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 $\frac{45}{144}$.

Solution

Method 1. Using the repeated division for simplifying fractions.

The factors of 45 are: 1,3,5,9, 15 and 45.

The factors of 144 are: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and 144.

We notice that the lowest prime number that is a common factor of the numerator and the denominator is 3. Thus, we divide the numerator and the denominator by 3 to get

$\frac{45}{144}=\frac{3\times 15}{3\times 48}=\frac{15}{48}$

15 and 48 are both dividable by 3, so by dividing by 3 we get,

$\frac{15}{48}=\frac{3\times 5}{3\times 16}=\frac{5}{16}$

There are no more common prime factors between the numerator 5 and the denominator 16.

Hence $\frac{5}{16}$ is the simplest form of the expression.

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

The Greatest Common Divisor of the 45 and 144 is 9.

We divide both the numerator and denominator by 9 to get

$\frac{45}{144}=\frac{9\times 5}{9\times 16}=\frac{5}{16}$.

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 $\frac{240}{90}$

Solution

Method 1. Using the repeated division for simplifying fractions.

We first notice that both 240 and 90 are divisible by 10, hence dividing by 10 we get,

$\frac{240}{90}=\frac{24\times 10}{9\times 10}=\frac{24}{9}$

Now 24 and 9 are both dividable by 3, so dividing by 3 we get,

$\frac{24}{9}=\frac{3\times 8}{3\times 3}=\frac{8}{3}$

Next, 8 and 3 have no ore common factors, thus $\frac{8}{3}$is the simplest form of the given fraction.

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

The factors of 240 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240.

The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

We notice that the greatest common divisor of 240 and 90 is 30.

Dividing both the numerator and the denominator by 30, we get

$\frac{240}{90}=\frac{8\times 30}{3\times 30}=\frac{8}{3}$.

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 $4\frac{32}{60}$

Solution

Firstly, we must turn $4\frac{32}{60}$ into an improper fraction. To do this, we can again express the integer part of the mixed fraction as a fraction with the same denominator as the fractional part.

$4\frac{32}{60}=\frac{4\times 60}{60}+\frac{32}{60}=\frac{240}{60}+\frac{32}{60}=\frac{272}{60}$

Again, 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 $\frac{68}{15}$.

Therefore,

$4\frac{32}{60}=\frac{68}{15}$

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,

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