Memories: Percent Problems

Homeschool Memories…

Can your students solve this problem?

There are 20% more girls than boys in the senior class. What percent of the seniors are girls?

This is from an old discussion of the semantics of percent problems and why students have trouble with them, going on over at MathNotations. (Follow-up post here.)

Our homeschool co-op prealgebra class had just finished a chapter on percents, so I thought my son might have a chance at this one. Nope! He leapt without thought to the conclusion that 60% of the class must be girls.

After I explained the significance of the word “than”, he solved the follow-up problem just fine.

Percents: The Search for 100%

Percents are one of the math monsters, the toughest topics of elementary and junior high school arithmetic.

The most important step in solving any percent problem is to figure out what quantity is being treated as the basis, the whole thing that is 100%. The basis is whatever quantity to which the other things in the problem are being compared.

Notice the variety of phrases that can be used to signal 100%:

_____% of _____
100% = whatever is after the word “of.”

_____ as a percentage of _____
100% = whatever is after the word “of.”

_____% more/less/greater/fewer than _____
100% = whatever is after the word “than.”

_____% increase/decrease/discount/rebate
100% = the original amount or list price.

_____% gain/profit/loss
100% = the cost of the item to the seller.

_____% VAT/GST/sales tax/down payment/deposit/commission
100% = the cost of the item to the buyer.

_____% interest
100% = the principal of the account or loan.

_____% raise
100% = the worker’s previous wages.

_____% income tax
100% = the person’s annual income, or a portion of that income in the case of a progressive tax.

_____% enlargement/reduction
100% = (usually) the linear dimensions of the original photo or drawing.

a solution containing _____% of some chemical
100% = the volume (or mass, depending on the problem) of the entire solution, NOT the chemical listed after the word “of.”

…and there may be others I’ve missed. Is it any surprise that many of our students struggle with percent problems?

That Tricky Word “Than”

Of all these, I think the most difficult for students, and sometimes even for teachers, to recognize is the comparison word “than”. Consider the following typical sixth-grade problem:

The king of Rohan placed 1500 guards atop the outer wall at Helm’s Deep. 300 of the guards were swordsmen and the rest were archers. How many percent fewer swordsmen were there than archers?

It is natural to think the whole or 100% has to be ALL the guards.

The most common mistake with a problem of this sort is to calculate that 20% of the guards are swordsmen and 80% are archers. Thus there must be 80 – 20 = 60% fewer swordsmen than archers, right?

Wrong! In this problem, the words “than archers” tell us that we are comparing the swordsmen to the number of archers alone. Whatever we are comparing to, that is what we treat as 100%. So in this problem, the whole = the number of archers.

Also Tricky: “More” and “Fewer”

This problem contains one more potential stumbling block. Remember that the basic percent proportion is:

part / whole = percent / 100

Careless students will ignore the word “fewer” and use the number of swordsmen as the part in the proportion. But “fewer” means that we are talking about the difference between the swordsmen and the archers. We need to find that difference by subtracting, and then we are ready to calculate:

whole = archers = 1200
part = how many fewer swordsmen = 1200 – 300 = 900

part / whole = percent / 100
900/1200 = percent/100

…and simplifying the proportion:
(3/4) x 100 = percent = 75

There are 75% fewer swordsmen than archers.

Help your students learn to recognize 100% in all of its disguises, and especially to think their way through “than” comparisons. Be careful in determining what is the whole and what is the part in each math problem. Then set up a percent proportion, and most of the time your solution will fall into place.