Understanding Percentages

Figures expressed in percentages are ubiquitous, and interpreting and doing calculations with them is an important part of the basic mathematical literacy useful in everyday life. Yet few points of arithmetic are as confusing and full of traps for the unwary as percentages. I recently got fed up with being unable to think clearly about percentages myself, so I figured I’d write this article to force myself to understand them completely. Hopefully it will help you, too!

This article is written for adults and secondary-school students who generally understand arithmetic and basic algebra, but could use a refresher on percentages. I’m going to explain why each fact about percentages is true to make it more memorable, so if you’ve never heard of percentages, the order of operations, or the distributive property, this is probably not the right introduction for you.

What is a percentage?

This may seem like a silly thing to need to remind yourself of, but before we think about anything else, it really is worth taking a minute to remind yourself of the definition of a percentage, because thinking clearly about the definition will often allow you to work out the answers to otherwise difficult questions. N% is simply the number N divided by 100, i.e., a fraction with N as the numerator and 100 as the denominator. Or put another way, it’s a decimal, times 100. So 10% means \(\frac{10}{100}\) or 0.1; 250% is \(\frac{250}{100}\) or 2.5.

Percentages are commutative

Here’s my favorite unexpected fact about percentages: 8% of 50 is the same thing as 50% of 8. While counterintuitive at first glance, this is easy to understand if you keep the definition of a percentage in mind: these simplify to \((8 \times \frac{1}{100}) \times 50\) and \((50 \times \frac{1}{100}) \times 8\), and multiplication is commutative and associative (the order and grouping don’t matter).

This often comes in handy when doing mental math: 8% of 50 feels awkward enough that you might go looking for a calculator if you didn’t notice the shortcut, but 50% of 8 is trivial.

Percent of, percent increase, and percent decrease

These small words make a huge difference to the meaning of a percentage.

Basics

• Percent of is the simplest: this simply means to multiply some number by a percentage. So if we take 20% of 50, that’s \(50 \times 0.2 = 10\).
  • Generally, P% of N is \(N\left(\frac{P}{100}\right)\).
• Percent increase means to multiply some number by a percentage (or, take that percent of the number) and then add the result to the original figure. So a 20% increase from a baseline of 50 is \(50 + 50(0.2) = 60\).
  • Generally, a P% increase to N is \(N + N\left(\frac{P}{100}\right)\).
• Percent decrease means to multiply some number by a percentage and then subtract the result from the original figure. So a 20% decrease from a baseline of 50 is \(50 - 50(0.2) = 40\).
  • Generally, a P% decrease to N is \(N - N\left(\frac{P}{100}\right)\).
• Percent change can be used to refer to both percent increase and percent decrease. A positive percent change is an increase; a negative percent change is a decrease of the absolute value of the percent change (e.g., a −15% change is a 15% decrease).

Converting percent change to percent of

Percents increase and decrease can be converted to percents of by adding the percentage change to 100%. You can see this equivalence clearly by applying the distributive property to the formula for percent increase given above:

\[\begin{align} N\text{ increased by }P\% &= N + N\left(\frac{P}{100}\right)\\ &= N\left(1 + \frac{P}{100}\right)\\ &= N(100\% + P\%)\\ \end{align}\]

So a 150% increase is \(100\% + 150\% = 250\%\) of the original. To convert in the opposite direction, switch the sign on the 100%: if a figure is 120% of something, it’s a \(120\% - 100\% = 20\%\) increase.

The same formula works for decreases if you treat them as negative percent change: a 30% decrease is \(100\% - 30\% = 70\%\) of the original, and 60% of something is a \(60\% -100\% = -40\%\) change or 40% decrease.

Another mental math tip: The conversion from percent change to percent of is almost always beneficial. The most straightforward approach to percent change requires one multiplication and then one addition, and you have to hold onto the original number during the entire problem. If you convert the change to a percent of, the addition is guaranteed to be to or from 100 (easy), and you don’t have to remember anything during the subsequent multiplication.

Percentages equal to or greater than 100%

All this likely feels straightforward so far. The trick is what happens once we reach 100%. The math still works the same way, but the answers don’t make as much intuitive sense anymore:

• 100% of 50: This is just 50.
• 100% increase from 50: This is 100, \(50 + 50(1.00)\).
• 100% decrease from 50: This is 0, \(50 - 50(1.00)\).

In summary, 100% of a number is a no-op (though depending on context, there’s a good chance it means whoever gave you this figure was confused and intended a 100% increase). A 100% increase means the number doubles. A 100% decrease means there’s nothing left, no matter what the original number was. It does not mean the number halved, as you might naïvely expect if you invert the increase case without checking the math; people frequently mix this up. To halve a number, you need a 50% decrease, but to double it, you need a 100% increase.

Similarly with percentages greater than 100%:

• 150% of 50: This is 75, \(50 \times 1.5\).
• 150% increase from 50: This is 125, \(50 + 50 \times 1.5\).
• 150% decrease from 50: This is not well-defined. You could argue it’s −25, but that probably doesn’t make a whole lot of sense in context. A 100% decrease is normally the largest meaningful decrease.

Other words

When you see something that isn’t clearly phrased as of, increase, or decrease, you need to determine how to interpret it to avoid errors. Examples:

• 20% off or a 20% reduction is a 20% decrease.
• An 18% surcharge or tip is 18% of the amount it’s applied to, but represents an 18% increase to your bill.
• A 30% adjustment or change is either an increase or decrease, depending on context.
• 5% as many means 5% of.

Percentage points

When talking about changes in figures already expressed as percentages, things get dicey. If you say 62% of people voted in the last election, and this year there was a 20% decrease, technically you’re saying that the number represented by the percentage decreased by 20%, so that \(0.62 - 0.62 * 0.2 = 49.6\%\) of people voted. You probably meant to say that 42% of people voted; to express this idea, you should say that voting decreased by 20 percentage points (that is, 0.2 should be subtracted from the previous percentage).

This only applies to increases and decreases. If you say that 80% as many people voted in this election as last election, and last election 62% of people voted, there is no ambiguity. (That said, this is a terrible way to express statistics for people to read, as it’s hard to interpret these numbers; nobody wants to have to multiply 80% by 62% to figure out the all-important value of what proportion of people voted this election.)

Sequentially applied percentages

People are often tempted to take shortcuts with percentages that result in incorrect answers. Most of these involve incorrectly combining the effects of several percentages.

Percentage increases or decreases cannot be added

Suppose you’re in a store and see that something’s on sale for 20% off (i.e., its price at checkout will be 20% lower than quoted on the tag), and you also have a 10% off coupon. It’s very tempting to add these up and calculate a 30% decrease, but this is not the same thing.

To see this, suppose the item is $50.

• 20% off and then 10% off: \(\$50 - \$50 \times 0.2 = \$40\); then \(\$40 - \$40 \times 0.1 = \$36\).
• 30% off: \(\$50 - \$50 \times 0.3 = \$35\). (Fast way to do this in your head, applying our tricks from earlier: a 30% decrease from X is 70% of X, and 70% of 50 is the same as 50% of 70, and half of 70 is 35.)

The problem here is that percentage changes are relative to some other number, and after applying the first percentage decrease, the base number has become lower, so the second percentage decrease has a smaller effect. A good way to understand this intuitively is to consider what happens if you have a 50% off coupon and find a clearance rack at 50% off. You probably (correctly) expect you can get a total of 75% off any items from that rack; you don’t expect you can walk out the door with the entire rack for free (*50% + 50% = 100% off) and sell it on eBay for hundreds of dollars.

If you want to correctly combine two percentage decreases into a single percentage decrease, you need to “pre-apply” the first decrease to the second decrease prior to summing them up, to account for the baseline being lower after the application of the first decrease. So to combine 20% and 10%, we would decrease 10% by 20% for 8%, and then add our reduced 8% to 20% for a 28% decrease. Now \(\$50 - \$50 \times 0.28 = \$36\), as we got when we applied them separately. Similarly, if we decrease 50% by 50% and add it to 50%, we get 75%. In general, to combine two sequential decreases of \(A\%\) and \(B\%\) into a combined \(C\%\) decrease:

\[C\% = A\% + B\%(100\% - A\%)\]

Note however that we do get the same result, without doing anything special, if we first take 10% off and then 20% off, rather than 20% off and then 10% off, because percentages are commutative; we can apply the reductions in any order as long as we apply them separately or combine them as described above.

Tip: As you might expect, stores occasionally do this wrong. On the off chance that you notice, you generally should not complain unless an increase is involved somewhere, as the error comes out in your favor. If you’re feeling honest, let me just say this will be a loooong argument, and you’re not likely to find a chalkboard at the cash register.

An N% increase and an N% percent decrease do not cancel out

As a corollary, suppose you have a $50 item that’s not on sale, you have a 10% off coupon, and you know 10% tax will be added at the register. That makes your cost $50, right?

Not so, because the 10% change is applied to different baseline values: the decrease is applied to a higher figure than the increase. A 10% decrease followed by a a 10% increase comes out as \(\$50 - \$50 \times 0.1 = \$45\), then \(\$45 + \$45 \times 0.1 = \$49.50\).

Surprisingly, the order we apply the changes in still doesn’t matter. If we take the 10% increase first, then the 10% decrease, we end up with \(\$50 + \$50 \times 0.1 = \$55\), then \(\$55 - \$55 \times 0.1 = \$49.50\) still. This is easier to intuit if you realize that in either order, the subtraction is applied to the larger number. (It still feels wrong that it’s systematically biased downwards, though, at least to me! If you know of a clever proof that makes this intuitive, let me know.)

The “error” (percentage decrease) introduced by a successive \(N\%\) increase and \(N\%\) decrease, or vice versa, is exactly \(N\%^2\).

Proof: This surprisingly simple result can be derived as follows. Suppose we have a baseline value \(B\). An \(N\%\) decrease in \(B\) can be expressed as \(B(100\% - N\%)\), since \(B\) decreased by \(N\%\) is the same as \(100\% - N\%\) of \(B\). A subsequent \(N\%\) increase similarly multiplies this value by \((100\% + N\%)\), for a total of:

\[B(100\% - N\%)(100\% + N\%)\]

Now distribute (note that 100% times 100% is 100%, since \(100\% = 1\)):

\[B(100\% + 100\% \cdot N\% - 100\% \cdot N\% - (N\%)^2) = B(100\% - (N\%)^2)\]

\(B(100\% - X)\) is equivalent to an \(X\)% decrease in \(B\), so this is the same thing as an \(N\%^2\) decrease. ∎

Percentages by another order of magnitude

Two other concepts are much less used than percentages, but work very similarly and are worth familiarizing yourself with:

• per mille (‰): This is a number divided by 1000, rather than a number divided by 100. 25% = 250‰.
• basis point (bp): This is one-one hundredth of a percentage point. Basis points are most commonly used to describe changes in rates in financial markets. If an interest rate is 4.0% and it increases by 25 basis points, the new interest rate is 4.25%.