Question

Dear Mitch,

I hope this is in the category of SAT subjects, because I'm pretty sure it's tested on the SAT, but right now I'm just having trouble doing it in school, even though it's supposed to be a review!

Anyway, it's one of those percentage things and to give you an example of the type I can't seem to get right, here's one:

Something in a store is on sale and it is reduced by 20%. The sale price is now $240. How much was it before the sale?

What I did, which the teacher says is wrong, is I took 20% of 240 and added it to 240 to get the price back up to what it was before the sale. So, after you do the canceling out stuff, 2/10 x 240 = 48, and I added that to 240 and got 288. But when I circled $288 she said it was wrong. I think she explained it the next day, but I had to go to the orthodontist and I missed class.

What's wrong with the way I did it?

Sincerely,

Mark K.

Wausau, Wisconsin

Answer

Dear Mark,

You know, the orthodontist probably could have straightened things out for you.

(Though they generally try to stick to the teeth.)

Seriously, this is very much an SAT type question and I promise you an easy one once you take a new look at it.

First of all, as I wrote in a tip a few days ago, when it comes to 'percentage change' questions, direction is EVERYTHING. In other words, you cannot figure out what the price was before it was changed a certain percentage by taking the new number and going backwards in time. NO, THAT IS VERY WRONG, YET IS SO COMMONLY DONE THAT THE ANSWER YOU ARRIVE AT USING THAT METHOD IS ALWAYS ONE OF THE MULTIPLE CHOICE (incorrect) ANSWERS. (AND IT IS ALWAYS WRONG).

Why? Because you are taking a percent (the same percent) of a DIFFERENT number from the number the original change was based upon.

So, here's how I suggest thinking about problems like this:

They involve DIMMER SWITCHES.

WHAT????

A DIMMER SWITCH, the type of light switch that a lot of homes have in at least one room (often the dining room), that enables you to turn the lights down without turning them completely off. Sometimes they are round buttons that rotate, sometimes they are flat and slide along a little track, and there are other styles, but they all do the same thing: they allow you to make the room less bright using the same bulbs that are in use when the room is at full brightness.

O.K. Now, If you turn the lights down 20%, you are making them operate at 20% off. And if they are 20% off, that mean that they are 80% on. (Because the amount off and the amount on must equal the full amount possible, which is 100%. (Sometimes it helps to think of 100% as 'complete", as when a job is 100% done.)

Likewise, if the lights are turned 70% off, they are 30% on. Etc.

So, when a problem states that a price is reduced by 20% and is now $240, you should think of it as 20% OFF and, therefore, 80% ON. And to find out how much light 80% gives you, you would want to know what kind of bulb was in the socket to begin with.

Mathematically, it looks like this:

20% off something is 80% ON, or 80% of something.

For "something" in math we use a variable, so let's use V for "variable".

80% of something equals $240.

("Of" in math means 'multiply' or "times").

So:

80/100 x V = 240 (remember that percent is Latin for 'out of 100', which is how we knew to change 80% to 80/100.

Simplifying:

8/10(V) = 240

4/5 (V) = 240

5/4 x 4/5 (V) = 240 x 5/4

(using 'reciprocals' to make the mysteries land on the right side of the equation)

V = 1200/4

V = 300

SO: 80% of 300 = 240

AND: 20% OFF 300 = 240

The answer is $300.

Sorry, but your teacher was correct in saying that $288 is not it.

So, I guess there's good news and bad news: bad news is you were wrong, and good news is that it should now be easy. Either way, I hope your orthodontist was able to get you ready to brace yourself for the upcoming exam.

Hope this helps.

Truly,

Mitch