Divide Fractions by Other Fractions
We know that fractions have two parts, a numerator and a denominator. When we have to divide a fraction (the dividend) by another fraction (the divisor), we can use the multiplication operation. Here we will multiply the reciprocal of the divisor by the dividend. The reciprocal of a fraction obtained by interchanging the values of the numerator and the denominator.
Dividing Fractions by Whole Numbers
All positive integers including zero form the set of whole numbers. We can divide a fraction by a whole number in a way similar to dividing fractions by another fraction.
In this case we will write the divisor, that is, the whole number as an improper fraction and then follow the usual process of division. So we will multiply the dividend, that is, the fraction by, \( \frac{1}{\text{Whole number}} \) .
Let us divide the fraction \( \frac{a}{b} \) by a whole number \( W \) :
\( \frac{a}{b} \div W \)
\( = \frac{a}{b} \times \frac{1}{W} \)
Or
\( = \frac{a}{b~ \times ~W} \)
So in order to divide a fraction by a whole number we can also multiply the denominator by the whole number.
Rapid Recall
Solved Examples
Example 1 : John poured \( 3 \) liters of milk in \( \frac{3}{4} \) liter cups. How many cups will John need to pour all of the \( 3 \) liters of milk?
Solution:
As stated, John has \( 3 \) liters of milk.
Let’s assume the number of \( \frac{3}{4} \) liter cups filled by John are ‘\( n \)’.
In order to find the number of cups, we will divide \( 3 \) by \( \frac{3}{4} \).
Then,
\( n = \frac{3}{1} \div \frac{3}{4} \) [Write whole number as an improper fraction]
\( n = \frac{3}{1} \times \frac{4}{3} \) [Multiply by the reciprocal]
\( n = 4 \) [Simplify]
Therefore, John has filled \( 4 \) cups with the \( 3 \) liters of milk.
Example 2 : Simplify \( \frac{7}{6} + \frac{5}{6} \div 5 \).
Solution:
The given equation is, \( \frac{7}{6} + \frac{5}{6} \div 5 \)
\( = \frac{7}{6} + \frac{5}{6} \div \frac{5}{1} \) [Write the whole number as an improper fraction]
\( = \frac{7}{6} + \frac{5}{6} \times \frac{1}{5} \) [Multiply by the reciprocal]
\( = \frac{7}{6} + \frac{1}{6} \) [Multiply]
\( = \frac{8}{6} \) [Add]
\( = \frac{4}{3} \) [Simplify]
Hence, when solved \( \frac{7}{6} + \frac{5}{6} \div 5 \) results in \( \frac{4}{3} \) .
Example 3 : Kathy baked \( 4 \) cakes for her friends. If she used \( \frac{17}{2} \) cups of sugar for baking the cakes then how much sugar did she use in one cake?
Solution:
To find the amount of sugar used to bake one cake, we will divide \( \frac{17}{2} \) by \( 4 \), that is,
\( \frac{17}{2} \div 4 \)
\( = \frac{17}{2} \times \frac{1}{4} \) [Multiply by the reciprocal of the whole number]
\( = \frac{17}{8} \) cups [Multiply]
Hence, Kathy needs \( \frac{17}{8} \) cups of sugar to bake one cake.
Division is the process of breaking a number into equal parts and calculating the number of equal parts that can be created.
To divide a whole number by a fraction, we multiply the whole number by the reciprocal of the given fraction. To divide a fraction by a fraction, multiply the reciprocal of the second fraction by the first fraction.
If any fraction or number is divided by zero, the result is undefined.
A fraction is used to represent a portion of the whole. A fraction has two parts called a numerator and a denominator. Fractions are usually written in the p/q form.
Any fraction or number multiplied by zero results in zero.