Pick what you want, print the page. The answer key comes with it, on its own sheet, and it tells you the mistake behind every wrong choice — not just which letter is right.
This exact URL always produces this exact sheet — bookmark it and the answer key will still match next term. Print with your browser; the controls and the site navigation are left off the page.
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.Two students are chosen from a group of six to represent the class. How many different pairs are possible?[4]
A12
B15
C21
D30
E36
5.Each of the four edges of a square is painted either black or white. Two paintings count as the same if one can be rotated onto the other. How many genuinely different paintings are there?[4]
A6
B16
C4
D8
E5
6.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
7.Five beads of five different colours are threaded on a circular bracelet. Two bracelets count as the same if one can be turned or flipped into the other. How many different bracelets are there?[5]
A12
B20
C24
D60
E120
8.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
9.How many squares of any size can be found on a 3 by 3 board?[5]
A9
B10
C12
D13
E14
10.An ordinary six-sided die is rolled three times. What is the probability that at least one roll shows a six?[5]
A91/216
B1/2
C125/216
D3/216
E1/6
Kangaroo Atlas · 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.B — 15KA-0074Selections of a few objects
AI doubled the group size instead of counting pairs.
CI allowed a student to be paired with themselves, adding six impossible pairs.
DI counted each pair twice, once in each order, and forgot to halve.
EI multiplied 6 by 6, counting every ordered pair including a student with themselves.
5.A — 6KA-0173Counting up to symmetry
BI counted every colouring of the four edges and forgot that rotations make some of them identical.
CI divided 16 by 4, but that only works when no painting is left unchanged by a rotation, and some are.
DI halved 16, treating only the 180 degree turn as a symmetry.
EI listed by how many edges are black but forgot that two black edges can be adjacent or opposite.
6.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.
7.A — 12KA-0071Overcount, then correct
BI divided the 120 arrangements by 6 rather than by the 10 movements that leave a bracelet looking the same.
CI allowed for turning the bracelet but forgot it can also be flipped over.
DI halved the 120 for flipping but forgot that turning also gives the same bracelet.
EI counted every arrangement in a line, as if the bracelet had a fixed first bead.
8.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.
9.E — 14KA-0133Counting shapes hidden inside a figure
AI counted the nine small squares and stopped, missing every larger one.
BI counted the small squares and the whole board, forgetting the middle size entirely.
CI found the 2 by 2 squares by splitting the board into blocks, getting two instead of four.
DI slid the 2 by 2 square across and down but missed one of its four positions.
10.A — 91/216KA-0168Complementary counting
BI added 1/6 three times, which counts the overlapping cases more than once and would exceed 1 for seven rolls.
CI worked out the probability of NO six and forgot to subtract it from 1.
DI found the probability of three sixes instead of at least one.
EI gave the probability for a single roll and ignored that there are three.