Questions: Order of Operations Introduction

Question 1

What is the value of 2 + 3 × 4?

• A. 20 — add 2 + 3 first to get 5, then multiply by 4
• B. 14 — multiply 3 × 4 first to get 12, then add 2
• C. 24 — multiply all three numbers together
• D. 9 — add all three numbers: 2 + 3 + 4

Answer: (B)

The correct answer is 14. By convention, multiplication is performed before addition. So 3 × 4 = 12 first, then 2 + 12 = 14. Option A (20) is the result of adding first — the most common error, and the exact wrong answer the order-of-operations convention was created to prevent.

Question 2

Why did mathematicians agree on an order of operations?

• A. Because multiplication is more important than addition in real-world applications
• B. So that everyone computes the same answer from the same expression — without a shared convention, the same problem has multiple 'correct' answers
• C. Because calculators were invented and needed a built-in rule to follow
• D. Because addition must always be done last so it can include all the multiplication results

Answer: (B)

Order of operations is a convention — an agreed-upon standard, not a mathematical law derived from deeper truths. Without it, 2 + 3 × 4 could legitimately equal either 14 or 20, depending on which operation you do first. The convention eliminates that ambiguity so that mathematical expressions communicate the same meaning to everyone.

Question 3

In the expression 12 ÷ 4 × 3, you should divide first (left to right), giving 3 × 3 = 9.

• T. True
• F. False

Answer: True

When operations have the same priority level (both multiplication and division, or both addition and subtraction), you evaluate left to right. So 12 ÷ 4 = 3, then 3 × 3 = 9. Doing the multiplication first (4 × 3 = 12, then 12 ÷ 12 = 1) gives the wrong answer. Left-to-right is the tiebreaker for equal-priority operations.

Question 4

Parentheses in a math expression are optional — the answer is the same whether or not you include them.

• T. True
• F. False

Answer: False

Parentheses override the default order of operations. (2 + 3) × 4 = 20, while 2 + 3 × 4 = 14. These are different expressions with different values. Parentheses are not decoration — they are punctuation that changes meaning, just as punctuation changes meaning in written language.

Question 5

Why does 2 + 3 × 4 equal 14 and not 20? Explain the rule that determines which operation is done first.

Model answer: By the order-of-operations convention, multiplication is performed before addition. So in 2 + 3 × 4, you compute 3 × 4 = 12 first, then add 2 to get 14. If you added first (2 + 3 = 5, then 5 × 4 = 20), you would get a different answer. The convention ensures everyone computes the same result from the same expression.

The order of operations is a shared grammatical rule for mathematical expressions. Multiplication before addition is the convention, not a logical necessity — but it is universally agreed upon so that expressions are unambiguous. Parentheses can override this order when a different sequence is intended.