Hi Mr. Mcclenathan,

I found this on the No Calc section of the March 2018 SAT, can you solve it the “traditional way” and with shortcuts (if there are any)? (it’s a grid-in question)

Tamika is ordering desktop computers for her company. The desktop computers cost $375 each, and tax is an additional 6% of the total cost of the computers. If she can spend no more than $40,000 on the desktop computers, including tax, what is the maximum number of computers that Tamika can purchase?

Thank you!!

The “shortcut” here is to remember the distributive property of multiplication, which applies to sales tax. 🙂 That is to say, instead of trying to figure out a total and then applying sales tax, figure out the cost of one computer with sales tax. That may seem like it’s a pain without a calculator, but the numbers here are easy enough to work with. I’m going to lay out how my scratch work would look on this without a calculator:

\dfrac{6}{100}\times \$375\\\\=\dfrac{6\times 375}{100}\\\\=\dfrac{(6\times 300)+(6\times 75)}{100}\\\\=\dfrac{1800+450}{100}\\\\=\$ 22.50

That means the total cost of a computer including sales tax is $375 + 22.50 = $397.50. For our purposes, that’s essentially the same as $400. So how many $400 computers can Tamika buy for $40,000? 100 of them.

Now, you may be worried that by rounding $397.50 up, I’m not being precise enough. Easy enough to check! In that rounding, I’m only adding $2.50 to the total, so at 100 computers, she’ll really spend $250 less than the full $40,000. That’s not enough for a 101st computer, though, so she can still only afford 100 computers.

The shortcut to the shortcut—if you’re brave enough (or out-of-time enough)—is to take one look at the numbers in the problem and assume that, because this is a no-calculator question, the price of a computer with tax is going to be just under $400, which lines up nicely with Tamika’s $40,000 budget.

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