# Order of operations

**What are the order of numbers?**

**Whole numbers**

A set of natural numbers and $$0$$ is called whole numbers. Negative numbers and fractional numbers are not whole numbers. For example: $$0$$, $$1$$, $$2$$, $$3$$, $$4$$ etc. are whole numbers.

**Fraction**

Fraction can be defined as a part of a whole. It includes numerator which shows the part and denominator which shows the whole. For example: In the fraction $$\frac{5}{9}$$, $$5$$ is numerator and $$9$$ is denominator.

**Decimal numbers**

Decimal numbers are the numbers that have decimal points in them. The decimal point separates the whole part and the fractional part in the decimal numbers. For example: $$20.78$$, $$20$$ is a whole part and $$78$$ is a fractional part.

**E1.8A: Use the four rules of calculations with whole numbers including correct ordering of operations and use of brackets.**

The whole number rules of calculation contain addition, subtraction, multiplication and division. If all four of these operations are there in a question, then division is performed first, followed by multiplication, then addition and finally subtraction.

For instance, consider the following example: evaluate $$2 \div 2 + 3 - 4 \times 2$$, first, we have to apply the division operation to get $$1 + 3 - 4 \times 2$$. Then multiplication operation to get $$1 + 3 - 8$$. Then apply the addition operation to get $$4 - 8$$ and lastly, apply subtraction operation to get the final answer equal to $$ - 4$$.

**Worked example**

Solve: $$16 - 3 + \left( {12 \div 2 \times 3} \right)$$ using order of operations.

**Step 1: Apply the division operation.**

$$16 - 3 + \left( {6 \times 3} \right)$$** **

**Step 2: Apply the multiplication operation.**

$$16 - 3 + 18$$

**Step 3: Apply the addition operation.**

$$16 + 15$$

**Step 4: Apply the subtraction operation.**

$$31$$

**Step 5: Write the final answer.**

$$16 - 3 + \left( {12 \div 2 \times 3} \right) = 31$$

**E1.8B: Use the four rules of calculations with decimal and including correct ordering of operations and use of brackets.**

The rules for the whole numbers also apply to the decimal numbers in the same order. Let us learn how to evaluate an expression using the correct order of operations with the help of examples.

**Worked example**

**Example 1:** Solve: $$0.16 - 0.3 + \left( {0.12 \div 2 \times 3} \right)$$ using order of operations.

**Step 1: Apply the division operation.**

$$0.16 - 0.3 + \left( {0.6 \times 3} \right)$$** **

**Step 2: Apply the multiplication operation.**

$$0.16 - 0.3 + \left( {1.8} \right)$$

**Step 3: Apply the addition operation.**

$$0.16 + 1.5$$

**Step 4: Apply the subtraction operation.**

$$1.66$$

**Step 5: Write the final answer.**

$$0.16 - 0.3 + \left( {0.12 \div 2 \times 3} \right) = 1.66$$

**E1.8C: Use the four rules of calculations with fractions including correct ordering of operations and use of brackets.**

The rules for whole numbers also apply to fractional numbers as well. The order of applying operations in the fractions is same as in the whole numbers and the decimals. Let us understand how to apply operations in the correct order with the help of examples.

**Worked example**

**Example 2:** $$\frac{{16}}{3} - \frac{{16}}{3} \times \left( {\frac{4}{2} \div \frac{4}{2}} \right)$$ using order of operations.

**Step 1: Apply the division operation.**

$$\frac{{16}}{3} - \frac{{16}}{3} \times 1$$** **

**Step 2: Apply the multiplication operation.**

$$\frac{{16}}{3} - \frac{{16}}{3}$$

**Step 3: Apply the subtraction operation.**

$$0$$

**Step 4: Write the final answer.**

$$\frac{{16}}{3} - \frac{{16}}{3} \times \left( {\frac{4}{2} \div \frac{4}{2}} \right) = 0$$