Grade 5 Math Puzzle 4: Count overlapping groups using totals

A free grade 5 math puzzle that teaches Count overlapping groups using totals. Try it first — the hint and the full solution open when you are ready.

Problem 4 · Count overlapping groups using totals · 4 points

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Eight kids line up: the first five average 11 points, the last five average 15 points, and all eight average 12 points. What is the average score of the middle two kids?

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Draw eight boxes in a row. Mark the first five and the last five in different colors.
Compare the two groups’ combined total with the total for all eight kids.

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Answer: C — 17

The first five scored 5 × 11 = 55 points in total, and the last five scored 5 × 15 = 75. Together, those totals count the middle two kids twice, while the total for all eight—8 × 12 = 96—counts them once. So the middle two scored 55 + 75 − 96 = 34 points together, and their average is 34 ÷ 2 = 17. The tempting answer 34 is their total, not their average.

Grade 5 puzzle 4: Count overlapping groups using totals

Problem 4 · Count overlapping groups using totals · 4 points