Part 5: Fractions
One way to think of a fraction is as a division that hasn't been done yet. Why do we even use
fractions? Why don't we just divide the two numbers and use the decimal instead? In this day of
cheap calculators, that's a very good question. Fractions were invented long before decimal
numbers, as a way of showing portions less than 1, and they're still hanging around. They're used
in cooking, in building, in sewing, in the stock market - they're everywhere, and we need to
understand them.
Just to review, the number above the bar is called the numerator, and the number below the bar
is called the denominator.
We can read this fraction as three-fourths, three over four, or three divided by four.
Every fraction can be converted to a decimal by dividing. If you use the calculator to divide 3 by 4,
you'll find that it is equal to 0.75.
Here are some other fractions and their decimal equivalents. Remember, you can find the decimal
equivalent of any fraction by dividing.
Here are some terms that are very important when working with fractions.
Proper fraction
When the numerator is less than the denominator, we call the expression a proper fraction. These
are some examples of proper fractions.
Improper fraction
An improper fraction occurs when the numerator is greater than or equal to the denominator.
These are some examples of improper fractions:
Mixed number
When an expression consists of a whole number and a proper fraction, we call it a mixed number.
Here are some examples of mixed numbers:
We can convert a mixed number to an improper fraction. First, multiply the whole number by the
denominator of the fraction. Then, add the numerator of the fraction to the product. Finally, write
the sum over the original denominator. In this example, since three thirds is a whole, the whole
number 1 is three thirds plus one more third, which equals four thirds.
Convert 1-1/3 to an improper fraction:
Equivalent fractions
There are many ways to write a fraction of a whole. Fractions that represent the same number are
called equivalent fractions. This is basically the same thing as equal ratios. For example, �, 2/4,
and 4/8 are all equivalent fractions. To find out if two fractions are equivalent, use a calculator and
divide. If the answer is the same, then they are equivalent.
Reciprocal
When the product of two fractions equals 1, the fractions are reciprocals. Every nonzero fraction
has a reciprocal. It's easy to determine the reciprocal of a fraction since all you have to do is
switch the numerator and denominator--just turn the fraction over. Here's how to find the
reciprocal of three-fourths.
To find the reciprocal of a whole number, just put 1 over the whole number. For example, the
reciprocal of 2 is 1/2.
Reducing Fractions
A fraction is reduced to lowest terms, or simplified, when its numerator and denominator have no
common factors. It's easier to multiply, divide, add and subtract fractions when they're simplified.
To simplify a fraction, we find an equivalent fraction where the numerator and denominator have
no common factors. Here are the steps.
1. List the prime factors of the numerator and denominator.
2. Find the factors common to both the numerator and denominator.
3. Divide the numerator and denominator by all common factors (called canceling).
Example 1
In step 1, we list the prime factors of the numerator and denominator. In step 2, we divide by, or
cancel, the factor of two that is common to both the numerator and denominator. That is, since
two divided by two equals one, we say the twos cancel each other. In step 3, we're left with a 3 in
the numerator and we multiply the remaining factors in the denominator to get 4. Here's another
example.
Example 2
In Step 1, we find all the prime factors. In Step 2, we can cancel the common factors 3 and 5.
In Step 3, since the only factors remaining are 2 in the numerator and 3 in the denominator, the
final answer is 2/3.
Adding and Subtracting Fractions
It's easy to add and subtract like fractions, or fractions with the same denominator. You just add
or subtract the numerators and keep the same denominator. The tricky part comes when you add
or subtract fractions that have different denominators. To do this, you need to know how to find
the least common denominator. In an earlier lesson, you learned how to simplify, or reduce, a
fraction by finding an equivalent, or equal, fraction where the numerator and denominator have
no common factors. To do this, you divided the numerator and denominator by their greatest
common factor.
In this lesson, you'll learn that you can also multiply the numerator and denominator by the same
factor to make equivalent fractions.
Example 1
In this example, since 12 divided by 12 equals one, and any number multiplied by 1 equals itself,
we know 36/48 and 3/4 are equivalent fractions, or fractions that have the same value. In
general, to make an equivalent fraction you can multiply or divide the numerator and denominator
of the fraction by any non-zero number.
Since only like fractions can be added or subtracted, we first have to convert unlike fractions to
equivalent like fractions. We want to find the smallest, or least, common denominator, because
working with smaller numbers makes our calculations easier. The least common denominator, or
LCD, of two fractions is the smallest number that can be divided by both denominators. There are
two methods for finding the least common denominator of two fractions:
Example 2
Method 1:
Write the multiples of both denominators until you find a common multiple.
The first method is to simply start writing all the multiples of both denominators, beginning with
the numbers themselves. Here's an example of this method. Multiples of 4 are 4, 8, 12, 16, and
so forth (because 1 × 4=4, 2 × 4=8, 3 × 4=12, 4 × 4=16, etc.). The multiples of 6 are 6, 12,…-that's the number we're looking for, 12, because it's the first one that appears in both lists of
multiples. It's the least common multiple, which we'll use as our least common denominator.
Method 2:
Use prime factorization.
For the second method, we use prime factorization-that is, we write each denominator as a
product of its prime factors. The prime factors of 4 are 2 times 2. The prime factors of 6 are 2
times 3. For our least common denominator, we must use every factor that appears in either
number. We therefore need the factors 2 and 3, but we must use 2 twice, since it's used twice in
the factorization for 4. We get the same answer for our least common denominator, 12.
Example 3
prime factorization of 4 = 2 × 2
prime factorization of 6 = 2 × 3
LCD = 2 × 2 × 3 = 12
Now that we have our least common denominator, we can make equivalent like fractions by
multiplying the numerator and denominator of each fraction by the factor(s) needed. We multiply
3/4 by 3/3, since 3 times 4 is 12, and we multiply 1/6 by 2/2, since 2 times 6 is 12. This gives the
equivalent like fractions 9/12 and 2/12. Now we can add the numerators, 9 + 2, to find the
answer, 11/12.
Multiplying Fractions
To multiply fractions, first we simplify the fractions if they are not in lowest terms. Then
we multiply the numerators of the fractions to get the new numerator, and multiply the
denominators of the fractions to get the new denominator. Simplify the resulting
fraction if possible.
Note that multiplying fractions is frequently expressed using the word "of." For example, to find
one-fifth of 10 pieces of candy, you would multiply 1/5 times 10, which equals 2. Study the
example problems to see how to apply the rules for multiplying fractions.
Example 1
Example 2
Example 3
Dividing Fractions
Dividing by fractions is just like multiplying fractions, except for one additional step.
To divide any number by a fraction:
First step: Find the reciprocal of the fraction.
Second step: Multiply the number by the reciprocal of the fraction.
Third step: Simplify the resulting fraction if possible.
Fourth step: Check your answer: Multiply the result you got by the divisor and be sure it equals
the original dividend.
Note that you can only divide by non-zero fractions.
Example 1
Example 2
Adding and Subtracting Mixed Number
A mixed number consists of an integer and a proper fraction. Any mixed number can also be
written as an improper fraction, in which the numerator is larger than the denominator, Example:
Example 1
To add mixed numbers, we first add the whole numbers together, and then the fractions.
If the sum of the fractions is an improper fraction, then we change it to a mixed number. Here's
an example. The whole numbers, 3 and 1, sum to 4. The fractions, 2/5 and 3/5, add up to 5/5, or
1. Add the 1 to 4 to get the answer, which is 5.
Example 2
If the denominators of the fractions are different, then first find equivalent fractions with a
common denominator before adding. For example, let's add 4 1/3 to 3 2/5. Using the techniques
we've learned, you can find the least common denominator of 15. The answer is 7 11/15.
Subtracting mixed numbers is very similar to adding them. But what happens when the fractional
part of the number you are subtracting is larger than the fractional part of the number you are
subtracting from?
Here's an example: let's subtract 3 3/5 from 4 1/3. First you find the LCD; here it's 15.
4 1/3 - 3 3/5
4 5/15 - 3 9/15
3 + 1 5/15 - 3 9/15
3 + 20/15 - 3 9/15
3 20/15 - 3 9/15
11/15
Write both fractions as
equivalent fractions with a
denominator of 15.
Since you're trying to subtract a
larger fraction from a smaller
one, you need to "borrow" a one
from the integer 4, change it to
15/15, and add it to the fraction.
Now the problem becomes 3
20/15 minus 3 9/15 and the
answer is 11/15.
Multiplying Mixed Numbers
Here are the steps for multiplying mixed numbers.
1. Change each number to an improper fraction.
2. Simplify if possible.
3. Multiply the numerators and then the denominators.
4. Put answer in lowest terms.
5. Check to be sure the answer makes sense.
Dividing Mixed Numbers
Here are the steps for dividing mixed numbers.
1. Change each mixed number to an improper fraction.
2. Multiply by the reciprocal of the divisor, simplifying if possible.
3. Put answer in lowest terms.
4. Check to be sure the answer makes s