## PERCENTAGE COMPARISONS

With percentages, the wording is very important.

Let me illustrate this with a couple of examples:

- Fred has 10 candy bars. Sally has 8 candy bars. Fred has 25% more candy bars than Sally, but Sally has 20% fewer candy bars than Fred. In one case it’s 25%; in the other case it’s 20%. The distinction is important.
- Jenny has 65 dollars saved. Mike has 50 dollars saved. Jenny has saved 30% more money than Mike, yet we could also say that Jenny’s savings is 130% of Mike’s savings. In one case it’s 30%; in the other case it’s 130%. The wording makes all the difference.

## REVIEW OF PERCENTS

A percentage is an alternate way to express a decimal or a fraction.

100% corresponds to one unit.

So, for example, if Linda ate 50% of the donuts, this means that Linda ate half of the donuts, since 50% is half of 100%.

Here are a few more examples:

- 200% means double, since 200% is twice 100%.
- 25% is one-fourth, as 25% is a quarter of 100%.
- 150% is one and one-half, since it’s 1.5 times 100%

The purpose of this article isn’t to *teach* percents, but to explain the importance of how it is worded. This quick review was intended just as a brief refresher to illustrate the basic concept.

## INTERPRETING PERCENTS

Let’s look at the two original examples more closely.

(1) Fred has 10 candy bars. Sally has 8 candy bars.

When Fred compares his candy bars to Sally’s, Fred divides 10 by 8 to get 1.25, which equates to 125%.

(Recall that any decimal value can be converted into a percentage by multiplying by 100%, since 100% means one.)

Since 125% is 25% more than 100%, this means that Fred has 25% more candy bars than Sally.

However, when Sally compares her candy bars to Fred’s, Sally divides 8 by 10 to get 0.8, which equates to 80%.

Since 80% is 20% less than 100%, this means that Sally has 20% fewer candy bars than Fred.

The distinction here is that in the first case Fred used Sally’s candy bars for the comparison, so Fred divided by Sally’s number (8) to see how his compared to hers.

In the second case, Sally used Fred’s candy bars for the comparison, so Sally divided by Fred’s number (10) to see how hers compared to his.

In either case, divide by the number that you’re comparing with.

(2) Jenny has 65 dollars saved. Mike has 50 dollars saved.

This time, we’ll only compare Jenny’s savings to Mike’s savings, so we’ll definitely divide by Mike’s 50 dollars.

Therefore, we divide 65 by 50 to get 1.3, which equates to 130%.

This means that Jenny’s savings is 130% of Mike’s savings. That’s one way to put it.

There is another way to say the same thing. Since 130% is 30% more than 100%, we could instead say that Jenny has 30% more savings than Mike has.

In the second case, we used the word ‘more.’ If you follow the percentage by the word ‘more’ (or by the word ‘less’βor their synonyms, like ‘fewer’) you’re comparing the overall percentage (which we obtained by dividing the two values) to 100%.

Saying that Jenny has 130% of Mike’s savings means to multiply Mike’s savings by 1.3 ($50 times 1.3 equals $65).

Saying that Jenny has 30% more than Mike means to find 30% of Mike’s savings and then add that to Mike’s savings ($15 plus $50 equals $65).

## COMPARISON EXAMPLES

Here are a few more examples:

- 3 bananas is 75% of 4 bananas.
- 4 bananas is 133% of 3 bananas. (Technically, it’s 133 and 1/3 percent, but I rounded.)
- 3 bananas is 25% less than 4 bananas.
- 4 bananas is 33% more than 3 bananas. (Really, 33 and 1/3 percent.)
- $800 is 400% of $200.
- $800 is 300% more than $200.
- $200 is 25% of $800.
- $200 is 75% less than $800. (Subtract 25% from 100%.)
- 90 cents is 150% of 60 cents.
- 90 cents shows a 50% improvement over 60 cents.

Copyright Β© 2014 Chris McMullen, author of the *Improve Your Math Fluency* Series

I just might learn something here. π

I hope to convince several science students to admit as much tomorrow. π

I cannot even begin to think about this, Chris. You used to be a safe haven for me. Now, I get nervous just coming into your class. I’ll try this again tomorrow. π

Don’t worry. There won’t be any tests. π

I’m taking the worst pop quiz at the moment, so thanks for going easy on me. There are far too many numbers up there, Sir. They started dancing…

Gee, I could replace them with letters, like x and y. π

:} No, they dance even faster…