Operations with Fractions Study Guide (page 2)
Introduction to Operations with Fractions
Five out of four people have trouble with fractions.
—Steven Wright (1955– )
This article explores the different types of fractions and how to order them. A discussion on how to perform operations of these types of numbers comes later.
Fractions are used to represent parts of a whole. You can think of the fraction bar as meaning "out of" or "divided by."
- means 1 out of 2 or 1 divided by 2.
- means 5 out of 8 or 5 divided by 8.
Although you may call the top part of the fraction the "top" and the bottom part of a fraction the "bottom," their technical names are numerator and denominator.
You are never allowed to have a zero in the denominator. Anything over zero is undefined. In other words, this is not a real number and should never be done.
A proper fraction has a numerator that is smaller than its denominator:
Improper fractions have numerators that are bigger than their denominators:
A mixed number is a number that is represented as an integer and a fraction. The following are all mixed numbers:
To change a mixed number into an improper fraction, follow these steps:
- Multiply the denominator of the fraction by the number.
- Add that sum to the numerator.
- Put that amount over the original denominator.
Try these steps with .
Step 1: 4 × 7 = 28
Step 2: 28 + 3 = 31
is equal to . The only difference is that is easier to work with.
Negative fractions are fractions that have a minus sign in front of them. Just like integers and numbers, a fraction multiplied by a negative also becomes negative. For example:
A fraction can be considered negative if either its numerator or denominator is negative. In this example, it might look like only the numerator is negative, but in fact, the entire fraction is receiving the negative value.
When a fraction receives any type of sign, particularly the negative sign, it can appear in three different places—in the numerator, denominator, or right before the fraction:
Any way you write it, you are indicating the same number. No matter where the negative is placed within the fraction, it has the same value.
Adding and Subtracting Like Fractions
To add or subtract fractions, the denominators have to match. To add fractions with like denominators, just add the numerators. To subtract fractions with like denominators, just subtract the numerators.
Adding and Subtracting Unlike Fractions
To find the sum or difference of two fractions with unlike denominators, rename the fractions with a common denominator. Then, add or subtract and simplify your answer.
Let's put this into practice. What would equal?
To subtract these fractions, you first need to find a common denominator. A good method for finding a common denominator is the bowtie method. In the bowtie method, multiply the two denominators. Then, multiply up and diagonally to get the numerators. You will be left with two fractions that have the same denominator.
To multiply fractions, multiply the numerators, then multiply the denominators, and finally simplify, if possible and necessary.
Try this with . First, multiply the numerators: 4 × 2 = 8. Now, do the same with the denominators: 5 × 3 = 15. So, >, which cannot be simplified.
If you are asked to find the fraction of a number, multiply that number by the fraction. In other words, of means ×, or "multiply." For example, of 16 means .
To divide one fraction by another, you need to flip the second fraction and then multiply the fractions. This flip of the second fraction is called the multiplicative inverse of a number or the reciprocal.
It's easier than it may seem. Try this problem:
First, find the reciprocal of. Now, multiply: .
Sometimes, you may need to find the greatest fractions or put fractions in order from least to greatest or from greatest to least.
Let's try putting the following fractions from least to greatest:
To do this, you should first give all the fractions a common denominator. The least common multiple of 3 and 4 is 12, so let's make 12 the new denominator for all four fractions:
All that's left to do is to put the numbers in order from least to greatest:
So, this means the order of the fractions from least to greatest would be and .
Find practice problems for these concepts at Operations with Fractions Practice Questions.
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