Math Core

What is $6 + 4 \times 5$ ? If you add first, you get 50. If you multiply first, you get 26. Both can't be right! Mathematicians agreed on one order for doing operations, so everyone gets the same answer.

The order to follow

Order of operations

Do the work inside grouping symbols first: parentheses $(\ )$ , brackets $[\ ]$ and braces $\{\ \}$ .
Multiply and divide, working from left to right.
Add and subtract, working from left to right.

So $6 + 4 \times 5$ means multiply first: $4 \times 5 = 20$ , then $6 + 20 = 26$ .

If you really want to add first, use parentheses: $(6 + 4) \times 5 = 10 \times 5 = 50$ . Parentheses are like a sign that says "do me first."

Left to right

Multiplication and division are partners. Neither one always goes first. You do them in the order they appear, reading left to right. Addition and subtraction work the same way.

\begin{aligned} 18 \div 3 \times 2 &= 6 \times 2 && \text{divide first (it is on the left)} \\ &= 12 \end{aligned}

Common mistake

Don't multiply just because "M comes before D" in a memory trick like PEMDAS. In $18 \div 3 \times 2$ , if you multiply first you get $18 \div 6 = 3$ , which is wrong. The division is on the left, so it goes first, and the answer is 12.

Brackets and braces

Sometimes one group sits inside another. To make it easier to read, we use different symbols for each layer:

parentheses $(\ )$ on the inside,
brackets $[\ ]$ around those,
braces $\{\ \}$ on the very outside.

Always start with the innermost group and work your way out, like peeling an onion from the middle.

Worked example: Multiply before you subtract

Evaluate $20 - 3 \times 4 + 1$ .

There are no grouping symbols. Multiply first: $3 \times 4 = 12$ .

Now add and subtract from left to right: $20 - 12 = 8$ , then $8 + 1 = 9$ .

The answer is 9.

Worked example: Parentheses inside brackets

Evaluate $2 \times [5 + (6 - 2)]$ .

\begin{aligned} 2 \times [5 + (6 - 2)] &= 2 \times [5 + 4] && \text{parentheses} \\ &= 2 \times 9 && \text{brackets} \\ &= 18 && \text{multiply} \end{aligned}

Worked example: Three layers

Evaluate $50 - \{4 \times [3 + (10 \div 2)]\}$ .

\begin{aligned} 50 - \{4 \times [3 + (10 \div 2)]\} &= 50 - \{4 \times [3 + 5]\} && \text{parentheses} \\ &= 50 - \{4 \times 8\} && \text{brackets} \\ &= 50 - 32 && \text{braces} \\ &= 18 && \text{subtract} \end{aligned}

Tip

Rewrite the whole expression after every step, doing only one step at a time. It takes a little longer, but it keeps you from skipping or mixing up operations.

Practice

Practice 1

Evaluate $6 + 4 \times 5$ .

Type a number.

Practice 2

Evaluate $(6 + 4) \times 5$ .

Type a number.

Practice 3

Evaluate $24 \div 4 \times 2$ .

Type a number.

Practice 4

Evaluate $30 - 12 \div 3 + 1$ .

Type a number.

Practice 5

Evaluate $3 \times [10 - (2 + 5)]$ .

Type a number.

Practice 6

Where should the parentheses go so that $2 + 6 \times 3 - 1$ equals 16?

Practice 7

Evaluate $40 - \{2 \times [3 + (8 - 4)]\}$ .

Type a number.

Practice 8

Evaluate $\dfrac{1}{2} \times (8 + 4) - 2.5$ .

Type a number.