Problem
Find empty container weight by subtracting fractions
A bottle full of water weighs 4 1/6 kg. After half the water is drunk, the bottle weighs 2 1/2 kg. I need the weight of the empty bottle.
Fractions
Your answer
How to solve
Strategy Work Backwards — The drop in weight from the full bottle to the half-empty bottle is exactly the weight of half the water. Working backwards from that lost amount gives the weight of all the water, and then subtracting it from the full weight leaves the empty bottle.
1STEP 1
Find the weight of half the water
Only water left the bottle, so 25/6 - 15/6 = 5/3 kg is half of the water.
4 1/6 - 2 1/2 = 25/6 - 15/6 = 10/6 = 5/3 kg
Only the water changed, so the weight that disappeared must be the half-glass of water that was drunk.
5.NF.A.1Work BackwardsThe difference between the two weighings is exactly half of the water.
Why?
Both weighings hold the very same bottle, so comparing them leaves the bottle's weight out of the difference entirely.
Why?
Each weighing is the bottle together with the water inside it, with nothing left out and nothing counted twice.
🧱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?
The only thing that changed between the two weighings was drinking half the water, so the difference measures just that half.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Find the weight of all the water
Half the water weighs 5/3 kg, so the whole amount of water is twice that.
5/3 × 2 = 10/3 kg
Two equal halves make a whole, so double the half to get all the water.
5.NF.A.2Identify Subproblems3STEP 3
Subtract the water from the full bottle
The empty bottle is the full bottle minus all of its water. Write both amounts as sixths and subtract.
4 1/6 - 10/3 = 25/6 - 20/6 = 5/6 kg
Take away everything that is water and what is left is just the bottle.
5.NF.A.2Work BackwardsAnswer
kg
Check by adding the bottle back to the water: empty bottle 5/6 + half water 5/3 = 5/6 + 10/6 = 15/6 = 2 1/2 kg, which matches the half-empty weight, and 5/6 + full water 20/6 = 25/6 = 4 1/6 kg, which matches the full weight. An empty bottle weighing less than the water it holds is sensible.
Takeaway
The weight that vanished is the water that was drunk, so work backwards to peel the water off and find the bottle.
- Find the weight of half the water
- Find the weight of all the water
- Subtract the water from the full bottle
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested