Problem
Bouncing ball: repeated fraction-ratio heights
A ball always rebounds to 5/6 of the height it just fell from. It starts by being dropped from 24 m. I need the height it reaches after its third bounce.
Fractions
Your answer
How to solve
Strategy Look for a Pattern — Each bounce repeats the same action: multiply the current height by 5/6. Spotting this repeated structure (Look for a Pattern) lets me solve the easy first bounce, then reuse the same step two more times. I keep meters as the unit throughout to check the setup.
1STEP 1
First bounce
The ball falls from 24 m and rebounds to 5/6 of that. Multiply 24 by 5/6.
24 × 5/6 = (24 × 5)/6 = 120/6 = 20 m
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 Problem2STEP 2
See the repeating rule
Every bounce takes 5/6 of the height just reached, so the rule repeats: x 5/6 each time.
bounce = previous height × 5/6
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.
Why?
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.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
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.
Why?
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.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Second bounce
Apply the rule to the first-bounce height of 20 m.
20 × 5/6 = 100/6 = 50/3 m
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 Pattern4STEP 4
Third bounce
Apply the rule one more time to the second-bounce height of 50/3 m.
50/3 × 5/6 = 250/18 = 125/9 = 13 8/9 m
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 PatternAnswer
m (= m ≈ 13.89 m)
The bounces 20 m, 16.67 m, 13.89 m steadily decrease and each is a bit more than 4/5 of the one before, which matches 5/6 of a shrinking height; the unit stays meters and the answer is well below the 24 m drop, so it is reasonable.
Takeaway
Each bounce is just 5/6 of the last height, so keep multiplying by 5/6 and the ball climbs a little less every time.
- First bounce
- See the repeating rule
- Second bounce
- Third bounce
Where next?
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Step back and solidifysuggested