Solving problems using DMAS

This lesson explains the DMAS rule, which sets the order in which we simplify a numerical expression, and then works through several examples. DMAS stands for division, multiplication, addition, and subtraction.

The DMAS rule

When an expression mixes these operations, we do not just work straight across. We scan from left to right and follow this order:

• D division
• M multiplication
• A addition
• S subtraction

So we clear every division and multiplication first, and only then do the additions and subtractions.

Example 1

Simplify $20 \times 3 - 25 \div 5 + 2 \times 5$.

Division and multiplication come first:

$20 \times 3 - 25 \div 5 + 2 \times 5 = 60 - 5 + 10$

Then the addition and subtraction:

$60 - 5 + 10 = 65$

Example 2

Simplify $12 \div 4 + 8 \times 1 + 6 - 3$.

Do the division and multiplication first:

$12 \div 4 + 8 \times 1 + 6 - 3 = 3 + 8 + 6 - 3$

$= 14$

Example 3: a longer expression

Simplify $102 - 12 \times 6 + 12 \div 2 + 3 - 2 + 7$.

Clear the division and the multiplication first:

$102 - 12 \times 6 + 12 \div 2 + 3 - 2 + 7 = 102 - 72 + 6 + 3 - 2 + 7$

Now add the positives and subtract:

$102 + 6 + 3 + 7 = 118, \qquad 72 + 2 = 74$

$118 - 74 = 44$

Example 4

Simplify $64 \div 16 \times 3 + 2$.

Division first, then multiplication:

$64 \div 16 \times 3 + 2 = 4 \times 3 + 2 = 12 + 2 = 14$

Example 5: two divisions in a row

Simplify $14 \div 1 \times 2 \div 2 + 16 - 2$.

Both divisions can be cleared first:

$14 \div 1 = 14, \qquad 2 \div 2 = 1$

Then the multiplication and the rest:

$14 \times 1 + 16 - 2 = 14 + 16 - 2 = 28$

Example 6

Simplify $21 + 3 \div 3 - 2 \times 3 + 7$.

Division and multiplication first:

$21 + 3 \div 3 - 2 \times 3 + 7 = 21 + 1 - 6 + 7$

Then add and subtract:

$21 + 1 + 7 = 29, \qquad 29 - 6 = 23$

Example 7

Simplify $3 \div 3 \times 0 + 11 - 8$.

Division then multiplication:

$3 \div 3 \times 0 + 11 - 8 = 1 \times 0 + 11 - 8 = 0 + 11 - 8 = 3$

Example 8

Simplify $7 \times 3 + 2 \times 1 - 8 \div 2$.

Clear the division and the multiplications:

$7 \times 3 + 2 \times 1 - 8 \div 2 = 21 + 2 - 4 = 19$

Example 9

Simplify $100 \div 5 + 10 \times 5 - 70 + 10$.

Division and multiplication first:

$100 \div 5 + 10 \times 5 - 70 + 10 = 20 + 50 - 70 + 10$

Then add and subtract:

$20 + 50 + 10 = 80, \qquad 80 - 70 = 10$

Example 10: clearing two divisions in one step

Simplify $7 + 2 \div 2 + 8 \times 20 \div 2$.

Do both divisions first. Under DMAS the division in $8 \times 20 \div 2$ is done before the multiplication, so we compute $20 \div 2 = 10$ first and only then multiply by $8$:

$2 \div 2 = 1, \qquad 20 \div 2 = 10$

Then the multiplication and addition:

$7 + 1 + 8 \times 10 = 7 + 1 + 80 = 88$

Key takeaways

• DMAS sets the order: division, then multiplication, then addition, then subtraction.
• Scan the expression from left to right and clear every division and multiplication before any addition or subtraction.
• Reduce the expression one operation at a time, and the final value follows safely.