Percents: The Search for 100%

[Rescued from my old blog.]

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 whole 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.

_____% GST (Goods & Services Tax)/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?

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.

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
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.

4 thoughts on “Percents: The Search for 100%

  1. Woo! I love Tolkien math. If only the word problems in my fifth grade class had *orcs* in them, I might have grasped these concepts long ago. 😀 Ah, but it was always trains and crops.

  2. Lydia, I have more Tolkien math problems that I’ll probably post some day. Our PUFM group (for teachers learning math) had an assignment to write word problems, so I set all mine in the world of Middle Earth. They are fun to write: Just take any page of homework calculations and write a story for each one. Then throw out the boring ones, and keep the gems.

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