Lesson 3: Multiplying, Dividing, and Order of Operations

A cinema trip: four friends buy tickets at $9 each and share a $7 popcorn bucket, so the total is 4 × 9 + 7 = $43. Now swap the order of the numbers: one $9 ticket plus 4 bags of popcorn at $7 each is written 9 + 4 × 7. Worked strictly left to right, that gives 13 × 7 = 91 — clearly wrong, since the popcorn alone costs only 4 × 7 = $28. The correct total is 9 + 28 = 37, found by multiplying before adding. Mathematicians agreed on an order of operations precisely so that everyone gets the same answer. This lesson locks in that order, plus the sign rules for multiplying and dividing integers.

The order of operations

Work through every expression in this order:

• Parentheses (and other grouping symbols) first — innermost groups first.
• Exponents next.
• Multiplication and Division next, working left to right — they are equal partners, neither goes first.
• Addition and Subtraction last, also left to right.

Many students remember this as PEMDAS ("Please Excuse My Dear Aunt Sally"), but the left-to-right rule for the pairs is where marks are won and lost.

Worked example 1. Evaluate 6 + 4 × 3. Multiplication before addition: 6 + 12 = 18 (not 30 — that is the answer to (6 + 4) × 3, a different question).

Worked example 2. Evaluate 20 − 12 ÷ 4. Division first: 20 − 3 = 17.

Worked example 3. Evaluate 2 × (5 + 3)² − 10 ÷ 5. Parentheses: (5 + 3) = 8. Exponent: 8² = 64. Multiply and divide: 2 × 64 = 128 and 10 ÷ 5 = 2. Subtract: 128 − 2 = 126.

Worked example 4. Evaluate 24 ÷ 4 × 2. Multiplication and division are equal, so go left to right: 24 ÷ 4 = 6, then 6 × 2 = 12. (Grouping it as 24 ÷ 8 = 3 is a classic trap.)

Multiplying and dividing integers

The sign rules are short and mighty:

• Same signs → positive answer. (−5) × (−7) = 35, and −48 ÷ (−6) = 8.
• Different signs → negative answer. (−4) × 6 = −24, and 36 ÷ (−4) = −9.

Why? Think of (−5) × (−7) as the opposite of 5 × (−7) = −35, which is +35. Or picture a video playing backwards of a person walking backwards: two reversals put you back facing forwards.

Worked example 5. A submarine descends 45 metres every 10 minutes. After 40 minutes at that rate, its depth changes by 4 × (−45) = −180 metres. The negative sign reports direction (down), not a mistake.

In the real world: splitting and scaling

A coach hires a minibus for $240, split equally among the players who show up. If 12 show up, each pays 240 ÷ 12 = $20; if only 8 show up, each pays 240 ÷ 8 = $30. Recipes scale the same way: doubling a recipe that uses 3 eggs and 250 g of flour means computing 2 × 3 = 6 eggs and 2 × 250 = 500 g — multiplication before any adding of extra ingredients. And when a shop assistant totals 3 shirts at $15 and a $5 discount, the register computes 3 × 15 − 5 = $40, multiplication first, exactly as PEMDAS demands.

Common mistakes

• Adding before multiplying. 9 + 4 × 7 is 9 + 28 = 37, not 13 × 7 = 91. Multiply and divide before you add and subtract (unless parentheses say otherwise).
• Treating multiplication as always before division. 24 ÷ 4 × 2 = 12 working left to right, not 24 ÷ 8 = 3.
• Two negatives make a positive — only sometimes. (−5) × (−7) = 35, yes; but −5 + (−7) = −12. The "two negatives" rule belongs to multiplication and division, not addition.
• Ignoring exponents until the end. In 3² + 5 × 2, the exponent comes before the multiplication: 9 + 10 = 19.

Exam Experience

1. What is the value of 8 + 6 ÷ 2? A) 7 B) 11 C) 14 D) 16

2. What is the value of (−9) × (−3)? A) −27 B) −12 C) 12 D) 27

3. What is the value of 3 + 2 × (8 − 5)²? A) 21 B) 45 C) 57 D) 75

4. (Constructed response) Amara buys 3 notebooks at $2.50 each and 2 pens at $1.25 each. She hands over a coupon for $2 off her total. (a) Write a single numerical expression for the amount she pays. (b) Evaluate your expression, showing the order in which you worked. (c) Amara pays with a $20 bill. How much change does she receive?

Answer key and explanations

• 1. B — Divide first: 6 ÷ 2 = 3, then 8 + 3 = 11.
• 2. D — Same signs multiply to a positive: 9 × 3 = 27.
• 3. A — Parentheses: (8 − 5) = 3. Exponent: 3² = 9. Multiply: 2 × 9 = 18. Add: 3 + 18 = 21.
• 4. (a) 3 × 2.50 + 2 × 1.25 − 2. (b) 7.50 + 2.50 − 2 = $8.00. (c) 20 − 8 = $12 change.

Scoring guide for question 4 (3 points)

• 1 point for a correct expression with multiplication used for the repeated items and the coupon subtracted once.
• 1 point for the correct total, $8.00, with working that multiplies before adding/subtracting.
• 1 point for the change, $12, found by subtracting the total from $20.

Going further: longer expressions and real temperatures

With the order of operations and the sign rules both in hand, longer expressions hold no terrors — just work the hierarchy and keep the signs attached to their numbers.

Worked example 6. Evaluate 5 × (3 + 4) − 18 ÷ 3 + 2². Parentheses: 5 × 7. Exponent: 2² = 4. Multiply/divide: 35 and 6. Left to right: 35 − 6 + 4 = 33.

Worked example 7. Evaluate −4 × (5 − 8). Inside the parentheses: 5 − 8 = −3. Then −4 × (−3) = 12 — same signs, positive answer.

Worked example 8. Evaluate −60 ÷ 5 ÷ (−3), strictly left to right: −60 ÷ 5 = −12; then −12 ÷ (−3) = 4. Chained divisions are where sign errors breed; writing each stage down is armour.

Worked example 9. Overnight, a cold front drops the temperature by 3°F every hour for 5 hours, starting from 41°F. The final temperature is 41 + 5 × (−3) = 41 − 15 = 26°F. Notice how the situation writes itself as multiplication of a negative — "drops by 3, five times" is 5 × (−3).

Quick practice

• 1. 7 + 3 × (10 − 4)
• 2. (−8) × (−5) ÷ 10
• 3. 100 − 6² × 2
• 4. 15 ÷ 3 × 4

Answers. 1. Parentheses first: 7 + 3 × 6 = 7 + 18 = 25. 2. (−8) × (−5) = 40, then 40 ÷ 10 = 4. 3. Exponent, then multiply: 100 − 36 × 2 = 100 − 72 = 28. 4. Left to right: 5 × 4 = 20 (not 15 ÷ 12).

Investigation: bracket detective

Two expressions arrive with their answers missing and one pair of brackets to place. (a) Make it true: 2 + 3 × 4 − 1 = 19. Without brackets the left side is 13. Placing brackets around the first pair, (2 + 3) × 4 − 1 = 5 × 4 − 1 = 19 ✓. The brackets promoted the addition above the multiplication — that is their entire job. (b) Evaluate 12 ÷ 2 × 3 + 1 with no brackets: work left to right through the equal partners: 12 ÷ 2 = 6, 6 × 3 = 18, 18 + 1 = 19. The same target reached by obedience rather than brackets. (c) Error hunt: a classmate writes 4 + 3² × 2 = 26, reasoning "(4 + 9) × 2". The error is adding before multiplying. Correctly: 4 + 9 × 2 = 4 + 18 = 22. (d) Finally, a sign workout: −2 × (−3)². Exponent first — and note the parentheses make (−3) the base: (−3)² = 9. Then −2 × 9 = −18. Four small problems, and every rule in the lesson used at least once: that is what fluency feels like.

Challenge corner

Evaluate 18 − 2 × (4 + 3²) + (−5), narrating every decision. Exponent first, even inside brackets: 3² = 9. Parentheses: 4 + 9 = 13. Multiplication: 2 × 13 = 26. Now the line reads 18 − 26 + (−5). Left to right: 18 − 26 = −8, and −8 + (−5) = −13. Notice the two traps you avoided: computing 2 × 4 before the exponent (wrong — exponents outrank multiplication), and treating "− 26 + (−5)" as −21 by dropping a sign (wrong — subtracting a positive and adding a negative both move you left). A final flourish: the expression's answer is negative, and that is fine — expressions, like temperatures, are allowed to end below zero. Confidence is finishing the calculation and trusting the rules over your instincts.