Dividing two fractions between them might seem somewhat difficult at first, but in reality it is a simple operation. All you have to do is flip the divisor fraction, replace the division symbol with the multiplication symbol, and finally simplify! This article will walk you through the process and show you how easy it is.
Steps
Part 1 of 2: How to Divide a Fraction by Another Fraction
Step 1. Think about what splitting between fractions implies
The operation 2 ÷ 1/2 means: "How many halves are there in number 2?" The answer is four because each unit (1) is made up of two halves, and since 2 corresponds to two units, the answer is: 2 halves in each unit * 2 units = 4 halves.
- Try to think of the same operation in terms of cups of water. How many half cups are there in 2 cups of water? You can pour 2 half cups into each cup, if you have two cups the answer is 4 halves.
- This means that when the divisor fraction is between 0 and 1, the quotient will be a number larger than the dividend! This is true whether the dividend is an integer or a fraction.
Step 2. Remember that division is the opposite of multiplication
So dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is simply the upside-down fraction itself, where the denominator takes the place of the numerator and vice versa. With this simple step you go from division to multiplication. For the moment we list some examples of reciprocal fractions:
- The reciprocal of 3/4 is 4/3.
- The reciprocal of 7/5 is 5/7.
- The reciprocal of 1/2 is 2/1 i.e. 2.
Step 3. Memorize these steps to divide the fractions together
In order they are:
- Leave the fraction as it is by dividing.
- Transform the division sign into the multiplication sign.
- Flip the divisor fraction to find its reciprocal.
- Multiply the numerators together. The product is the numerator of the solution.
- Multiply the denominators together. The product is the denominator of the solution.
- Simplify the resulting fraction by reducing it to its lowest terms.
Step 4. Try to apply the method described to solve the division 1/3 ÷ 2/5
Let's start by simply transcribing the dividend and changing the division sign to the multiplication sign:
- 1/3 ÷ 2/5 = it becomes:
- 1/3 * _ =
- Now flip the second fraction (2/5) and find its reciprocal 5/2:
- 1/3 * 5/2 =
- Multiply the numerators together, 1 * 5 = 5.
- 1/3 * 5/2 = 5/
- Multiply the denominators together, 3 * 2 = 6.
- You can write that: 1/3 * 5/2 = 5/6
- This particular fraction cannot be simplified further and represents the final solution.
Step 5. Try to remember a nursery rhyme:
"Dividing fractions is not a big deal, just turn the second and then multiply. In the end, don't forget that you have to simplify."
You can come up with any rhyme or mnemonic trick to remember the process
Part 2 of 2: Practical Examples
Step 1. Let's start with an example
Let's consider the division 2/3 ÷ 3/7. This problem is asking you how many parts corresponding to 3/7 of an integer can we find in the value 2/3. Do not worry! The practical side is much simpler than it looks.
Step 2. Change the division sign to the multiplication sign
You should now have: 2/3 * _ (leave the space blank for now).
Step 3. Now find the reciprocal of the second fraction
This means flipping 3/7 so that the numerator and denominator swap places. The reciprocal of 3/7 is 7/3. Now write it down in your equation:
2/3 * 7/3 = _
Step 4. Multiply the fractions
First find the product between the numerators: 2 * 7 = 14. 14 is the numerator of the solution. Now do the same for the denominators: 3 * 3 = 9. 9 is the denominator of the solution. Now you know that 2/3 * 7/3 = 14/9.
Step 5. Simplify the fraction
In this case, since the numerator of the fraction is greater than the denominator, we know that its value is greater than 1 and we can convert it to a mixed fraction (an integer and a fraction combined together as 1 2/3).
-
First divide the numerator
Step 14. for 9.
9 goes into 14 only once with the remainder of 5, so your fraction can be written as: 1 5/9 ("One and five ninths").
- Stop, you have found the solution! You can understand that the quotient fraction cannot be simplified further because the denominator is not divisible by the numerator and this is also a prime number (an integer that is only divisible by 1 and itself).
Step 6. Try another example
Let's consider the division 4/5 ÷ 2/6 =. First replace the division symbol with the multiplication symbol (4/5 * _ =), find the reciprocal of 2/6 which is 6/2. Now you have the equation: 4/5 * 6/2 =_. Multiply the numerators together, 4 * 6 = 24 and denominators 5* 2 = 10. You can transcribe the equation as 4/5 * 6/2 = 24/10.
Now simplify the fraction. Since the numerator is greater than the denominator, you know you can convert it to a mixed fraction.
- Divide the numerator by the denominator, (24/10 = 2 with the remainder of 4).
- Write the solution as 2 4/10. You can still simplify the fractional part!
- Since 4 and 10 are both even numbers, the first thing to do is divide them by 2 to get 2/5.
- Since the denominator is not divisible by the numerator, and both are prime numbers, then you know that no other simplification is possible and your definitive answer is: 2 2/5.
Step 7. Find other aids for reducing fractions
You've probably spent a lot of time practicing simplifying fractions before moving on to divisions, however, if you need a refresher, you can find many guides online.
