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Dear Mitch,

You know those questions they have on the SAT test that start out with something like "If 6 pencils cost g dollars, then how many can you buy for..."

Well, I seem to get them wrong a LOT. Do you know a way to make all the variables go away so the answer can pop out or something like that?

And even if you don't get to this letter, thanks anyway, because I enjoy reading the ones you do answer.

M.R. Ginsberg

High School Junior

Palo Alto, CA


Dear M. R.,

Of course I know those questions!

And I happen to like them because they're the type that students find perplexing until you just get them to see them in a new way. Then, they become MUCH easier.

Here's one:

If 20 paintbrushes cost b dollars, how many paint brushes can be purchased for 6 dollars?

A) 3b/20

B) 120b

c) b/120

D) 120/b

E) 20b/6


First think of this dialog:

"Henry, if you don't mind my asking, how much was that television?"

"Oh, I remember exactly. I bought seven of them for my brother's motel, and that cost 1800 dollars."

What's wrong?

This: A more helpful response would have been the price of one television. Correct?

And the next step is to divide the total number of dollars by the number of televisions to figure out the price of one.

In the above problem, when they say 20 paintbrushes cost b dollars, what you want to do is get the cost of one paintbrush.

And to turn a number into one, you divide the number by itself. And as long as you divide the other side of the equation by the same number, there's no new mathematical problem:

20 = b

20/20 = b/20

1 brush costs b/20 dollars.

And now you are being asked to figure out how many can be bought for 6 dollars. Well, whenever you wish to figure out how many of an item you can purchase for a certain amount of dollars, you take the certain amount of dollars and divide it by the cost of one item (one paintbrush). (just like when you have sixteen dollars and you want to know how many movie tickets you can buy, and each ticket is eight dollars, you divide the total (16) by 8 and you get two tickets.

So, you divide the 6 dollars by b/20 (which was the price of one brush).

And when you divide by a fraction you simply flip that fraction over and multiply.

So, 6/1 divided by b/20 =

6/1 x 20/b =

(which was choice D)


By the way, in the real world, instead of asking how much something costs, often people will ask how much something 'is'. Well, in math language, 'is' means equals, as in '3 +4 is 7'. So the translating the price part to an equation can be very easy when you think of it like that!

Hope this helps,