Divide Fractions by Converting to Multiplication of Fractions

To divide fractions, we need to know these 3 basic parts. Suppose we want to divide \Large{a \over b} by \Large{c \over 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.
in a/b ÷ c/d, the quotient of a and b or a/b is the dividend while the quotient of c and d or c/d is the divisor. we divide both fractions by using the division symbol in the middle, ÷, which obviously indicates the division operation.

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

General Steps on How to Divide Fractions

  • Step 1: Find the reciprocal of the divisor (second fraction) by flipping it upside down. The reciprocal of \Large {a \over b} is \Large {d \over c}.
  • Step 2: Multiply the dividend (first fraction) by the reciprocal of the divisor.
when dividing fractions, we get the reciprocal of the divisor then replace the division operation with multiplication. here we have d/c as the reciprocal of the fraction c/d. thus in a/b÷c/d, we proceed by writing (a/b)(d/c) = (a)(d)/(b)(c).
  • Step 3: Simplify the “new” fraction that comes out after multiplication by reducing it to lowest term.

Examples of How to Divide Fractions

Example 1: Divide the fractions below.

the fractions 2 over 5 and 3 over 7. we write this as (2/5)÷(3/7).

The reciprocal of the divisor (second fraction) is

the reciprocal of the fraction 3 over 7 or 3/7 is 7 over 3 or 7/3.

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

the dividend 2/5 is multiplied to 7/3 which is the reciprocal of the divisor 3/7. therefore we have, (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 below.

four over six divided by eight over three which can be written as (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 a fraction. So let’s go ahead and find the inverse of the divisor (second fraction).

The inverse of \Large{8 \over 3} is just \Large{3 \over 8}.

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

the dividend, 4/6, multiplied to 3/8 which is the reciprocal of the divisor. thus, we have (4/6)(3/8) = / = 12/48.

The resulting answer is not simplified yet because the numerator and denominator have 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:

the common divisors of 12 and 48 are 2,3,4,6, and 12.

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.
to reduce the fraction 12/48 to its lowest term, we divide both the numerator and denominator by 12 which is the greatest common divisor. therefore we have, 12/48 = (12÷12)/(48÷12) = 1/4.

Example 3: Divide a fraction by a whole number, .

the fraction 2/3 or two-thirds divided by 10 which is a whole number. we can write this as (2/3)÷10.

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 \large10 = {{10} \over 1}.  In this form, it is easy to find its inverse or reciprocal.

flipping the fraction 10/1 upside down to get its reciprocal, we get 1/10.

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

we multiply the dividend 2/3 by the reciprocal of the whole number 10 which is 1/10. now we get (2/3)(1/10) = / = 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.

we can reduce our answer 2/30 to its simplest form by dividing it with their greatest common divisor, 2. we now write this as 2/30 = (2÷2)/(30÷2) = 1/15.

Example 4: Divide the fractions below.

the fraction 3 over 4 divided by the fraction 15 over 6, written as 3/4÷15/6.

Solution:

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

we multiply the dividend, 3/4, by the reciprocal of the divisor, 15/6. we now have (3/4)(6/15) = / = 18/60.

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

we divide our answer 18/60 by the GCF which is 6 to reduce it to its lowest term. so we have 18/60 = (18÷6)/(60÷6) = 3/10.

Example 5: Divide the fractions below.

the fraction 9 over 5 or 9/5, divided by the fraction 18 over 20 or 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.

the divisor, 18/20, reduced to its lowest term becomes 9/10. thus, 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.

we multiply the dividend, 9/5, by the reciprocal of the divisor which is 10/9. we can write this as (9/5)(10/9)= / = 90/45 = 2.

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


Example 6: Divide the fraction by a whole number.

the fraction 5 over 6, divided by the whole number 15 which can be written as 5/6÷15.

Solution:

The divisor can be rewritten with a denominator of 1. Thus, \large15 = {{15} \over 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) =  = 5/90 = 1/18

Practice with Worksheets

You might also be interested in:

Adding and Subtracting Fractions with the Same Denominator
Add and Subtract Fractions with Different Denominators
Multiplying Fractions
Simplifying Fractions
Equivalent Fractions
Reciprocal of a Fraction