Ratios and Rates: Reading Percentages Correctly

1.Ratios, Rates, and Proportional Sharing

A ratio sets two quantities side by side to compare them, as in "rice to water at 1 : 1.2" or "a male-to-female ratio of 3 : 2." A rate expresses that ratio as a single number: the compared quantity divided by the base. The rate for 3 : 2 is 3/2 = 1.5, and multiplying it by 100 gives a percentage (150%).

Proportional sharing splits a total according to a given ratio. Add the terms of the ratio to find how many shares there are in total, find the size of one share, then multiply.

2.Comparing with Rates: Unit Prices and Bundle Deals

Rates are most useful when you compare two things of different sizes on the same basis. If you want to know which of two products with different sizes at the supermarket is cheaper, divide the price by the volume to get a unit price, such as "price per 100 mL" or "price per 1 g." First make sure both products use the same unit. If one is labeled in L and the other in mL, convert using 1 L = 1,000 mL before dividing.

Bundle deals also become clear once you turn them into rates. "1+1" (buy one, get one free) means you get 2 items for the price of 1, so the price per item is cut in half: a 50% discount. But "2+1" means you get 3 items for the price of 2, so each item costs 2/3 of the original price, and the discount is 1 − 2/3 = 1/3, about 33%. It's easy to focus on the 1 free item and feel it is a 50% discount, but remember that the base is the total number of items you receive.

3.The Three Forms of Percentage Problems

Almost every percentage problem comes from a single equation: base × rate = compared quantity. Depending on what you don't know, it takes one of three forms. Asking for 15% of $80,000 means finding the compared quantity; asking what percent $12,000 is of $80,000 means finding the rate; and asking for the whole when 15% is $12,000 means finding the base.

Converting percentages to decimals before multiplying cuts down on mistakes. 15% is 0.15, 8% is 0.08, and 120% is 1.2.

4.Percent vs. Percentage Point

If an interest rate rises from 2% a year to 3% a year, how much did it rise? Both of these answers are correct. Looked at as the difference between the two percentages, it rose by 1 percentage point (1 pp); looked at relative to the original rate, it rose by (3 − 2) ÷ 2 = 0.5, or 50%.

A percentage point is simply the difference between two percentages, while a percent is the rate of change relative to the original value. In news stories and reports, "up 50%" and "up 1 percentage point" can describe the same event in different ways. The choice changes the impression a lot, so whenever someone describes a change in a percentage, check which of the two they mean.

5.If You Cut 20% and Then Add 20%, Are You Back Where You Started?

A $10,000 item with a 20% discount costs $8,000. Raising that by 20% seems like it should give $10,000, but you get $9,600. The base for the discount was $10,000, while the base for the increase was $8,000. 20% of $8,000 is $1,600.

Problems like this become obvious once you turn percentages into multipliers. A 20% discount is × 0.8, and a 20% increase is × 1.2. Doing one after the other gives × 0.8 × 1.2 = × 0.96, which is 4% lower than the original. Even if you reverse the order, raising first and cutting later, you still get 0.96.

For the same reason, a 30% discount followed by an extra 30% off is not a 60% discount. 0.7 × 0.7 = 0.49, so you pay 49% of the original price: a 51% discount.

6.Rates of Increase and Decrease

The rate of increase is "change ÷ previous value." If visitors went from 120 to 150, that is (150 − 120) ÷ 120 = 0.25, a 25% increase. Going the other way, from 150 to 120, is (120 − 150) ÷ 150 = −0.2, a 20% decrease. The difference is the same 30 people, but because the base differs, it comes out as 25% one way and 20% the other.

So "up 25% from last year" and "last year was 20% lower than this year" are the same fact. When you read a growth rate, always look at when the starting point was and what the base value was. If the base value is very small, even a small change becomes a large percentage: going from 2 cases to 4 cases is a 100% increase. We'll come back to this trap in Lesson 9.