posted by Laura on .
Alfonso got an 87.5% on his test. He got 21 right. How many questions were on the test??
Let's clarify some terms first:
1) %. This is a percent sign. "Cent" means "100." (A century is 100 years). "Per" means "for each one." (I can buy cans of coke for 50 cents per can). So what we have to do is divide.
If we say 50%, we take 50 divided by 100.
If we say 25%, we take 25 divided by 100.
2) Is/Are in math, if they are by themselves, almost always means equals.
If we take the phrase, "5 times 20 is 100," we can easily substitute the word "is" with an equal sign.
5 times 20 IS 100.
5 times 20 = 100.
See how that works? OK. This is important for this problem. More on that in a minute.
3) When we say "Of," it usually means we multiply a number. 10% OF 20 is 2. (We take 10/100 (since it's a %) TIMES (since it said "of") 20. Trust me...it will give you 2. Go ahead and try it).
Now, let's get down to solving this problem.
Let's delete most of the words. Math questions often have TOO MANY words and it confuses the crap outta' us. Let's just look at what we know:
87.5% on a test.
21 were right.
The question we need to know is "How many questions were on the test?" The question on everyone's mind (which we won't look at now) is, "who cares? The test is over!" Unfortunately, it's questions like this that leave math teachers to be left alone in the faculty lounge. Who wants to talk to someone that asks THESE questions all day long? Anyway...back to our problem.
Since we don't know how many questions there are, that becomes what we call our unknown thing-a-ma-bobby. The cool thing about math is that we can take these unknown thing-a-ma-bobbies and give them just a letter. The most normal letter used is "x," but I'm not a normal person. So I'm going to call it "q." You can pick your favorite letter....I don't care.
So let's look again. Here's what we know:
--87.5% was his score.
--21 were right.
--There were "q" number of questions on the test.
Now, let's put it all into a few sentence:
--Alfonso correnctly answered 87.5% OF the number of questions.
--The number of questions he correctly answered IS 21.
AHA! A percent, an "of" and an "is."
So let's substitute from the rules I taught you before.
87.5% = 87.5/100 = .875
.875 OF q
.875 x q
Or as it's more easily written:
That .875q IS 21. So...
.875q = 21 (Is = equals)
Now we just have to get "q" by itself. Since we're multiplying .875 times q, we have to divide to make it be 1q. To know how to do this, let's think back.
If you have 5 apples and give them to 5 friends, each friend gets 1 apple.
If you have 3 dogs and give them to three friends, each friend gets 1 dog.
AHA! If we divide something by itself, we get 1. So we can divide .875/.875 and get ... yes....1.
But if we do that to one side, we have to do it to the other. So:
----- = ----
q = 24
Now you can go to Alfonso...after all this work...and say to him, "I figured it out. There were 24 questions on your test." Guess what....he won't care and think you're crazy for spending all that time to figure it out.
(As a side note: there are practical reasons to learn this. So don't let my sarcasm towards the math problem shine through too much).
Hope that helped. It was a lot to type, but I tried to think of it step by step and explain WHY for some of those things.