Problem 13 · 2012 Math Kangaroo
Medium
Algebra & Patterns
substitutionsum-constraint
Barbara wants to complete the grid shown on the right by inserting three numbers into the empty spaces. The sum of the first three numbers should be 100, the sum of the middle three numbers 200 and the sum of the last three numbers 300. Which is the middle number in this grid?
| 10 | 130 |
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Answer: B — 60
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Hint 1 of 2
Write the five numbers as 10, then three unknowns, then 130.
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Hint 2 of 2
Use the overlapping sums of three to peel off one number at a time.
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Approach: overlapping sums
- Call the row 10, a, b, c, 130 with 10+a+b = 100, a+b+c = 200, b+c+130 = 300.
- From 10+a+b=100: a+b = 90. From b+c+130=300: b+c = 170.
- Then a+b+c = 200 gives c = 200−90 = 110, and b = 170−110 = 60.
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