AMC Sample Questions with Worked Solutions

The best way to get a feel for the Australian Mathematics Competition (AMC) is to try some problems. Below are five original, AMC-style questions across the divisions (Years 3–10). Each rewards insight over memorised procedure — the hallmark of the AMC. Have a real go before reading the solution; each one shows the key idea, the method, and a quick check.

1. Middle Primary (Years 3–4) — counting outfits

Q. Ava has 3 different T-shirts and 2 different pairs of shorts. Wearing one T-shirt and one pair of shorts, how many different outfits can she make?

Insight: for each T-shirt she has a full choice of shorts, so you multiply the choices rather than add them.
Method: 3 T-shirts × 2 shorts = 6 outfits.
Check: list them — (T1,S1), (T1,S2), (T2,S1), (T2,S2), (T3,S1), (T3,S2). That's 6. ✓

2. Upper Primary (Years 5–6) — handshakes

Q. Six friends meet up, and each one shakes hands exactly once with every other friend. How many handshakes happen in total?

Insight: a handshake involves a pair of people. If you count "each person shakes 5 others", every handshake gets counted twice — so halve it.
Method: 6 × 5 = 30, then ÷ 2 = 15 handshakes.
Check: with 3 friends the formula gives 3 × 2 ÷ 2 = 3, and you can see 3 handshakes by hand. ✓

3. Upper Primary (Years 5–6) — working backwards

Q. A number is doubled, and then 5 is added. The result is 27. What was the original number?

Insight: undo the steps in reverse order — subtract before you halve.
Method: 27 − 5 = 22, then 22 ÷ 2 = 11.
Check: 11 doubled is 22, plus 5 is 27. ✓

4. Junior (Years 7–8) — divisibility

Q. How many whole numbers from 1 to 100 are divisible by both 3 and 4?

Insight: a number divisible by both 3 and 4 is divisible by their lowest common multiple, 12 — so you only need to count multiples of 12.
Method: multiples of 12 up to 100 are 12, 24, 36, 48, 60, 72, 84, 96 — that's 8.
Check: 96 ÷ 12 = 8, and the next multiple (108) is over 100. ✓

5. Intermediate (Years 9–10) — consecutive numbers

Q. Three consecutive whole numbers add up to 72. What is the largest of the three?

Insight: for consecutive numbers, the middle one is the average — so it's the total divided by 3.
Method: middle number = 72 ÷ 3 = 24, so the numbers are 23, 24, 25 and the largest is 25.
Check: 23 + 24 + 25 = 72. ✓

Notice the pattern

None of these needed a formula you had to memorise — each turned on spotting the right idea: multiply independent choices, a handshake is a pair, undo steps in reverse, "both" means the LCM, the middle of consecutive numbers is the average. That's exactly what the AMC trains, and what makes it different from a school test.

Practise more like these

PrepPath offers unlimited AMC-style practice across the Middle Primary, Upper Primary, Junior and Intermediate divisions (Years 3–10) — full 30-question mock papers with a real difficulty ramp, plus a practise-with-hints mode that nudges your child toward the key idea one step at a time. Every question is explained in the same three layers you see above: the insight, the method, and the check. For the bigger picture, read What Is the AMC? and How to Prepare for the AMC.

PrepPath is an independent educational platform and is not affiliated with, endorsed by, or associated with the Australian Maths Trust (AMT). "AMC" and "Australian Mathematics Competition" are used here for description only. These are original, AMC-style practice questions written by PrepPath — not real AMC papers.