How to Divide a Fraction by a Fraction?
Division of fractions (or any other numbers or variables) may be indicated either by a division signs `\div` or `:` between two fractions or by writing the division in the form of the complex fraction. A complex fraction is a fraction where the numerator, denominator, or both contain a fraction. So, division of fractions can be represented as in the following examples:
$$\frac 2 3\div \frac 5 4, \quad \frac 2 3: \frac 5 4,\quad \frac{\frac 23}{\frac 54}$$
The first fraction, in this case `2/3`, is the dividend, the second fraction, in this case `5/4`, is the divisor, and the result is the quotient.
One property of fractions that do not valid for whole numbers and integers is well known as inverse property of fractions.
Two fractions whose product is `1` are called multiplicative inverses or reciprocal.This means, if `\frac ab\times \frac cd=1` for `b,d\ne0`, then `\frac{a}{b}` and `\frac{c}{d}` are reciprocal. The product of a number and its reciprocal is `1`. In other words, for every fraction `frac ab`, where `a, b \ne0`,
there is exactly one number `\frac b a` such that
$$\frac ab\times \frac ba=1$$
For example, `5/7` and `7/5` are reciprocal because `\frac {5}{7}\times \frac {7}{5}=1.`
When we deal with dividing fractions, there are some important cases which deserve to be mentioned here:
Division of a fraction by another fraction
Dividing by the fraction $\frac c d$ is equal to the multiplying by the fraction `\frac d c`, its reciprocal. This is valid for any fraction.
Hence, to divide `a/b` by `c/d` multiply `a/b` by `d/c`, i.e. multiply their numerators and multiply their denominators.
In other words, the quotient of two fractions `a/b` and `c/d` for `b,c,d\ne0` is
$$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{a\times d}{b\times c},\quad b,c,d\ne0$$
To summarize, to find the quotient of two fractions, it is necessary to follow three steps:
- Find the reciprocal of the divisor
- Multiply the dividend by this reciprocal
- Simplify the product if needed.
For example, let us divide the fractions: `8/3\divide7/2`. Using the division rule explained above, we obtain
$$\frac{8}{3}\div\frac{7}{2}=\frac{8}{3}\times\frac{2}{7}=\frac{8\times2}{3\times7}=\frac{16}{21}$$
To write the quotient in simplest form, find the GCF of the numerator and denominator of the quotient. Since 16 and 21 are relatively prime numbers, the GCF of 16 and 21 is 1. So, the quotient for fraction
`\frac{8}{3}` divided by `\frac{7}{2}` is `\frac{16}{21}`
Division of a fraction by a whole number
Since a whole number can be rewritten as itself divided by $1$, we can apply the previous rule of division of a fraction by another fraction.
The quotient for the fraction `\frac{a}{b},b\ne0` divided by the whole number `c, c\ne0`, is the product of `\frac{a}{b}` and `\frac 1 c`, the reciprocal of `c`. This can be written in the following way
$$\frac a b\div c=\frac a b\times \frac{1}{c}$$
Division of a fraction by a mixed numbers
Firstly, it is necessary to convert the mixed number to improper fractions then divide the fractions. For example, find `\frac{3}{4}\div3\frac {1}{3}`. Since `3\frac{1}{3}` is equal to `\frac{10}{3}`, the quotient for the fraction `\frac{3}{4}` divided by the mixed number `3\frac {1}{3}` is equal to the quotient for the fraction `\frac{3}{4}` divided by the fraction `\frac {10}3`. Then we will continue with the rule for division of a fraction by another fraction.
The similar consideration can be applied in the division of algebraic fractions.
The Fractions Division work with steps shows the complete step-by-step calculation for finding quotient for the fraction `\frac{8}{3}` divided by `\frac{7}{2}` using the division rule. For any other fractions, just supply two proper or improper fractions and click on the "GENERATE WORK" button. The grade school students may use this fraction division calculator to generate the work, verify the results of dividing numbers derived by hand or do their homework problems efficiently.