Simplification of algebraic expressions with fractional coefficients

Simplification of algebraic expressions with fractional coefficients

Algebraic expressions with fractional coefficients can be simplified like ordinary fractions. As with simplification of algebraic expressions that have whole number coefficients, like terms are collected together and simplified to get a single term. For example: $\frac{2}{3}x$, $\frac{1}{2}xy$, $\frac{21}{50}x$, $\frac{3}{5}x^2y$ are terms with fractional coefficients.

The algebraic expressions $\frac{2}{3}x$ and $\frac{21}{50}x$ are examples of like terms, whereas $\frac{1}{2}xy$ and $\frac{3}{5}x^2y$ are examples of unlike terms.

Addition and subtraction of algebraic expressions with fractional coefficients

Example 1

Simplify: $\frac{2}{3}y + \frac{3}{4}y$

Solution

Since the variable is the same, add the coefficients:

The LCM of 3 and 4 is 12.

Thus, $\frac{2}{3}y + \frac{3}{4}y = \frac{2y}{3} + \frac{3y}{4}$

$= \frac{4 \times 2y + 3 \times 3y}{12}$

$= \frac{8y + 9y}{12}$

$= \frac{17y}{12}$

$= 1\frac{5}{12}y$

Therefore, $\frac{2}{3}y + \frac{3}{4}y = 1\frac{5}{12}y$

Example 2

Simplify: $\frac{1}{2}a + \frac{2}{3}a$

Solution

Add the coefficients of the given algebraic expression as follows:

The LCM of 2 and 3 is 6.

Add and simplify the answer:

Thus, $\frac{1}{2}a + \frac{2}{3}a = \frac{3 \times 1 + 2 \times 2}{6a}$

$= \frac{3 + 4}{6a}$

$= \frac{7}{6}a$

Therefore, $\frac{1}{2}a + \frac{2}{3}a = \frac{7}{6}a$

Example 3

Simplify: $\frac{2x}{5t^2} - \frac{x}{t^2}$

Solution

Subtract the coefficients as shown:

The LCM of $5t^2$ and $t^2$ is $5t^2$

Thus, $\frac{2x}{5t^2} - \frac{x}{t^2} = \frac{1 \times 2x - 5 \times x}{5t^2}$

$= \frac{2x - 5x}{5t^2}$

$= -\frac{3x}{5t^2}$

Therefore, $\frac{2x}{5t^2} - \frac{x}{t^2} = -\frac{3x}{5t^2}$

Multiplication and division of algebraic expressions with fractional coefficients

Example 1

Simplify: $\frac{1}{2}a \times \frac{3}{5}b$

Solution

Expand the algebraic expression:

$\frac{1}{2}a \times \frac{3}{5}b = \frac{1}{2} \times a \times \frac{3}{5} \times b$

Collect the like terms and multiply:

$\frac{1}{2} \times a \times \frac{3}{5} \times b = \frac{1}{2} \times \frac{3}{5} \times a \times b$

$= \frac{3}{10}ab$

Therefore, $\frac{1}{2}a \times \frac{3}{5}b = \frac{3}{10}ab$

Example 2

Simplify: $\frac{1}{2}m \div \frac{1}{4}m$

Solution

$\frac{1}{2}m \div \frac{1}{4}m$ can be written as $\frac{m}{2} \div \frac{m}{4}$

Using the rule $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ then simplify:

$\frac{m}{2} \div \frac{m}{4} = \frac{m}{2} \times \frac{4}{m} = \frac{4m}{2m} = 2$

Therefore, $\frac{1}{2}m \div \frac{1}{4}m = 2$

Example 3

Simplify: $\frac{15}{21}a^3c \div \frac{1}{14}ac^3$

Solution

$\frac{15}{21}a^3c \div \frac{1}{14}ac^3 = \frac{15a^3c}{21} \div \frac{ac^3}{14}$

Apply the rule of fraction division:

$\frac{15a^3c}{21} \div \frac{ac^3}{14} = \frac{15a^3c}{21} \times \frac{14}{ac^3}$

Collect like terms and simplify:

$\frac{15a^3c}{21} \times \frac{14}{ac^3} = \frac{15}{21} \times 14 \times \frac{a^3c}{ac^3}$

$= 10 \times \frac{a^2}{c^2}$

$= \frac{10a^2}{c^2}$

Therefore, $\frac{15}{21}a^3c \div \frac{1}{14}ac^3 = \frac{10a^2}{c^2}$