Early on in mathematics, dividing by fractions doesn’t come up very often; we’re more accustomed to dividing things into a discrete number of parts. A pizza, for example, might be divided into eight slices.
As it turns out, we never actually have to learn a new technique; because dividing is the same as multiplying by the reciprocal, there are two simple steps to dividing fractions. (If you’re not familiar with some of the words used in this article, you should take a minute to read the rules for fractions).
Step one: Take the reciprocal of the fraction you want to divide by. This means to swap the numerator and the denominator, at the reciprocal of a number is the number it multiplies by to get a product of one. Thus, the reciprocal of 4/5 is 5/4, and the reciprocal of 3, which is the same as 3/1, is 1/3.
Step two: Take the number that was originally going to be divided and multiply it by your new number.
Suppose, for example, that we want to divide 4/5 by 2/3. Unlike with adding, subtracting, or comparing fractions, there’s no need to find a common denominator. Simply reverse 2/3 to get 3/2 and then multiply. We now have:
4/5 * 3/2 = (4 * 3) / (5 * 2) = 12/10 = 6/5 = 1.2.
This works with improper fractions as well. Want to divide three by nine-sixth?
3 / (9/6) = 3/1 * 6/9 = (3 * 6) / (1 * 9) = 18/9 = 2
Follow the above steps to dividing fractions to answer the problems below; the solutions are in the following section.
1) What is 2/3 divided by 4/5?
2) If you have three small chocolate cakes and divided them into pieces such that each piece is 1/8 of a cake, how many pieces do you have?
3) A taxi ride costs $3 for the first quarter mile and 25 cents for each additional quarter mile. If the fare is $5.25, how long was the ride?
4) After eating 2/3 of the chocolate chip cookies, there are 47 cookies left. How many did we start with?
Solutions to the Practice Problems
1) 2/3 divided by 4/5 is the same as two times five, divided by the product of three and four: (2*5) / (3*4) = 10/12 = 5/6
2) Three divided by one eighth is the same as three times eight (divided by one), or 24. Twenty four happy people get cake!
3) This problem is solved in multiple steps. The first quarter mile costs $3, so we subtract $3 from $5.25 to find that the additional quarter miles cost a total of $2.25. $2.25 divided by $0.25 is the same as 9/4 dollars divided by 1/4 dollars, which is the same as 9/4 * 4, or 9 quarter miles. Added to the original $3 quarter mile, we've driven 10 quarter miles, or two and a half miles altogether.
4) This is a trick question – no division is required! We ate 2/3 of the cookies and have 1/3 left, so multiplying 47 by 3 tells us that we started with 141 cookies. We can check our work as follows: we ate 141 * 2/3 = 94 cookies, which leaves us with 141 – 94 = 47 cookies left over.
This post is part of the series: Explore and Understand Fractions
- How to Divide Fractions: Step by Step
- How to Find the Least Common Denominator
- Steps to Subtracting Fractions