Find container weight after adding or removing

3.MD.A.2 · adapt · grade 3

Archetype: Track a Quantity Through Changes · step in a 7-type progression

A basket full of apples is weighed and comes to $9\ \text{lb}\ 8\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $5\ \text{lb}\ 4\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

Understand

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.

Givens

Basket with all apples weighs 9 lb 8 oz.
Basket with half the apples weighs 5 lb 4 oz.
Every apple weighs the same.
1 lb = 16 oz.

Unknowns

The weight of the empty basket in ounces.

Constraints

Exactly half the apples were eaten between the two weighings.

Plan

#11 Work Backwards · also uses: #8 Analyze the Units

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.

Execute

#8 Analyze the Units 3.MD.A.2

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 \times 16 + 8 = 152,\quad 5 \times 16 + 4 = 84

Working in a single unit (ounces) makes the differences easy to subtract.

#11 Work Backwards 3.MD.A.2

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\ \text{oz (half the apples)}

The basket itself never changed weight, so the whole drop is just the apples that left.

#11 Work Backwards 3.OA.A.3

If half the apples weigh 68 oz, then all the apples weigh twice that: 68 x 2 = 136 oz.

68 \times 2 = 136\ \text{oz (all apples)}

Doubling half of something gives the whole.

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 152 - 136 = 16 oz.

152 - 136 = 16\ \text{oz}

Take everything off the basket and only the basket's own weight is left.

Answer: 16 oz

Review

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.

Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 152 and b + a/2 = 84; subtracting gives a/2 = 68, so a = 136 and b = 16 oz.

Standards · min grade 3

3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting pound-ounce weights and subtracting masses to isolate the basket.
3.OA.A.3 Solve multiplication and division word problems within 100 — Doubling the half-apple weight to get the full apple weight.

💡 This only needs Grade 3 thinking: the weight that disappeared is half the apples, so double it, then subtract from the full basket!