MathBench > Miscellaneous

Tricks with Division

Section 1: Ratios and Fractions

Finding Ratios

Whenever you do division, you are comparing 2 numbers – essentially you are asking “how many times bigger (or smaller) is the first number compared to the second number”.

gas pricesSo, let’s say you are comparing gasoline prices. Here is some data for the price of a gallon of gas:

Summer 1998: $1.30

October 2001: $1.90

There are 2 ways you could compare these two numbers:

Subtraction: $1.90 – $1.30 = $0.60

that is, a gallon of gas costs 60 cents more in 2001 compared to 1998. This is called the absolute difference, and it is in dollars (or cents, if you make a quick conversion). Here's another way to compare prices:

Division: $1.90 / $1.30 = 1.46

so, gas costs 1.46 TIMES as much in 2001 compared to 1998. Normally we would say, “gas was about one-and-a-half times as expensive”. Sometimes this is called the relative change.

For those of you with sharp eyes, YES, the dollar sign disappeared in front of the answer. In fact, the units (dollars) get cancelled in a fraction just like you can cancel numbers. Technically the answer is known as “unitless” – its not in dollars, or cents, or anything else, it’s just a number.

Let’s extend this example a little:

summer 1998: $1.30

October 2001: $1.90

Summer 2005: $2.50

What is the absolute change from 2001 to 2005?

What is the relative change from 2001 to 2005?

windmills and carsNotice that although the absolute change in price is the same (60 cents in each case) the relative change is different (smaller in the later period because gas prices were already higher).

In fact there is no necessary relationship between absolute and relative change. For example, in some versions of the future, gasoline could reach $20 a gallon:

Summer 2035: $20

Summer 2036: $23

This is a big absolute jump, but in relative terms, it is still only an increase of 23/20 = 1.15 times, actually SMALLER than the increases discussed above.