Depth · one concept from first principles · step 2 of 4
The ball falls from 24 ft and rebounds to 5/6 of that. Multiply 24 by 5/6.
Taking a fraction of a height is exactly multiplying by that fraction, a fifth-grade fraction-times-whole-number skill.
5.NF.B.6Solve An Easier Related ProblemEvery bounce takes 5/6 of the height just reached, so the rule repeats: x 5/6 each time.
Multiplying by 5/6 (a number less than 1) shrinks the height each time, so the bounces get lower in a predictable pattern.
5.NF.B.5Look For A PatternEvery bounce reaches 5/6 of the height the ball reached just before it.
Taking 5/6 of a height means cutting that height into 6 equal parts and keeping 5 of them, and the rule is stated about whatever height it is applied to.
After each bounce the ball falls again from the height it just reached, so the same rule runs once more starting from that new height.
One bounce is one fall followed by one rise, so the height a bounce starts from is exactly the height the bounce before it reached.
Apply the rule to the first-bounce height of 20 ft.
Even when the number stops dividing evenly, the same multiply-by-5/6 step still works; I just keep it as a fraction.
5.NF.B.4Look For A PatternApply the rule one more time to the second-bounce height of 50/3 ft.
Multiplying two fractions multiplies the tops and the bottoms; the result is the same as 24 times 5/6 three times in a row.
5.NF.B.4Look For A PatternEach bounce is just 5/6 of the last height, so keep multiplying by 5/6 and the ball climbs a little less every time.
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The same shape shows up later on the AMC, as Compound Area by Decomposition / Difference. Why the shapes repeat →