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Author:
Christopher Danielson

To demonstrate the importance of paying attention to the whole when using percents.

Through real-world examples and through diagrams, several percent problems are solved to demonstrate an important principle of percents-that the whole is important.

Tutorial

Making sense of a percent problem depends on keeping track of the whole.

*Percent* means *for each one hundred*. We use percents to compare (typically) a part to a whole, and we treat that whole as if it were one hundred. This is helpful when we want to compare two different sized wholes.

But it gets in the way sometimes.

Let's see how...

My wife bought some cough drops the other day, and I took a picture of the package:

...because I was wondering *33% more than what?*

More than there used to be in a package, right? But how many was that?

Here's why it matters, and here's what I mean about the whole.

The only number we have handy to find 33% of is the 40 drops that are now in the package. So we find 33% of 40 drops...

So there must have been drops originally.

But there are two problems with this:

Why would there be a fractional number of drops in the bag?

Maybe it's a rounding error. Maybe they rounded off the 33%, and it's really supposed to be 27 drops in the original bag.

But then...

When we find 33% of 27 to add the extra drops back, we don't get 13 drops anymore:

Even if we round this up to 9 drops, we still don't get back to 40 drops.

So what happened?

What's wrong is that we used the wrong whole. When we found 33% of 40, we were finding 33% of the wrong total. It's 33% of *what was in the bag* that we need to find.

But we don't know what was in the bag. Catch-22, right?

Here's the original bag:

33% is about , so we'll cut the original bag into thirds:

And we'll add an extra one on:

This is the new bag, which has 40 drops in it. Therefore, each part must have 10 drops:

And we can see that there were 30 drops in the original bag.

The symbolic/algebraic version of this reasoning requires the insight that the new bag is 133% of the old bag. So if *x* is the number of drops in the old bag, we can write:

Ignoring rounding, we get 30 when we solve for x.

The main idea here is that we have to pay attention to the whole when we are working with percents. We cannot just assume that the percent applies directly to the numbers we know.

Here are some other problems this principle applies to:

Your group has $60 altogether for pizza. The tax is 5% and you want to leave a 15% tip on the price of the food before sales tax. What is the maximum amount your group can spend and not go over $60? (Remember that tax and tip are on the food, not on the money you have available to spend)

Here's a coupon from the now-defunct Border's book store:

If I can add together my 10% discount and the 40% discount, why is it *nearly 47% *instead of *exactly 50%*?

Solutions below...

Tax and tip are each on the same original whole-the amount of food we ordered. So we can add these together to get an extra 20% total. Twenty percent is . So we need to cut our food total into fifths:

And add one:

That totals $60, so each piece must have had $10:

And we can order $50 worth of food.

The second discount is taken off the discounted price, not the original price. So a $100 book is discounted by $40 to $60. Then the 10% discount is applied to the $60 price (not 10% of the original $100 price).

Ten percent of $60 is $6, so we end up paying $54.

Incidentally, a better description of this total discount is *exactly 46%*. And it comes out that way no matter what the original price of the book was.

While 46% is in fact nearly 47%, it seems strange to sell it that way. Is this why Border's went under? They couldn't do percents?