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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 20 points

1.How many of the numbers from 1 to 20 are not multiples of 3?[3]
- A6
- B13
- C14
- D17
- E20
2.A cafe offers 3 main courses and 2 puddings. How many different two-course meals can be ordered?[3]
Text description of the figure
A tree diagram. One starting point branches into three main courses, and each of those branches into two puddings, giving six paths in all.
- A2
- B5
- C6
- D9
- E12
3.In a class of 30, 18 play football and 14 play chess. 6 play both. How many play neither?[4]
- A2
- B4
- C8
- D10
- E26
4.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
5.Ten different whole numbers, each bigger than zero, add up to 100. What is the largest that the biggest of them can be?[5]
- A45
- B50
- C55
- D91
- E100

Math Kangaroo Star · https://kangaroo-atlas.vercel.app · seed 1 · 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.C — 14KA-0051Complementary counting
- AI counted the multiples of 3 and gave that instead of the numbers that are not multiples.
- BI counted 21 as a multiple of 3 inside the range, so I removed one too many.
- DI removed only the multiples of 3 that are also even, so I subtracted three instead of six.
- EI gave the size of the whole range and forgot to remove anything at all.
2.C — 6KA-0067Tree diagrams
- AI counted the puddings only and forgot that the main course is also a choice.
- BI added the number of mains to the number of puddings instead of multiplying them.
- DI used three puddings as well as three mains, without reading how many puddings there are.
- EI doubled the answer, as if the order of the two courses made a different meal.
3.B — 4KA-0069Inclusion–exclusion with two sets
- AI added 18 and 14 without removing the overlap, then subtracted from 30.
- CI subtracted the 6 who play both twice instead of once.
- DI subtracted only the footballers from the class and forgot the chess players entirely.
- EI found how many play at least one sport and gave that instead of how many play neither.
4.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.
5.C — 55KA-0078Extremal / worst-case counting
- AI found the smallest possible total of the other nine and gave that instead of what is left.
- BI assumed the biggest number could be at most half the total.
- DI made the other nine numbers all equal to 1, forgetting that they have to be different from each other.
- EI ignored the other nine numbers entirely, as if the biggest could take the whole total.

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