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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 25 points

1.A three-digit code uses only the digits 1, 2 and 3, and digits may repeat. How many such codes contain at least one 3?[5]
- A8
- B9
- C12
- D19
- E27
2.How many three-digit whole numbers contain at least one digit 7?[5]
- A243
- B252
- C271
- D280
- E648
3.A box holds 4 red, 5 blue and 6 green balls, mixed up. How many must be taken out without looking to be sure of having 3 of the same colour?[5]
- A3
- B6
- C7
- D9
- E13
4.The numbers 1 to 8 are on a board. A move rubs out two of them and writes their difference, larger minus smaller. After seven moves one number is left. Can it be 1?[5]
- AYes, and there are many ways
- BYes, but only one way
- CNo, the last number is always even
- DNo, the last number is always 0
- EIt depends on the order of the moves
5.A class of 12 scored an average of 8. Another class of 18 scored an average of 13. What is the average for all 30 students together?[5]
- A10.5
- B11
- C11.5
- D12
- E21

Kangaroo Atlas · https://kangaroo-atlas.vercel.app · seed 2 · CC BY-NC-SA 4.0 · Independent project, not affiliated with Math Kangaroo in USA, NFP.

Answer key — Math Kangaroo practice

Each wrong choice carries the thinking that produces it. When a student picks C, this is what they were doing.

1.D — 19KA-0018Complementary counting
- AI counted the codes that avoid 3 entirely and gave that as my answer without subtracting.
- BI counted the codes with a 3 in the first position only and forgot the other two positions.
- CI counted the codes with exactly one 3 and forgot the ones with two or three of them.
- EI counted every possible code and forgot to remove the ones with no 3 at all.
2.B — 252KA-0075Complementary counting
- AI counted the numbers made entirely of digits other than 7 in every position, including a leading zero.
- CI counted the numbers with a 7 in each position separately and forgot that some were counted twice.
- DI added three lots of 90 and one extra hundred, double counting the 700s.
- EI counted the numbers with no 7 at all and gave that instead of subtracting it.
3.C — 7KA-0077The pigeonhole principle
- AI answered with the luckiest case, where the first three balls happen to match.
- BI found the worst case of two of each colour and forgot to take one more.
- DI multiplied the three colours by the three balls I need, rather than thinking about the worst case.
- EI worked from the number of balls of each colour rather than from the number of colours.
4.C — No, the last number is always evenKA-0098Invariants and monovariants
- AI found sequences ending in small numbers and assumed 1 was among the reachable ones.
- BI assumed a hard-looking question must have exactly one answer.
- DI found one sequence ending in 0 and assumed every sequence must end there.
- EI thought the order could change the outcome, when the quantity that decides it never changes.
5.B — 11KA-0112Mixtures and weighted averages
- AI averaged the two averages, ignoring that the classes are different sizes.
- CI weighted the averages but used the wrong class size against each one.
- DI leaned towards the larger class's average without computing the totals.
- EI added the two averages together instead of combining them.

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