprinklers by the Sanders family is (55 - x) hours.

25x + 20*(55 - x) = 1250,

25x + 1100 - 20x = 1250,

5x = 1250 - 1100,

5x = 150.

x = {{{150/5}}} = 30.

55 - 30 = 25.

<U>Answer</U>. Same as in the <U>Solution 1</U>:  30 hours of use by the Robinson family, and 25 hours of use by the Sanders family.
</pre>

<H3>Problem 2</H3>The &nbsp;Johnson family and the &nbsp;Taylor family each used their sprinklers last summer. 
The &nbsp;Johnson family' sprinkler was used for &nbsp;20 &nbsp;hours. &nbsp;The &nbsp;Taylor family' sprinkler was used for &nbsp;30 &nbsp;hours. 
There was a combined total output of &nbsp;1450 L of water. 
What was the water output rate for each family' sprinklers if the sum of the two rates was &nbsp;55 &nbsp;L per hour?

<B>Solution</B>

<pre>
Let x = the water output rate for the Taylor family' sprinklers, in liters per hour.

Then the water output rate for the Johnson family' sprinklers is (55-x).

The combined total output water equation is

30x + 20*(55-x) = 1450 liters.

Isolate the terms with x

30x - 20x = 1450 - 20*55

and find x

x = {{{(1450-20*55)/(30-20)}}} = 35 liters per hour.

<U>ANSWER</U>.  Taylor family' sprinklers output rate is 35 L/hour.

Johnson family' sprinklers output rate is 55-35 = 20 L/hour.
</pre>

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