Problem 21 · 2020 Math Kangaroo
Stretch
Arithmetic & Operations
sum-constraint
The teacher wrote the numbers 1 to 8 on the board. Then he covered the numbers with triangles, squares and one circle (see picture). The sum of the numbers covered by the triangles equals the sum of the numbers covered by the squares, and the number covered by the circle is a quarter of that sum. What is the sum of the numbers covered by the triangles and the circle?
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Answer: C — 20
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Hint 1 of 3
First add up all the hidden numbers: 1 + 2 + 3 + ... + 8.
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Hint 2 of 3
The triangle pile and the square pile weigh the same, and the circle is just a small extra equal to a quarter of one of those piles.
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Hint 3 of 3
Try to split the total into two equal big piles plus a small piece that is a quarter of one big pile.
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Approach: split the total 36 into two equal piles plus a quarter-size circle
- The hidden numbers are 1 through 8, and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.
- The triangles and the squares make two equal piles, and the circle adds a quarter of one of those piles, so 36 splits as one pile + one equal pile + a quarter-pile.
- That is the same as four-and-a-quarter quarter-piles making 36, so each quarter-pile is 4; one full pile (the triangles) is four of them, which is 16, and the circle is one quarter-pile, which is 4.
- The triangles cover 16 and the circle covers 4, so together they cover 16 + 4 = 20, choice C.
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