Problem
Find container weight after adding or removing
A basket of apples weighs 9 lb 8 oz. After half the apples are eaten, it weighs 5 lb 4 oz. All apples weigh the same. Find the weight of the empty basket in ounces.
Measurement & data
Your answer
oz
Watch out It's easy to treat the 68 oz that disappeared as the weight of all the apples instead of realizing it's only half — that slip gives a wrong basket weight of 84 oz instead of the true 16 oz.
How to solve
Strategy Work Backwards — The weight lost equals exactly half the apples. Doubling that loss gives all the apples; removing all apples from the full weight leaves the basket. Convert to ounces first so subtraction is clean.
1STEP 1
Change both weights to ounces
With 1 lb = 16 oz: full basket 9 lb 8 oz = 9 x 16 + 8 = 152 oz; half-eaten basket 5 lb 4 oz = 5 x 16 + 4 = 84 oz.
9 × 16 + 8 = 152, 5 × 16 + 4 = 84
Working in a single unit (ounces) makes the differences easy to subtract.
3.MD.A.2Analyze The Units2STEP 2
Find the weight of the apples that were eaten
Eating half the apples dropped the weight from 152 oz to 84 oz, a loss of 152 - 84 = 68 oz. That 68 oz is exactly half of all the apples.
152 - 84 = 68 oz (half the apples)
The basket itself never changed weight, so the whole drop is just the apples that left.
3.MD.A.2Work BackwardsThe 68-ounce weight lost between the two weighings is exactly half of all the apples.
Why?
The 68 ounces that vanished is the weight of the apples that were eaten, because the basket and the uneaten apples were never touched between the two weighings.
Why?
A weight on the scale is the basket and the apples added together, so taking one part away drops the total by exactly that part's weight and leaves the rest unchanged.
🧱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?
Those eaten apples are exactly half of all the apples' weight, because half the apples were eaten and every apple weighs the same.
Why?
Eating half means each eaten apple pairs with one uneaten apple, so the eaten group and the kept group hold the same number of apples.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The same count of equal-weight apples weighs the same, so the eaten group and the kept group each come to half of all the apples' weight.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Find the weight of all the apples
If half the apples weigh 68 oz, then all the apples weigh twice that: 68 x 2 = 136 oz.
68 × 2 = 136 oz (all apples)
Doubling half of something gives the whole.
3.OA.A.3Work Backwards4STEP 4
Subtract all apples from the full weight
The empty basket weighs the full weight minus all the apples: 152 - 136 = 16 oz.
152 - 136 = 16 oz
Take everything off the basket and only the basket's own weight is left.
3.MD.A.2Work BackwardsAnswer
16 oz
Check: basket 16 oz + half apples 68 oz = 84 oz, which matches the second weighing; and 16 oz + 136 oz = 152 oz matches the first weighing. So 16 oz is consistent.
Takeaway
This only needs Grade 3 thinking: the weight that disappeared is half the apples, so double it, then subtract from the full basket!
- Change both weights to ounces
- Find the weight of the apples that were eaten
- Find the weight of all the apples
- Subtract all apples from the full weight
Where next?
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