Simplifying Fractions

The fraction $\frac{8}{11}$ is in its simplest form.

In fact, the factors of 8 are 1, 2, 4 and 8, and the factors of 11 are 1 and 11.

We see that 1 is the highest (and only) factor of the numerator and denominator.

Hence$\frac{8}{11}$ is indeed in its simplest form.

However, the fraction $\frac{20}{88}$ is not in its simplest form.

In fact, the factors of 20 are 1,2,4,5,10, and 20, and the factors of 88 are 1,2,4,22,44,and 88. We notice that there are two common factors between 20 and 88 which are 2 and 4. Hence we deduce that our fraction is not in its simplest form and hence can be simplified furthermore.

We will see this in detail hereafter in the article.

Simplify $\frac{48}{216}$

Solution

Method 1. Using the repeated division for simplifying fractions.

We notice first both the numerator 48 and the denominator 216 are even numbers, so they are both dividable by 2,

$\frac{48}{216}=\frac{2\times 24}{2\times 108}$

We divide by 2 to get

$\frac{48}{216}=\frac{\cancel{2}\times 24}{\cancel{2}\times 108}=\frac{24}{108}$

The same goes for 24 and 108, both numbers are even, so they are dividable by 2,

$\frac{24}{108}=\frac{2\times 12}{2\times 54}$

We divide by 2 to get,

$\frac{24}{108}=\frac{\cancel{2}\times 12}{\cancel{2}\times 54}=\frac{12}{54}$

12 and 54 are both even numbers so they are dividable by 2 too, so we have

$\frac{12}{54}=\frac{2\times 6}{2\times 27}$

We divide by 2 to get,

$\frac{12}{54}=\frac{\cancel{2}\times 6}{\cancel{2}\times 27}=\frac{6}{27}$

Now 6 and 27 have 3 as their lowest common prime factor. Dividing by 3, we get

$\frac{6}{27}=\frac{2\times 3}{3\times 9}$

Dividing by 3, we get,

$\frac{6}{27}=\frac{\cancel{3}\times 2}{\cancel{3}\times 9}=\frac{2}{9}$

There are no more common prime factors between the numerator 2 and the denominator 9.

So $\frac{2}{9}$ is the fraction expressed in its simplest form.

Method 2. Using the greatest common divisor of the numerator and the denominator.

The factors of 48 are: 1,2,4,6,24, and 48.

The factors of 216 are: 1,2,3,4,6,8,9, 12, 18, 24,27,36,54, 72, 108 and 216.

Thus, the Greatest Common Divisor of the 48 and 216 is 24.

In fact, by dividing both the numerator and denominator by 24, we get

$\frac{48}{216}=\frac{2\times 24}{9\times 24}=\frac{2}{9}$

Simplify $3\frac{10}{15}$

Solution

Firstly, we must turn $3\frac{10}{15}$ into an improper fraction. We can do this by writing the integer part of the mixed fraction as a fraction with the same denominator as the fractional part.

$3\frac{10}{15}=\frac{3\times 15}{15}+\frac{10}{15}=\frac{45}{15}+\frac{10}{15}=\frac{55}{15}$

The final step is to simplify the improper fraction using either the repeated division method or the greatest common divisor method. Using either of these methods, we find that the simplified fraction is $\frac{11}{3}.$

Simplify $12\frac{12}{30}$

Firstly, we turn $12\frac{12}{30}$ into an improper fraction. We do this by expressing the integer part of the mixed fraction as a fraction with the same denominator as the fraction part.

$12\frac{12}{30}=\frac{12\times 30}{30}+\frac{12}{30}=\frac{360}{30}+\frac{12}{30}=\frac{372}{30}$

Finally, we simplify the improper fraction with either the repeated divisor method or the greatest common divisor method. Using either of these methods we find that the simplified fraction is $\frac{62}{5}$.

Therefore,

$12\frac{12}{30}=\frac{62}{5}$

Simplify $\frac{2^8{3}^2{7}^5{11}^2}{2^5{3}^2{5}^5{11}^3}$.

Solution

As stated earlier in the article, when simplifying fractions with exponents in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the Greatest Common Divisor of $2^8{3}^2{7}^5{11}^2$ and $2^5{3}^2{5}^5{11}^3$ is $2^5{3}^2{11}^2$.

Next, dividing both the numerator and denominator by the GCD, we get

$\frac{2^8{3}^2{7}^5{11}^2}{2^5{3}^2{5}^5{11}^3}=\frac{\frac{2^8{3}^2{7}^5{11}^2}{2^5{3}^2{11}^2}}{\frac{2^5{3}^2{5}^5{11}^3}{2^5{3}^2{11}^2}}=\frac{2^3{7}^5}{5^5{11}^1}=\frac{2^3{7}^5}{5^{5}11}$

Thus, the simplest form of the given fraction is

$\frac{2^3{7}^5}{5^{5}11}$.

Simplify $\frac{3^4{5}^2{8}^4{9}^8}{3^2{8}^{2}5}$

Solution

The greatest common divisor of the numerator and denominator is $3^2{8}^{2}5$. Dividing both the numerator and denominator by the greatest common divisor gives

$3^2{8}^2{9}^{8}5$

Simplify $\frac{4^{15}{3}^4{10}^3{5}^4}{4^{12}{3}^2{10}^2{9}^3}$

Solution

The greatest common divisor of the numerator and denominator in this case is $4^{12}{3}^2{10}^2$. Dividing both the numerator and denominator by the greatest common divisor gives

$\frac{4^3{3}^2{5}^{4}10}{9^3}$

Simplify $\frac{12b^5{c}^2}{30a{b}^{3}c}$

Solution

As stated earlier in the article, when simplifying fractions with variables in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the Greatest Common Divisor of $12{\mathrm{b}}^5{\mathrm{c}}^2$ and $30{\mathrm{ab}}^3\mathrm{c}$ is $6{\mathrm{b}}^3\mathrm{c}$.

Next, by dividing both the numerator and denominator by $6{\mathrm{b}}^3\mathrm{c}$, we get

$\frac{12b^5{c}^2}{30a{b}^{3}c}=\frac{\frac{12b^5{c}^2}{6b^{3}c}}{\frac{30a{b}^{3}c}{6b^{3}c}}=\frac{2b^{2}c}{5a}$

Thus, the simplest form of the given fraction is

$\frac{2b^{2}c}{5a}$

Simplify $\frac{14a^{3}b}{21ac}$

Solution

As stated earlier in the article, when simplifying fractions with variables in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the GCD of$14{\mathrm{a}}^3\mathrm{b}$and $21\mathrm{ac}$ is $7a$.

Next, by dividing both the numerator and denominator by $7a$ we get,

$\frac{14a^{3}b}{21ac}=\frac{\frac{14a^{3}b}{7a}}{\frac{21ac}{7a}}=\frac{2a^{2}b}{3c}$

Hence, the simplest form of the given fraction is,

$\frac{2a^{2}b}{3c}$

Simplify $\frac{49x^2{y}^3}{35y^2{z}^2}$

Solution

As stated earlier in the article, when simplifying fractions with variables in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the Greatest Common Divisor of $49{\mathrm{x}}^2{\mathrm{y}}^3$and $35{\mathrm{y}}^2{\mathrm{z}}^2$is $7{\mathrm{y}}^2.$

By dividing both the numerator and denominator by the GCD, we get

$\frac{49x^2{y}^3}{35y^2{z}^2}=\frac{\frac{49x^2{y}^3}{7y^2}}{\frac{35y^2{z}^2}{7y^2}}=\frac{7x^{2}y}{5z^2}$

Hence, the simplest form of the given fraction is,

$\frac{7x^{2}y}{5z^2}$