Problem
Undo repeated scaling by a fraction
A ball bounces back to only two-thirds of the height it fell from. Three bounces after the first drop it reaches 16 inches. The story must be run backwards. Find the height it was dropped from.
Reasoning · Grade 6-2 Fraction Calculation in Action
Your answer
Watch out Undoing two-thirds once is not enough: three bounces need three undos.
How to solve
Strategy Work Backwards — The problem hands me the end of the story and asks for its beginning, which is the classic signal to Work Backwards: whatever multiplying by 2/3 did on the way forward, dividing by 2/3 undoes on the way back. Before doing that I draw the bouncing path so I can see exactly how many times the fraction has been applied — miscounting the bounces is the only real danger here. Looking for a Pattern then lets me collect the three separate multiplications into a single fraction of the starting height, so the whole thing can also be finished in one step as a check on the step-by-step version.
1STEP 1
Draw the bounce path and label every height
Draw the path and label every height.
□ → 1st bounce → 2nd bounce → 3rd bounce = 16 in
Because 2/3 is smaller than 1, each arch in the drawing has to be shorter than the one before it — so the drawing itself warns you that the answer must be bigger than 16, not smaller.
5.NF.B.5Draw A Diagram2STEP 2
Write each bounce height as a fraction of the starting height
The third bounce is 8/27 of the drop.
2/3×2/3×2/3×□ = 8/27×□ = 16
Taking two-thirds of two-thirds is like shading two-thirds of an already-shaded strip: multiply the tops together and the bottoms together, and the repeated pattern turns three separate steps into one fraction.
5.NF.B.4Look For A Pattern3STEP 3
Undoing one bounce means dividing by 2/3, which is multiplying by 3/2
Undoing one bounce is multiplying by three-halves.
height ÷ 2/3 = height × 3/2
If a height is two-thirds of something, then one third of that something is half of the height, so the whole of it is three halves of the height — which is why flipping the fraction over is the right way back.
6.NS.A.1Work BackwardsUndoing one bounce means dividing by two thirds, which is the same as multiplying by three halves.
Why?
Dividing by a number undoes multiplying by it, so running the bounce backwards reverses exactly what the bounce did.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Dividing by two of three equal shares is the same as taking three of them for every two, which is what the flipped fraction says.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
4STEP 4
Climb back up the picture one bounce at a time
Climbing back gives 24, 36, 54.
16 × 3/2 = 24, 24 × 3/2 = 36, 36 × 3/2 = 54
Undoing one step at a time keeps every number whole and easy — 16, 24, 36, 54 — and each result can be sanity-checked on the spot, since two-thirds of 24 really is 16.
6.NS.A.1Work Backwards5STEP 5
Do it in one move as a check
Undoing in one move also gives 54.
□ = 16 ÷ 8/27 = 16 × 27/8 = 2 × 27 = 54
Cancelling the 8 into the 16 before multiplying keeps the arithmetic to one small times-table fact instead of working out 16 × 27 and then dividing 432 by 8.
6.EE.B.7Work BackwardsAnswer
54 inches
16 × 27 ÷ 8 = 54
Run the story forwards from 54 inches to confirm it: two-thirds of 54 is 36, two-thirds of 36 is 24, and two-thirds of 24 is 16 — the third bounce lands exactly on the given 16 inches. The magnitude is sensible too. Three bounces shrink the height to 8/27 of the start, which is a little less than a third, so the drop height should be a bit more than three times 16 inches, that is a bit over 48 — and 54 fits. The units stay in inches throughout, since multiplying a length by a plain fraction leaves a length. It is also a good sign that 54 is a multiple of 27: it has to be, or the three bounce heights could not all come out as whole numbers of inches.
Takeaway
When something has been shrunk by the same fraction over and over, flip the fraction over and grow it back the same number of times.
- Draw the bounce path and label every height
- Write each bounce height as a fraction of the starting height
- Undoing one bounce means dividing by 2/3, which is multiplying by 3/2
- Climb back up the picture one bounce at a time
- Do it in one move as a check