Operations
Choose the right calculation, then check it
Knowing when to add, subtract, multiply, or divide — and using mental strategies to get there quickly and accurately without a calculator.
See It in Action
The Problem
A box holds 8 books. There are 376 books. How many complete boxes are needed to hold ALL the books?
Common Mistake
376 ÷ 8 = 47. So 47 boxes.
The calculation is correct but the reasoning is incomplete. 376 ÷ 8 = 47 exactly, so 47 boxes hold exactly all 376 books. In this case the answer IS 47 — but students often make this error when there's a remainder. If it were 380 books: 380 ÷ 8 = 47.5, which means 48 boxes are needed (you can't have half a box).
Correct Approach
376 ÷ 8: think 8 × 40 = 320, so 376 − 320 = 56 remaining. 8 × 7 = 56. So 8 × 47 = 376. 376 ÷ 8 = 47 exactly. Since there's no remainder, 47 complete boxes hold all the books.
Answer: 47 boxes
47 × 8 = 376. ✓ If there had been a remainder, we would need 48 boxes.
The Core Concept
The NSW selective maths test has no calculator. That means two things matter equally: choosing the right operation and performing it efficiently. Most errors come from the first part — students calculate correctly but answer the wrong question.
Each operation has a real-world meaning. Addition combines quantities. Subtraction finds the difference or remaining amount. Multiplication is repeated addition (or scaling). Division is sharing equally or finding how many groups. When a word problem says "how many more?" that's subtraction. "How many groups?" is division.
Mental strategies save time. For multiplication, break numbers apart: 47 × 6 = (40 × 6) + (7 × 6) = 240 + 42 = 282. For division, think of related multiplication: 252 ÷ 9 — what times 9 is close to 252? 9 × 28 = 252. Always estimate first (47 × 6 ≈ 50 × 6 = 300) so you know if your answer is reasonable.
Common Mistakes to Avoid
Rounding down when the problem needs rounding up
"How many boxes to hold ALL the books?" with a remainder means you need one extra box. Always re-read whether the context needs ceiling (round up) or floor (round down).
Ignoring order of operations
3 + 4 × 2 = 11, not 14. Multiplication and division happen before addition and subtraction. Use BODMAS/BIDMAS as a reminder.
Using the wrong operation from misreading
"How many more" → subtract. "Shared equally" → divide. "Times as many" → multiply. Underline the key phrase before calculating.
Not checking if the answer is reasonable
Always estimate first. If 47 × 8 ≈ 50 × 8 = 400, and your answer is 47, it's in the right ballpark. An answer of 470 or 4.7 would immediately look wrong.
Try It Yourself
Eggs are sold in cartons of 12. A bakery needs 500 eggs. How many cartons must they buy?
Hint: Divide 500 by 12. Don't forget: "must they buy" means they need enough for ALL 500 eggs, so what happens if there's a remainder?
Key Tips
Estimate before calculating. It takes two seconds and tells you instantly if your answer is wildly wrong.
Underline the operation word in word problems: "more", "fewer", "share", "product", "times as many".
When dividing with a remainder, re-read the question to decide whether to round up or down.
Related Skills
Ready to practise Operations?
Reading is the start. The Maths Gym has exercises designed around this skill — with instant feedback and progress tracking.