84 Part 1: The World of Numbers
If you had two lamps, one 2 feet tall and one 3 feet tall, you could say one is 1.5 times taller than
the other, but more often, you’ll stay with the whole numbers and say the taller lamp is to the
shorter one as 3 is to 2. Or you could say the shorter lamp is to the taller one as 2 is to 3.Understanding Ratios and Extended Ratios
A ratio is a comparison of two numbers by division. If one number is three times the size of
another, you say the ratio of the larger to the smaller is “3 to 1.” This can be written as 3:1 or as
the fraction^3
1. You could also compare the smaller to the larger by saying the ratio is 1 to 3.
DEFINITION
A ratio is a comparison of two numbers by division.When you are told that the ratio of one number to another is 5:2, you are not being told that the
numbers are 5 and 2, but that when you divide the first by the second, you get a number equal
to^52. This happens when the first number is 5 times some number and the second is 2 times that
number. If that’s all you know about the numbers, you have a sense of their relative size, but that’s
about all. If you have some other information, you might be able to figure out what the numbers
actually are.
Suppose two numbers are in ratio 7:3 and their sum is 50. You know that the first of the numbers
you’re looking for is 7 times some number and the other is 3 times that same number, so that
they divide to^73. With the extra piece of information that they add to 50, you can try to find the
numbers by guess and test. If the numbers were actually 7 and 3, they would add to 10. Multiply
each one by 2 and you have 14 and 6, which add to 20. A little more experimentation will tell you
that 7 v 5, or 35, and 3 v 5, or 15, will add to 50.
That guess and test method can run into some problems. For example, most people won’t guess
anything but whole numbers, and the answer won’t always be a whole number, or it might be a
very large whole number. Try using a letter, maybe n, called a variable, for the number you don’t
know. If the numbers are in ratio 6:5, one is 6n and the other is 5n. If they add to 88, you can say
6 n + 5n = 88. That means that 11n = 88, and n must be 8. The numbers are 6 v 8 = 48 and
5 v 8 = 40.DEFINITION
A variable is a letter or other symbol that takes the place of an unknown number.