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Division of fractions is one among four operations involving fractions. Unlike the usual methods of long division where the dividend is divided by the divisor to get the quotient in the form of a whole number or decimal, the division of fractions usually results in another fraction. Division of two fractions has only one step different from the multiplication of two fractions. This will be explored in the following sections. ...Read MoreRead Less

Dividing two fractions is easy if you are thorough with the steps involved in the multiplication of fractions. The reciprocal of the divisor is multiplied with the dividend to get the answer. In the case of a mixed number, the mixed number is converted to an improper fraction before performing the division.

Formula:

If \(\frac{a}{b}\) and \(\frac{c}{d}\) are two fractions and b,c and d are not equal to 0.

\(\frac{a}{b}\div \frac{c}{d}=\frac{a\times d}{b\times c}\)

\(\frac{a}{b}\) and \(\frac{c}{d}\) are two fractions where the values of b, c and d are not 0. If we are asked to divide these two fractions, then the formula to find the result is as shown below:

\(\frac{a}{b}\div \frac{c}{d}\)

\(=\frac{a}{b}\times \frac{d}{c}\)

\(=\frac{a~\times~d}{b~\times~c}\)

In the case of mixed numbers, first convert them to improper fractions and then perform the division as shown above. Mixed numbers can be converted to improper fractions as shown below:

\(a\frac{b}{c}=\frac{a~\times~c~+~b}{c}\)

Here is an example to help us understand division of fractions better.

**Solve: \(\frac{5}{6}\div \frac{5}{12}\)**

**Solution:**

\(\frac{5}{6}\div \frac{5}{12}\)

**First, we multiply the dividend with the reciprocal of the divisor which in this case is \(\frac{12}{5}\)**

**\(\frac{5}{6}\times \frac{12}{5}=\frac{60}{30}\)**

Therefore, \(\frac{5}{6}\times \frac{12}{5}=\frac{60}{30}\)

But, the fraction should be presented in its simplest form.

In this case, the numerator and denominator are first divided by ten.

**\(\frac{60~\div~10}{30~\div~10}=\frac{6}{3}\)**

Then the numerator and denominator are divided by three to give the answer.

**\(\frac{6~\div~3}{3~\div~3}=\frac{2}{1}\)**

** = 2**

**Example 1:**

**Write the reciprocal of the following fractions.**

**1. \(\frac{8}{23}\) **

**2. \(12\frac{3}{5}\) **

**Answer: **

1. The reciprocal of \(\frac{8}{23}\) is \(\frac{23}{8}\)

2. We first simplify the mixed number **\(12\frac{3}{5}\) ** to make it an improper fraction.

**\(12\frac{3}{5}=\frac{12~\times~5~+~3}{5}\)**

**= \(\frac{63}{5}\)**

The reciprocal of **\(12\frac{3}{5}\)** or **\(\frac{63}{5}\)** is **\(\frac{5}{63}\)**

**Example 2:**

**Find \(\frac{1}{6}\div \frac{7}{13}\) **

**Solution:**

**Reciprocal of \( \frac{7}{13}\)** **is \( \frac{13}{7}\)**

Therefore,

** \(\frac{1}{6}\div \frac{7}{13}\) **

**= \(\frac{1}{6}\times \frac{13}{7}\) ** (multiplying with the reciprocal of the divisor)

**= \(\frac{13}{42}\) **

**Example 3: **

**Solve: \(4\frac{6}{7}~\div~2\frac{1}{2}\) **

**Solution:**

Converting the mixed fractions to improper fractions we get:

= \(\frac{34}{7}~\div~\frac{5}{2}\)

Reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\)

Therefore,

**\(4\frac{6}{7}~\div~2\frac{1}{2}=\frac{34}{7}~\div~\frac{5}{2}\)**

= \(\frac{34}{7}~\times~\frac{2}{5}\) (multiplying with the reciprocal of the divisor)

= \(\frac{68}{35}\)

**Example 4:**

A large container filled with \(200\frac{3}{4}\) cups of juice were placed in the middle of a class party. To avoid wasting paper cups, each student was given a water bottle which holds \(4\frac{1}{4}\)cups of juice. Was there enough juice to fill the bottles for 50 people?

**Solution:**

To find if there is enough juice to fill 50 bottles, we need to divide \(200\frac{3}{4}\) by \(4\frac{1}{4}\) . If the quotient of this division is 50, it suggests that we have enough juice. To divide \(200\frac{3}{4}\) by \(4\frac{1}{4}\) we need to convert them to improper fractions.

\(200\frac{3}{4}\) = \(\frac{200~\times~4~+~3}{4}\) or \(\frac{803}{4}\)

and, \(4\frac{1}{4}=\frac{4~\times~4~+~1}{4}\) or \(\frac{17}{4}\)

Therefore,

\(200\frac{3}{4}~\div~4\frac{1}{4}=\frac{803}{4}~\div~\frac{17}{4}\)

= \(\frac{803}{4}~\times~\frac{4}{17}\) **(multiplying with the reciprocal of the divisor)**

= \(\frac{803}{17}\)

We get the result as 47 with the remainder as 4. This means that there is not enough juice for 50 people.

**Example 5:**

**\(\frac{1}{5}th\)** **of a swimming pool is full. When 220 gallons of water is added to the pool, ****\(\frac{3}{4}th\) ****of the tank gets filled. With this information, answer the following questions:**

**1)How much water can the swimming pool hold? **

**2)How much water was in the swimming pool when \(\frac{1}{5}th\) **** of it was full? **

**3)How much water is in the swimming pool when it is \(\frac{1}{2}\) ****full?**

**Solution:**

1)

The swimming pool had \(\frac{1}{5}th\) **of the total capacity of water it could hold. The capacity went up to **\(\frac{3}{4}th\) **of the total capacity when 220 gallons of water was added. To find the increase in the capacity we must find the difference between \(\frac{1}{5}\) ** **and \(\frac{3}{4}\) **.

\(\frac{3}{4}-\frac{1}{5}\)

= \(\frac{3~\times~5}{4~\times~5}-\frac{1~\times~4}{5~\times~4}\) (Change the unlike fractions to like fractions)

= \(\frac{15}{20}-\frac{4}{20}\) (Subtract)

= \(\frac{11}{20}\)

This shows that \(\frac{11}{20}th\) of the pool is filled by **220** gallons.

Now to find the amount of water required to fill the entire swimming pool, we must divide **220** by

\(\frac{11}{20}\)

\(220~\div~\frac{11}{20}\)

= \(220~\times~\frac{20}{11}\) (multiplying with the reciprocal of the divisor)

= \(20~\times~20\)

= 400

Therefore, the amount of water the swimming pool can hold is 400 gallons.

** **

2)

To find the quantity of water that was in the swimming pool when** \(\frac{1}{5}th\)** of it was full**, **we multiply 400 with \(\frac{1}{5}\)

\(400~\times~\frac{1}{5}\)

= 80 gallons

Therefore, the quantity of water that was in the swimming pool when \(\frac{1}{5}th\) of it was full is 80 gallons.

3)

To find the amount of water the swimming pool can hold when it is **\(\frac{1}{2}\)** full, we need to divide 400 by 2.

\(400~\div~2\)

= 200

Therefore, the amount of water the swimming pool can hold when it is **\(\frac{1}{2}\)** full is 200 gallons.

So, originally there was 80 gallons of water and half of the swimming pool can be filled by 200 gallons of water.

Frequently Asked Questions

A mixed number is a combination of a whole number and a fraction. A mixed fraction can be converted into an improper fraction.

A fraction whose numerator is lesser in value than the denominator is a proper fraction. A fraction whose numerator is greater than the denominator’s value is an improper fraction.

Simplified fractions represent the simplest form of a fraction. There exist infinitely many equivalent fractions but to have a unique representation of them all we use the simplest form of a fraction.

Fractions, percents and decimals are all interconvertible.

Mixed numbers give a clear distinction of the number of whole parts it represents as well as the fractional part. Whereas, it is difficult to say the same when it is in the form of an improper fraction.