Multiplication and Division With Fractions
Multiplication and division with fractions are actually very similar processes, but they need to be explained one at a time. So, we'll start with multiplication, which is the simplest. To multiply two fractions together, treat the problem as two multiplication problems stacked on top of each other. For example:
2⁄3 * 3⁄4 = 6⁄12
How did we get this answer? Simple: we split up the fraction into two multiplication problems, then put it back together.
For the numerator, we did 2 * 3 = 6.
For the denominator, we did 3 * 4 = 12.
Finally, we put the two resulting numbers back into the fraction. There's one more step: it's always good to simplify your answer. To do this, think about common factors.
To find the factor that will let us create the most simplified fraction, we need to find the biggest number that we can divide both 6 and 12 by and still get a whole number. In this case, 6 is the factor that we need to divide both parts of the fraction by, because 6 * 1 = 6 and 6 * 2 = 12. There isn't any way we can divide 6 by something bigger than 6 and still get a whole number.
6 ÷ 6⁄12 ÷ 6 = 1⁄2
So, 1⁄2 is our final answer.
Now, what about division? Division with fractions is even simpler than normal division. To divide a fraction by another fraction, you multiply by the reciprocal. The reciprocal of a fraction is the fraction flipped upside down. For division problems, it's always the second fraction that should be turned into its reciprocal form.
So, if we wanted to solve this problem:
1⁄2 ÷ 3⁄7 = x
Our first step would be to find the reciprocal of 3⁄7 by flipping it over. Then, we would replace 3⁄7 with its reciprocal and replace the division symbol with the multiplication symbol, and do the problem as a multiplication.
The reciprocal of 3⁄7 is 7⁄3. So, our equation and solution will be:
1⁄2 * 7⁄3 = 7⁄6
Just like in the first multiplication problem, we multiplied the top and bottom of the fraction separately.
1 * 7 = 7
2 * 3 = 6
Then, we put the two numbers back together as a fraction. This is always how we multiply and divide with fractions.