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Dividing Fractions

by Ron Kurtus (11 January 2008)

A fraction is a part of the whole or the quotient of two whole numbers. Dividing a fraction by another fraction involves some trickery. You invert the second fraction and then multiply the two fractions. When you are working with a fraction and whole number, the whole number must be converted to a fraction. Likewise, mixed numbers must be changed into improper fractions before the division operation.

Questions you may have are:

• How do you divide one fraction bu another?
• How do you divide a fraction by a whole number?
• How do you divide mixed numbers?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Dividing one fraction by another

The way to divide one fraction by another fraction is to simply invert or flip the fraction you are dividing by (the second fraction) and then multiply the two fractions together. The following example demonstrates this method:

1/4 ÷ 3/7 =

(Flip over 3/7 and change sign to multiplication)

1/4 × 7/3 =

(Multiply top numbers and bottom numbers)

(1 × 7)/(4 × 3) =

7/12

Another example is:

12/13 ÷ 1/4 =

(Invert 1/4 and change sign to multiplication)

12/13 × 4/1 =

(Multiply numerators and denominators)

(12 × 4)/(13 × 1) =

48/13

Exercises

1. 1/2 ÷ 1/4

2. 3/5

Dividing fraction and whole number

Sometimes you encounter a division problem with a fraction and a whole number.

Divide fraction by whole number

When you divide by a whole number, you can put the number in the form of an improper fraction and then flip it over. You take the whole number and show it as divided by 1, such as 7 = 7/1.

An example of dividing a fraction by a whole number is:

3/4 ÷ 5 =

(Put 5 in fraction format)

3/4 ÷ 5/1 =

(Flip over 5/1 and change to multiplication)

3/4 × 1/5 =

(Multiply numerators and denominators)

(3 × 1)/(4 × 5) =

3/20

Divide a whole number by a fraction

You can also divide a whol number by a fraction in the same manner. Change the whole number into a fraction over 1 and then flip the fraction you are dividing by.

17 ÷ 2/3 =

(Put 17 in fraction format)

17/1 ÷ 2/3 =

(Flip over 2/3 and change to multiplication)

17/1 × 3/2 =

(Multiply numerators and denominators)

(17 × 3)/(1 × 2) =

51/2

Dividing with mixed numbers

When you divide using mixed numbers, you need to change the mixed number into a fraction first.

Divide a mixed number by a fraction

5 1/3 ÷ 2/7 =

(Change mixed number to a fraction)

16/3 ÷ 2/7 =

(Flip over 2/7)

16/3 × 7/2 =

(Multiply numerators and denominators)

(16 × 7)/(3 × 2) =

112/6 =

56/3 =

18 2/3

Divide a fraction by a mixed number

2/3 ÷ 1 2/7 =

(Change mixed number to a fraction)

2/3 ÷ 16/7 =

(Flip over 16/7)

2/3 × 7/16 =

(Multiply numerators and denominators)

(2 × 7)/(3 × 16)

= 6/112

(Reduce 6/112)

= 3/56

Divide a mixed number by a mixed number

2 1/2 ÷ 3 1/3 =

5/2 ÷ 3 1/3 =

5/2 ÷ 10/3 =

5/2 × 3/10 =

(5 × 3)/(2 × 10) =

15/20 = 3/4

Summary

Dividing a fraction by another fraction involves inverting the second fraction that you are dividing by and then multipling the two fractions. When you are working with a fraction and whole number, the whole number must be changed to an improper fraction. Likewise, mixed numbers must be changed into improper fractions before the division operation.

Answers to Readers' Questions

Learn some clever tricks

Resources

The following resources provide information on this subject:

Websites

Arithmetic Resources

Books

Top-rated Arithmetic books

Miscellaneous

Mini-quiz to check your understanding

1. What is 2/3 ÷ 3/2?

2. What is 1/5 ÷ 7?

3. What is 1 1/5 ÷ 2 1/2?

If you got all three correct, you are on your way to becoming a Champion in Arithmetic. If you had problems, you had better look over the material again.

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