Question

Janet worked 3 hours more than David and 7 hours less than Kate. David earned $7.50 per hour, Janet earned $6 per hour, and Kate earned $9 per hour. If Kate earned $27 more than David and Janet combined, how much did each person earn?​

Answer 1

We want to write and solve a system of equations to see how much each person earns.

What each one earned is:

• Janet = $78
• Kate =  $180
• David = $75

Let's define:

• J = number of hours that Janet worked.
• D = number of hours that David worked.
• K = number of hours that Kate worked.

The information is:

"Janet worked 3 hours more than David and 7 hours less than Kate"

From this we can write:

J = D + 3

J = K - 7

We know that:

• David earned $7.50 per hour.
• Janet earned $6 per hour.
• Kate earned $9 per hour.

Finally, we know that Kate earned $27 more than David an Janet combined, then we can write:

$9*K = $27 + ($7.50)*D + $6*J

Then we have a system of equations:

J = D + 3

J = K - 7

$9*K = $27 + ($7.50)*D + $6*J

We can rewrite the first and second equations to get:

D = J - 3

K = J + 7

Now we can replace these two in the other equation to get:

$9*(J + 7) = $27 + ($7.50)*(J - 3)+ $6*J

Now we can solve this for J:

$9*J + $63 = $27 + $7.50*J - $22.50 + $6*J

$9*J - $7.50*J - $6*J = $27 - $63 - $22.50

-$4.50*J = -$58.60

J = (-$58.60)/(-$4.50) = 13

This means that Janet worked for 13 hours.

And with:

K = J + 7 = 13 + 7 = 20

D = J - 3 = 13 - 3 = 10

We can see that Kate worked for 20 hours and David for 10 hours.

Then each one of them earned:

Janet = 13*$6 = $78

Kate = 20*$9 = $180

David = 10*$7.50 = $75

If you want to learn more, you can read:

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