Question

At his regular hourly rate, Don had estimated the labor cost of a repair job as $336 and he was paid that amount. However, the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate. What was the time Don had estimated for the job, in hours?

(A) 28
(B) 24
(C) 16
(D) 14
(E) 12

Answer 1

The time Don estimated for the job, in hour, is 24 hours

Explanation:

Based on his regular hourly rate, Don had estimated :

Estimated cost (C) = $336

Estimated hour of work =
`x`

So, estimated rate in hour (R) =
`(336)/(x)`

In fact, it took him 4 hours longer than he estimated, therefore

Factual hour of work =
`{x+4}`

Factual rate in hour (F) =
`(336)/(x+4)`

He earned $2 less than his regular hourly rate, hence the factual rate (F) is the subtraction of estimated rate (R) by 2

F = R - 2

Then, substitute F and R,

`(336)/(x+4)` =
`(336)/(x) - 2`

`(336)/(x+4)+2` =
`(336)/(x)`

`x((336)/(x+4)+ (2(x+4))/(x+4)) = 336`

`x(336+2(x+4))` =
`336 (x+4)`

`336x+2x^(2) +8x = 336x+336*4`

since at the both sides have
`336x`, we can removed it.

`2x^(2) + 8x = 336 * 4`

devide both sides by 2, thus

`x^(2) +4x = 336 * 2`

`x^(2) +4x = 672`

Make the algebra function equal to 0 by moving the 672 to the left side.

`x^(2) +4x - 672 = 0`

factorize the function above,

the factor of 672 is 24 and 28, because if it's multiplied it becomes 672 and if it's subtracted equal to 4. since
`4x` is positive, the higher number (28) should be in positive. Therefore we conclude that

`(x+28)(x-24)` = 0

`x+28=0, x = -28\\x-24=0, x = 24`

We takes the positive result, because the time is in positive. so, the estimated time is 24 hours