Order of operations with fractions: Problem type 1

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Definition

We combine the order operations $PEMDAS$ with adding, subtracting, multiplying, and dividing fractions.

Rules for Order of Operations with Fractions

• First, we simplify any parentheses if any in the expression.

• Next, we simplify any exponents if present in the expression.

• We do multiplication and division before addition and subtraction.

• We do multiplication and division based on order of appearance from left to right in the problem.

• Next, we do addition and subtraction based on order of appearance from left to right in the problem.

Consider the following problems involving PEMDAS with adding, subtracting, multiplying, and dividing fractions.

Example 1

Evaluate $\frac{4}{5}[17-32\left ( \frac{1}{4} \right )^{2}]$

Solution

Step 1:

As per the PEMDAS rule of operations on fractions we simplify the brackets or the parentheses first.

Step 2:

Within the brackets, the first we simplify the exponent as $\left ( \frac{1}{4} \right )^{2} = \frac{1}{16}$

Step 3:

Within the brackets, next we multiply as follows

$17-32\left ( \frac{1}{4} \right )^2 = 17-32 \times \frac{1}{16} = 17 - 2$

Step 4:

Within the brackets, next we subtract as follows

17 - 2 So, $[17-32\left ( \frac{1}{4} \right )^2] = 15$

Step 5:

$\frac{4}{5}[17-32\left ( \frac{1}{4} \right )^2] = \frac{4}{5}[15] = \frac{4}{5} \times 15$

So, simplifying we get

$\frac{4}{5} \times 15 = 4 \times 3 = 12$

Step 6:

So, finally $\frac{4}{5}[17-32\left ( \frac{1}{4} \right )^2] = 12$

Example 2

Evaluate $\left ( \frac{36}{7} - \frac{11}{7}\right ) \times \frac{8}{5} - \frac{9}{7}$

Solution

Step 1:

As per the PEMDAS rule of operations on fractions we simplify the brackets or the parentheses first.

Within the brackets, the first we subtract the fractions as follows

Step 2:

Next, we multiply as follows

$\left ( \frac{36}{7} - \frac{11}{7}\right ) \times \frac{8}{5} - \frac{9}{7} = \frac{25}{7} \times \frac{8}{5} - \frac{9}{7} = \frac{40}{7} - \frac{9}{7}$

Step 3:

We then subtract as follows

$\frac{40}{7} - \frac{9}{7} = \frac{(40-9)}{7} = \frac{31}{7}$

Step 4:

So, finally $\left ( \frac{36}{7} - \frac{11}{7} \right ) \times \frac{8}{5} - \frac{9}{7} = \frac{31}{7} = 4\frac{3}{7}$