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Question

Dear Mitch,

I know you must be getting sick of being asked for more and more 'tips' for the upcoming PSAT's and SAT's, but since it really is coming soon and is still considered one of the most important college entrance tests in the country, would you mind giving just a few more of your great tips?

Sincerely,

SAT and Studied

Answer

Dear Mr. or Ms. S. A. Studied,

YES, I will, and it will be my pleasure. (I never get 'sick' of sharing SAT tips, or, for that matter, discussing any other area of math). However, since it is still summer (though, just the last bit of it), we are still a bit busy handling programs for schools that are beginning AS WE SPEAK, so we will probably only be able to give one or two tips per request.)

(IT IS JUST A MATTER OF TIME CONSTRAINTS AND MEETING ALL OUR PREVIOUSLY AGREED OBLIGATIONS.). STILL, IF YOU KEEP CHECKING IN YOU WILL GET PLENTY!!!

I've recently glanced through some articles I've written on the subject and realized that some of the tips that are most helpful on the test (due to the frequency and ease with which they can be applied to different kinds of problems one is likely to encounter on the exam), are actually among the shortest and easiest to teach and recall. So here goes with today's TIP:

In "percentage change" problems, direction is EVERYTHING...

What does that mean? That means the direction of the change. If the price of something goes up by twenty-five percent, that is different than a situation in which an item goes down by twenty-five percent. And when I say different, I am not just referring to the up or down of it. I am referring to a hidden ingredient that a lot of people miss until they've had some practice with this type of problem.

The tricky part of this concept is most readily brought to light by looking at two different situations side by side:

Situation #1:

There is a shoe store in a fancy part of town. It takes a pair of shoes that normally sells for $30 and shines them up and attaches a big bow to the top.

It then raises the price to $45 and places them in the window.

That store, incidentally, is called "Shoe Store A".

Situation # 2:

Another store, called "Shoe Store B", which is located in a less fancy part of town, takes a pair of shoes that usually sells for $45, and puts it on sale for $30.

Question: In these two situations in which the price changes $15, how much of a percent change was made between 30 dollars and 45 dollars?

Choices:

A) 5%

B) 10%

C) 15%

D) 33 1/3%

E) 50%

Well, guess what? One of the shoe stores has one correct answer and the other shoe store has a different correct answer. Why? Because the amount of the change in price is determined by the price you start with. In one of the above examples, the store starts out with $30, but in the other example the store starts out with the number $45.

The way you figure out the amount that the percentage changed is you take the difference between the before price and the after price (a difference which, in BOTH of the cases, store A and store B, is $15, and you place it ABOVE THE ORIGINAL PRICE.

SO, in the store that begins with $45 and reduces their price $15, you get: 15/45.

And in the store that starts out with $30 and raises their price $15, you get: 15/30.

In other words, the magic formula is:

DIFFERENCE

ORIGINAL

("D /O" , or 'DO' IT RIGHT!!!)

Well, since the numbers 15 and 30 are both multiples of 15, 15/30 can be simplified to 1/2, which is 50%.

And since the numbers 15 and 45 are both multiples of 15, 15/45 can be simplified to 1/3 which is 33 1/3 %.

SO: the answer for shoe store A is choice E, as their price went up 50%.

And the answer to shoe store B is choice D, as their price went down 33 1/3 %.

NOTE: IN MATHEMATICS, I AM USUALLY OPPOSED TO MEMORIZATION. IT IS ALMOST ALWAYS BETTER TO UNDERSTAND THINGS THAN JUST HAVE THEM MEMORIZED (THOUGH BOTH UNDERSTANDING AND COMMITTED TO MEMORY IS NOT A BAD THING EITHER).

HOWEVER, WHEN IT COMES TO FRACTION/DECIMAL CONVERSIONS, THERE ARE A FEW PAIRS WHICH YOU MUST HAVE MEMORIZED IN ORDER TO MAKE YOUR LIFE ON THESE TESTS (AND IN REAL LIFE) EASIER THAN IT WOULD BE WITHOUT HAVING THEM MEMORIZED.

HERE ARE THE EASIEST AND MOST IMPORTANT ONES:

% FRACTION

5% 1/20

10% 1/10

20% 2/10 (OR) 1/5

25% 1/4

30% 3/10

33 1/3% 1/3

4 0% 4/10 (OR) 2/5

5 0% 1/2

60% 6/10 (OR) 3/5

66 2/3% 2/3

70% 7/10

75% 3/4

80% 8/10 (OR) 4/5

90% 9/10

100% 1

150% 1 1/2

200% 2

 

Hope this helps,

Mitch