60 Part 1: The World of Numbers
Arithmetic with Fractions
When you work with fractions, the methods of whole number arithmetic are no longer enough
to handle the job. You’ll need some new strategies for multiplying and adding. Once you have
learned those methods, however, subtraction and division will just be variations and so not
much more to learn. In fact, we’ll cover addition and subtraction together and multiplication
and division together, because they have so much in common. Before that, however, you need to
master the key to working happily with fractions: the art of disguise.Equivalent Forms
The key to working successfully with fractions is to be able to find equivalent forms of the same
fraction. All that means is that you can change the way a fraction looks without changing its
worth. The simplest way to do that is to multiply by 1, in disguise.
Disguise? Well, the number 1 can be written as a fraction in many different ways:^44 ,,,^19197373 or
any number over itself. Multiplying a number by one will never change its value, but multiplying
by 1 wearing the right disguise will change the number’s appearance.
You already know how to multiply by 1 when it appears as a whole number. In fact, you know
that there is nothing to do when multiplying by 1, because multiplying a number by 1 doesn’t
change it. But when the 1 puts on its fraction disguise, things are a little different. If, for example,
you multiply^12 by 1, you know you should get a number equal to^12. But if the 1 is disguised as^55 ,
the answer won’t look like^12 , even though that’s what it will be worth.There are lots of numbers that are worth
1
2. That fraction says the whole was divided into two
parts, and you have one of them. If 2 wholes were each divided into two parts, there would be a
total of 4 pieces. When the first whole is broken in 2, you should get 1 piece, and then when the
second whole is broken, you should get 1 piece of that. You get 2 of the 4 pieces.
1
22
4(^5). There
are lots of other fractions that are equal to
1
2 , and in fact, there are lots of names for any fraction.
They all have the same value but different appearances.
You don’t want to break things into parts and count every time you need a fraction to change
its appearance, so there’s a simple shortcut. For that shortcut, you need a little preview of
fraction arithmetic. The basic rule for multiplying fractions says to multiply numerator times
numerator and denominator times denominator. When you want to change the appearance of a
fraction, multiply it by a disguised 1.
1
2
5
5
(^3) is still worth^1
2 because you’re multiplying by 1, but its
appearance changes to
5
10. In the same way,
1
2
3
3
3
6
(^35) and^1
2
19
19
19
(^538)
(^3). The same fraction can have
many different looks.