Hi Mike…can you explain Question 26 from dSAT 10, math module 1?

Hi Mike…can you explain Question 26 from dSAT 10, math module 1? Thanks!
(And many thanks for your explanations on previous questions!)

Absolutely! This is one where it definitely makes sense to start working with real numbers before you start playing with variables, if you’re at all unsure of where to start working with variables directly.

We know that the bill for 5 hours of work is $400; let’s just start by seeing if we can eliminate any answer choices because they don’t give us $400 when $x = 5$ .

A) $f(x)=60x+100\\f(5)=60(5)+100\\f(5)=400$

B) $f(x)=60x+220\\f(5)=60(5)+220\\f(5)=520$

C) $f(x)=80x\\f(5)=80(5)\\f(5)=400$

D) $f(x)=80x+220\\f(5)=80(5)+220\\f(5)=620$

OK, so a few observations here.

First, if we’re paying attention when we look at choices A and B, and then C and D, they aren’t going to equal the same thing for any value of $x$ , so once we saw A and C came out to 400, we didn’t really need to write out B and D.

Second, even though C came out to 400, there’s no way that’s the answer. That’s almost too obviously there to trap us. This is number 26: it’s supposed to be hard. C would be way too easy.

OK OK cute Mike but why is it A?

Well, we know that for the first two hours, the repair specialist charges a flat $220. That means if he shows up at your hours and spends 20 mins fixing your window, it’s $220. If it takes him 2 hours, it’s still $220. After 2 hours, the meter starts running. The question doesn’t tell us the hourly rate after 2 hours, but we can figure it out from the fact that a 5-hour repair cost $400.

In that 5 hour repair, the first two hours cost $220, and then the next three hours cost a total of $180 to get to $400. Division tells us the hourly rate after the first two hours is $\$180 \div 3=\$ 60$ .

Now here’s where it gets tricky! You can’t just say $60x$ if you’re saying $x$ is the total number of hours on the job, because the first two hours the rate doesn’t apply. So we need to write an expression that has $(x-2)$ in it. When a job is $x$ hours long (and $x > 2$ ), you’re going to pay $60 times $(x-2)$ .