Problem
Work backwards in fraction word problems
The weed on a pond doubles every 24 hours. It covers the pond completely on day 20. Forward a day doubles it, so back a day halves it. Find the day it covered one eighth of the pond.
Reasoning · Grade 3-2 Applying Fractions
Your answer
Watch out One eighth suggests an eighth of the 20 days, but with doubling it is only three halvings back.
How to solve
Strategy Work Backwards — The end state is known (full pond on day 20) and I need an earlier state, which is exactly when to work backwards. Doubling forward means halving backward, so I step back day by day, halving the covered fraction until it reaches 1/8.
1STEP 1
Turn 'doubles forward' into 'halves backward'
A step back halves the cover.
day before = 1/2 × (that day's amount)
If something keeps doubling, running the clock backward means it keeps halving — the reverse of doubling is taking half.
3.NF.A.1Work BackwardsSince the cover doubles going forward, running the days backwards means halving each time.
Why?
Halving reverses exactly what doubling did, so stepping back a day undoes a day's growth.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Because the growth is the same multiple every day, the same single step backwards works no matter which day you start from.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
2STEP 2
Halve back from a full pond
From 1 to 1/2, 1/4, 1/8 takes three steps.
1 → 1/2 → 1/4 → 1/8 (days 20 → 19 → 18 → 17)
Each halving cuts the cover into one more pair of equal parts, so three halvings from the whole lands exactly on one-eighth.
3.NF.A.1Look For A Pattern3STEP 3
Count the days back
So it is day 20 − 3 = 17.
20 - 3 = 17
1/8 is the whole halved three times, so it happens three days before the pond is full.
3.NBT.A.2Work BackwardsAnswer
17 day
20 − 3 = 17
Checking forward from day 17: 1/8 → 1/4 (day 18) → 1/2 (day 19) → 1 (day 20). The pond is full on day 20, exactly as stated, so day 17 is correct. The answer is earlier than day 20, which makes sense since less cover means an earlier day.
Takeaway
When something keeps doubling, run the movie backwards by taking half each time — three halves from a full pond is just one-eighth!
- Turn 'doubles forward' into 'halves backward'
- Halve back from a full pond
- Count the days back