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Dividing Fractions

To divide fractions, we need to know these 3 basic parts. Suppose we want to divide a/b by c/d, the setup should look like this.

  • Dividend - the number being divided or partitioned by the divisor. It is found to the left of the division symbol.
  • Divisor - the number that is dividing the dividend. It is located to the right of the division symbol.
parts of division of fractions: dividend, divisor and division symbol

 

Now, apply the following simple steps to divide these fractions.

  • Step 1: Find the reciprocal of the divisor (second fraction) by flipping it upside down. The reciprocal of c/d is d/c.
  • Step 2: Multiply the dividend (first fraction) by the reciprocal of the divisor.

 

formula showing how to divide basic fractions

 

  • Step 3: Simplify the "new" fraction that comes out after multiplication by reducing it to lowest term.

 

 


Example 1: Divide the fractions divide the fractions: (2/5)/(3/7).

The reciprocal of the divisor (second fraction) is reciprocal of the fraction 3/7 which is 7/3.

Multiply the dividend (first fraction) to the reciprocal of the divisor.

(2/5) * (7/3) = 14/15

This is our final answer because the resulting fraction is already in its lowest term!

 


Example 2: Divide the fractions divide the fractions: (4/6) / (8/3).

Sometimes you may encounter the phrase "inverse of a fraction". That's pretty much the same when we find the reciprocal of fraction. So let's go ahead and find inverse of the divisor (second fraction).

The inverse of 8/3 is just 3/8.

 

Obviously, the next step is to find the product of the dividend and the inverse of the divisor.

(4/6) * (3/8) = 12/48

The resulting answer is not simplified yet because the numerator and denominator has a common divisor. Can you think of the common divisors of 12 (numerator) and 48 (denominator) ?

If we do some trial and error, the possible common divisors  of 12 (numerator) and 48 (denominator) are:

 

2  
3  
4  
6  
12 check mark

But we want the greatest common divisor to reduce our answer to the lowest term, which in this case is 12.

  • Divide the top and bottom by GCF = 12 to get the final answer.
(12/48) = (12/12) / (48/12) = 1/4

 


Example 3: Divide a fraction by a whole number, divide the fractions: (2/3) / (10/1).

This time we have a fraction being divided by a whole number. Notice that any nonzero whole number can be rewritten with a denominator of 1. Therefore, the number 10 is just 10 = 10/1.  In this form, it is easy to find its inverse or reciprocal.

reciprocal of (10/1) is (1/10)

 

Now we are ready to divide by multiplying the dividend to the reciprocal of the divisor.

(2/3) * (1/10) = (2*1) / (3*10) = 2/30

 

The greatest common divisor between the numerator and denominator is 2.  That means, we can reduce it to the lowest term by dividing both the top and bottom numbers by 2.

2/30 = 1/15

 


Example 4: Divide the fractions divide the fractions : (3/4) / (15/6).

Solution:

Multiply the dividend (first fraction) by the reciprocal of the divisor (second fraction).

(3/4) * (6/15) = (3*6)/(4*15)=18/60

 

The greatest common divisor is 6. Use that to reduce the answer to the lowest term.

18/60=(18/6)/(60/6)=3/10

 


Example 5: Divide the fractions divide the fractions: (9/5) / (18/20).

Solution:

Before we even divide the fractions, try to see if you can reduce the existing fractions to its lowest term. Observe that the divisor (second number) can be reduced using a common divisor of 2.

(18/20) = (9/10)

Rewriting the original problem with a reduced divisor, we have...

(9/5) / (9/10)

The fractions now are relatively smaller in size. Proceed with division by multiplying the dividend to the inverse of the divisor.

(9/5) * (10/9) = (9*10) / (5*9) = 2

The final answer is reduced to a whole number. Great!

 


Example 6: Divide the fraction by a whole number, divide the fractions: (5/6) / 15.

Solution:

The divisor can be rewritten with a denominator of 1. Thus, 15 = 15/1.

The problem becomes (5/6) / (15/1). We are now ready to divide the fractions.

Multiply the dividend to the inverted divisor.

 

(5/6) * (1/15)= (5*1)/(6*15) = 5/90=1/18

 


 

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