# At his regular hourly rate, Don had estimated the labor…

## Solution:

To solve this problem we can translate the problem with the given information into an equation. Since we don’t know Don’s hourly rate nor the time he had estimated for the job, we use two variables:

w = Don’s hourly rate

t = number of hours he estimated for the job

We are given that Don was paid $336, based on his original estimate, so we can say:

w x t = 336

Next we are given that the job took 4 hours longer and that, as a result, he earned 2 dollars less than his regular rate. This leads us to say:

(w – 2)(t + 4) = 336

We rewrite the equation w x t = 336 as w = 336/t. Now we substitute 336/t for w in the equation (w – 2)(t + 4) = 336. Thus, we have:

((336/t) – 2))(t + 4) = 336

After FOILing we have:

336 + (4×336)/t – 2t – 8 = 336

(4×336)/t – 2t – 8 = 0

Multiplying the entire equation by t we get:

4 x 336 – 2t^2 – 8t = 0

Dividing the entire equation by 2 we get:

2 x 336 – t^2 – 4t = 0 or 672 – t^2 – 4t = 0

We can also rewrite this as: t^2 + 4t – 672 = 0

Now this is where we should be strategic with our answer choices. To solve this quadratic we are looking for two numbers that sum to a positive 4 and multiply to a negative 672. Our answer choices are:

(A) 28

(B) 24

(C) 16

(D) 14

(E) 12

There are only two pairs of answer choices that are 4 units apart: 16 and 12, and 28 and 24. Since 24 multiplied by 28 is 672, we know that the numbers that are needed for the factoring are 24 and 28. Thus, we can say:

(t – 24)(t + 28) = 0

Thus, t = 24 is the correct answer.

**Answer: B**

Note that when we multiplied the entire equation by “t,” we might have introduced an extraneous solution into the equation. We would normally substitute both solutions (in this case, t = 24 and t = -28) back into the original equation to ensure that each value of t works. If we did this, we would find that t = -28 would not work. This makes practical sense, too, because the variable t represents the number of hours worked, and one cannot work for a negative number of hours.