Denominators tell you what kind of things you have, and it’s difficult to even think about adding unless you have the same kind of things, or a common denominator. When you have a common denominator, you simply add the numerators and keep the denominator. 3
11 these things plus 7 of these things equals 10 of these same things.
DEFINITION
A common denominator is one that is a multiple of each of the denominators of two or more fractions. The least common denominator is the least common multiple of the denominators.
Your real work comes when the fractions don’t have a common denominator to start out. It’s not impossible to add or subtract fractions with different denominators, but you must change their appearance first. You have to change them to equivalent fractions with a common denominator.
A common denominator is a number that is a multiple of each of the denominators you were given. Ideally, you should choose the lowest number that can be evenly divided by each of the denominators in your equation. That’s called the lowest or least common denominator. Larger multiples will work, but you will have to simplify at the end. Let’s look at some examples.
Suppose you want to subtract 1 3
2
27. In order to do this operation, you need to make the denomina-tors of the two fractions match. First, find the lowest common multiple of the two denominadenomina-tors.
The lowest common multiple of 3 and 7 is 21. Multiply 1 3 with the same value but a different look.
Then you are able to subtract.
Sometimes you’ll look at two denominators and know quickly what number can serve as a common denominator, but if the numbers are large, you may need more of a game plan. To find the lowest common denominator, take a moment first to factor each denominator.
Suppose you need to add 7 30
5
142. You look at 30 and 42 and don’t see a common denominator immediately. Think about the factors of each denominator. 30 = 2 v 3 v 5 and 42 = 2 v 3 v 7.
Both denominators have factors of 2 and 3, but 30 also has a factor of 5. 42 has a factor of 7. The lowest common denominator will be the product of the 2 and 3 they have in common and the 5 and 7 they don’t. factor of 5. Give each fraction what it’s missing, then add and simplify.
WORLDLY WISDOM
It is not always necessary to have the lowest common denominator. Sometimes you simply want to add or subtract the fractions as quickly as possible. You can fall back on this strategy. Multiply the two denominators for a common denominator. Multiply the lower right denominator times the upper left numerator and put the result in the upper left numerator’s position. Multiply the lower left denominator by the upper right numerator and put the result in the upper right numerator’s position. Add or subtract the numerators and simplify if necessary.
You can handle mixed numbers in addition and subtraction by changing them to improper fractions, as you did in multiplication, but you don’t always have to. In addition, you can just combine the fractions and add the whole numbers on to that sum at the end. 43
5 25
For subtraction, it can be trickier, because there may be regrouping, or borrowing, involved. Try to subtract the fraction parts, and if you can, then you can also just subtract the whole numbers.
If you try subtracting the fractions and realize the first fraction is smaller than the second, you need to borrow from the first whole number. To subtract 81
5 24
Complete each addition or subtraction problem.
16. 4
The Least You Need to Know
•
To change a fraction to an equivalent form, multiply (or divide) both the num erator and the denominator by the same number.•
To multiply fractions, multiply the numerators, multiply the denominators, and simplify if possible. You may be able to simplify before multiplying by dividing a numerator and a denominator by the same number.•
To divide fractions, invert the divisor and multiply.•
The lowest common denominator of two fractions is the LCM of the two denominators.•
To add or subtract fractions, change each fraction to an equivalent fraction with a common denominator, and add or subtract the numerators. Simplify if possible.6
Decimals
In This Chapter
•
The system of decimal fractions•
Writing very small num-bers in scientific notation•
Adding and subtracting decimals•
Multiplying and dividing decimals•
Converting between frac-tions and decimals Not every number system in history had a way to representparts of a whole but our system does. In fact, it has two.
You’ve already met common fractions, the ones that are written as a quotient of two integers. In this chapter, you’ll get to know decimal fractions.
A decimal fraction, or as most people call it, a decimal, is a representation of part of a whole written in a way that fits into our decimal place value system. In this chapter, you’ll see how to extend the place value system to include parts of a whole, and how to write very small numbers in scientific notation. You’ll look at each of the operations of arithmetic when decimals are involved, and you’ll see that one of the advantages of decimals is that they fit nicely into the methods you already know. Just in case you need to switch between the two ways of writing fractions, you’ll learn to convert from common fractions to decimal fractions and back.