Fractions are numerical expressions that represent one number divided by another to explain the part of a whole that is being measured. Each fraction is basically a ratio, for example one half is one out of two parts. The top number of a fraction is called the numerator, and the bottom number is called the denominator. Knowing how fractions work can be incredibly helpful to understanding how to arithmetically calculate them.
Let’s start with an addition problem: five-eighths plus three-sixteenths.
When adding fractions, if the denominators are different, you must first find a common denominator. A common denominator is a number that is a multiple of both or all denominators in a given set of fractions.
In this problem, the denominators are eight and sixteen. Because sixteen is a multiple of eight, we will use sixteen as our common denominator. We must determine what number is multiplied by the denominator, eight, to achieve the common denominator, sixteen, and multiply the numerator by the same number. Because eight times two is sixteen, we must multiply by a fraction of two over two to achieve the common denominator.
When we convert five-eighths to the common denominator, we find the numerator by multiplying five times two, which equals ten. We find the denominator by multiplying eight times two, which equals sixteen. The final step is to add the numerators, ten and three, together, for a total of thirteen. Because we converted to a common denominator, the denominator stays the same; sixteen.
Subtraction with fractions follows similar rules as addition. Let’s look at a subtraction problem: three-fourths minus one-third.
Similar to addition, when subtracting fractions with differing denominators, you must first find a common denominator.
In this problem, the denominators are four and three. Because twelve is a multiple of both three and four, we will use twelve as our common denominator. We must determine what numbers are multiplied by the denominators of both fractions, to achieve the common denominator, twelve, and multiply the numerators by the same numbers. Because three times four is twelve, we must multiply by three-fourths by a fraction of three over three and one-third by a fraction of four over four to achieve the common denominator.
When we convert three-fourths to the common denominator, we find the numerator by multiplying three times three, which equals nine. We find the denominator by multiplying four times three, which equals twelve. When we convert one-third to the common denominator, we find the numerator by multiplying one times four, which equals four. We find the denominator by multiplying three times four, which equals twelve.
The final step is to subtract the second numerator from the first, nine minus four, for a total of five. Because we converted to a common denominator, the denominator stays the same; twelve.
As surprising as it may sound, multiplying fractions is actually easier than adding or subtracting fractions. This is because you do not have to have a common denominator.
Let’s look at a multiplication problem: five-twelfths times three-fourths.
To multiply fractions, simply multiply the two numerators and then multiply the two denominators. When we multiply the numerators, five and three, we get fifteen. When we multiply the denominators, twelve and four, we get forty-eight. Next, we must determine if the total, fifteen-forty-eighths, can be reduced. Because both fifteen and forty-eight are divisible by three, we can reduce the fraction by dividing it by three over three.
We find the numerator by dividing fifteen by three, which equals five. We find the denominator by dividing forty-eight by three, which equals 16. Always look to reduce to the lowest terms possible. Even after reducing once, some answers can be reduced farther. In this example, the answer of five-sixteenths cannot be reduced any farther.
Once you know how to multiply fractions, learning to divide fractions is quite easy.
Let’s look at a division problem, three-eighths divided by four-sevenths, to find out why.
The first step in dividing fractions is to invert the divisor. The divisor is the number that is doing the dividing; in this case four-sevenths. When we invert the divisor, we turn it upside-down by switching the numerator and the denominator. After inverting the divisor, the operation symbol is changed from division to multiplication.
The rest of the calculation continues exactly like a multiplication problem: multiply the two numerators and then multiply the two denominators. When we multiply the numerators, three and seven, we get twenty-one. When we multiply the denominators, eight and four, we get thirty-two. As before, we check if the answer can be reduced. In this example, the answer of twenty-one thirty-seconds cannot be reduced.