In multistep word problems the story told you what to do first. A bare expression like $2 + 3 \times 4$ has no story, so everyone agrees on one order. That way the same expression always has the same value.
Key idea
Multiply and divide first, working from left to right. Then add and subtract, from left to right. Anything in parentheses comes before all of that.
Why it works
Multiplying is a shortcut for adding equal groups: $3 \times 4$ means $4 + 4 + 4$. In $2 + 3 \times 4$, the $3 \times 4$ is one chunk, three groups of 4. Write the chunk out and you get $2 + 4 + 4 + 4 = 14$. Doing the multiplication first keeps the chunk together.
Worked examples
Worked example: Multiply before adding
Find $2 + 3 \times 4$.
- Look for $\times$ or $\div$. There is one: $3 \times 4 = 12$.
- Now the expression is $2 + 12$.
- Add: $2 + 12 = 14$.
Answer: $14$
Worked example: Three operations
Find $20 - 12 \div 4 \times 2$.
- Multiplying and dividing come first. Of those two, the division is further left.
- $12 \div 4 = 3$, so the expression is $20 - 3 \times 2$.
- $3 \times 2 = 6$, so the expression is $20 - 6$.
- Subtract: $20 - 6 = 14$.
Answer: $14$
Worked example: Left to right
Find $18 \div 3 \times 2$.
- Multiplying and dividing are partners. Neither one always comes first: go from left to right.
- $18 \div 3 = 6$.
- $6 \times 2 = 12$.
Answer: $12$ (multiplying first would give $3$, which is not right)
Common mistake
Working straight from left to right: $2 + 3 = 5$, then $5 \times 4 = 20$. That adds the 2 into every group of 4, which the expression never asked for. Quick check: before you start, circle each $\times$ and $\div$ with the numbers on both sides. The circles are worked first.
- Parentheses first.
- Then multiply and divide, from left to right.
- Then add and subtract, from left to right.