Pet & Animal: How to Divide Fractions by Fractions

Sunday 26 October 2014

How to Divide Fractions by Fractions

Dividing fractions by fractions isn't as hard as it looks. To divide one fraction by another, all you have to do is flip the numerator and denominator of the second fraction and switch the division sign to a multiplication sign. After that, you just multiply the two remaining numbers and you're good to go. If you want to know how to multiply fractions by fractions, see Step 1 to get started.

Steps

Flip the numerator and denominator of the second fraction. Remember that the numerator is the number on top of the fraction and the denominator is the number on the bottom. You can also think of this as changing the second fraction to its reciprocal. Let's say you're working on the following problem: 2/3 ÷ 3/7. After you write it down, the first thing you should do is to flip the numerator and denominator of the second fraction, 3/7, so that it becomes 7/3.
- You can rewrite the problem as 2/3 ÷ 7/3.

Change the division sign to a multiplication sign. Now, just change the division sign to a multiplication sign so that the problem reads like this: 2/3 x 7/3. You can do this because dividing a number by a fraction is the same as multiplying by a fraction's reciprocal, also known as its inverse.
- For example, take the problem, 1 x 1/2. The answer here is simply 1/2. If you rewrite the problem as 1 ÷ 2/1, which is the inverse of 1/2, then you will still get 1/2.

Reduce the numerators and denominators of both fractions to their lowest terms by canceling out any common factors. In this example, the problem 2/3 x 7/3 is already reduced to its lowest terms. Before you move on to the next stage of a problem like this, see if you can find a common factor either between the numerator and denominator of one fraction, or between the numerator of one fraction and the denominator of the other. A common factor of two numbers is evenly divisible into each number. For example:
- In the problem 2/4 x 4/8, you can reduce the problem to 1/2 x 1/2, because the numerators and denominators of both fractions are evenly divisible by 2. Just divide each part of each fraction by the common factor to get the new reduced problem.
- In the problem 4/3 x 9/2, you can divide 3 and 9 by the common factor, 3, because these numbers are diagonal from each other. 3 ÷ 3 = 1 and 9 ÷ 3 = 3, so your new problem reads 4/1 x 3/2. However, you can also then divide the 4 and the 2 by 2, because it is a common factor of both numbers. So, 4 ÷ 2 = 2 and 2 ÷ 2 = 1, so your new problem should read 2/1 x 3/1.

Multiply the numerators and denominators of the fractions. Now, you must simply multiply the numerators and denominators of the fractions, just as you would do in an ordinary multiplication problem. Here's what you need to do:
- Multiply the numerators. In this example, you should multiply 2 x 7 to get 14.
- Multiply the denominators. In this example, you should multiply 3 x 3 to get 9.
- Place the new numerator over the new denominator. The result is 14/9.

Convert the fraction to a mixed number if it's necessary. 14/9 is considered an improper fraction because the numerator is larger than the denominator. If the problem — or your teacher — calls for the problem to be written in mixed number form, then you should rewrite the improper fraction as such. Here's what you need to do to convert an improper fraction to a mixed number:
- See how many times the denominator evenly goes into the numerator. In this case, see how many times 9 will fit into 14. 9 goes into 14 just one time (with some left over).
- Write this number as a whole number to the left of the fraction. Now, you have 1 whole number and a fraction remaining.
- Find the remainder. The remainder is the number that is left over when you divide the denominator into the numerator. When you divide 9 into 14, you have 5 left over. This is your remainder.
- Write this number over the original denominator. Now, you should write 5/9.
- Write the fraction next to the whole number and you're all done. Just put it all together to get 1 5/9 and you're all done. You've converted an improper fraction to a mixed number.

Use a recipe for a real-life example. Though this isn't mandatory, you can use a recipe to get a better sense of how this process works. It's a great way of learning for yourself and if you're trying to help a friend or student learn, it's also a great way to teach it. For example, say that you're making a cake. The recipe calls for 1/3 of a pint of milk and you need to know how many batches of batter you can make from the entire pint. The answer is three––and here is why:
- 1/3 of a pint goes into one pint three times.
- Just as "2 goes into 8 four times" is 8/2 = 4, then 1/3 goes into 1 three times and is expressed as 1 ÷ 1/3 = 3, which can also be rewritten as 1 x 3/1. 1 ÷ 1/3 is 3 thirds.

Video

Tips

Your tutor or teacher may have you do this a different way, but this is the most common way.

When multiplying, you should cancel numbers whenever possible. Like 1/4 x 2/1, you can cancel the 4 (denominator) and 2 (numerator) out to become 2 and 1, respectively, because each number is divisible by 2.

Follow any specific directions given as part of the problem. If it says simplify, then do so. If it doesn't, then ask the person tutoring or teaching you before you do.