← Find container weight after adding or removing · Track a Quantity Through Changes

Find container weight after adding or removing · 10 practice problems

3.MD.A.2

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 24 oz

A basket full of apples is weighed and comes to $5\ \text{lb}\ 4\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $3\ \text{lb}\ 6\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 5 lb 4 oz. After half the apples are eaten, it weighs 3 lb 6 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 5 lb 4 oz.
Basket with half the apples weighs 3 lb 6 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 5 lb 4 oz = 5 x 16 + 4 = 84 oz; half-eaten basket 3 lb 6 oz = 3 x 16 + 6 = 54 oz.

5 \times 16 + 4 = 84,\quad 3 \times 16 + 6 = 54

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 84 oz to 54 oz, a loss of 84 - 54 = 30 oz. That 30 oz is exactly half of all the apples.

84 - 54 = 30\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 30 oz, then all the apples weigh twice that: 30 x 2 = 60 oz.

30 \times 2 = 60\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 84 - 60 = 24 oz.

84 - 60 = 24\ \text{oz}

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

Answer: 24 oz

4 · Reviewdoes it hold up?

Check: basket 24 oz + half apples 30 oz = 54 oz, which matches the second weighing; and 24 oz + 60 oz = 84 oz matches the first weighing. So 24 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 84 and b + a/2 = 54; subtracting gives a/2 = 30, so a = 60 and b = 24 oz.

Standardsmin 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.

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

Variant 2 easy answer: 32 oz

A basket full of apples is weighed and comes to $8\ \text{lb}\ 0\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $5\ \text{lb}\ 0\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 8 lb 0 oz. After half the apples are eaten, it weighs 5 lb 0 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 8 lb 0 oz.
Basket with half the apples weighs 5 lb 0 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 8 lb 0 oz = 8 x 16 + 0 = 128 oz; half-eaten basket 5 lb 0 oz = 5 x 16 + 0 = 80 oz.

8 \times 16 + 0 = 128,\quad 5 \times 16 + 0 = 80

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 128 oz to 80 oz, a loss of 128 - 80 = 48 oz. That 48 oz is exactly half of all the apples.

128 - 80 = 48\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 48 oz, then all the apples weigh twice that: 48 x 2 = 96 oz.

48 \times 2 = 96\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 128 - 96 = 32 oz.

128 - 96 = 32\ \text{oz}

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

Answer: 32 oz

4 · Reviewdoes it hold up?

Check: basket 32 oz + half apples 48 oz = 80 oz, which matches the second weighing; and 32 oz + 96 oz = 128 oz matches the first weighing. So 32 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 128 and b + a/2 = 80; subtracting gives a/2 = 48, so a = 96 and b = 32 oz.

Standardsmin 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.

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

Variant 3 easy answer: 24 oz

A basket full of apples is weighed and comes to $6\ \text{lb}\ 8\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $4\ \text{lb}\ 0\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 6 lb 8 oz. After half the apples are eaten, it weighs 4 lb 0 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 6 lb 8 oz.
Basket with half the apples weighs 4 lb 0 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 6 lb 8 oz = 6 x 16 + 8 = 104 oz; half-eaten basket 4 lb 0 oz = 4 x 16 + 0 = 64 oz.

6 \times 16 + 8 = 104,\quad 4 \times 16 + 0 = 64

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 104 oz to 64 oz, a loss of 104 - 64 = 40 oz. That 40 oz is exactly half of all the apples.

104 - 64 = 40\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 40 oz, then all the apples weigh twice that: 40 x 2 = 80 oz.

40 \times 2 = 80\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 104 - 80 = 24 oz.

104 - 80 = 24\ \text{oz}

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

Answer: 24 oz

4 · Reviewdoes it hold up?

Check: basket 24 oz + half apples 40 oz = 64 oz, which matches the second weighing; and 24 oz + 80 oz = 104 oz matches the first weighing. So 24 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 104 and b + a/2 = 64; subtracting gives a/2 = 40, so a = 80 and b = 24 oz.

Standardsmin 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.

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

Variant 4 medium answer: 32 oz

A basket full of apples is weighed and comes to $9\ \text{lb}\ 0\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $5\ \text{lb}\ 8\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 9 lb 0 oz. After half the apples are eaten, it weighs 5 lb 8 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 9 lb 0 oz.
Basket with half the apples weighs 5 lb 8 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 9 lb 0 oz = 9 x 16 + 0 = 144 oz; half-eaten basket 5 lb 8 oz = 5 x 16 + 8 = 88 oz.

9 \times 16 + 0 = 144,\quad 5 \times 16 + 8 = 88

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 144 oz to 88 oz, a loss of 144 - 88 = 56 oz. That 56 oz is exactly half of all the apples.

144 - 88 = 56\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 56 oz, then all the apples weigh twice that: 56 x 2 = 112 oz.

56 \times 2 = 112\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 144 - 112 = 32 oz.

144 - 112 = 32\ \text{oz}

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

Answer: 32 oz

4 · Reviewdoes it hold up?

Check: basket 32 oz + half apples 56 oz = 88 oz, which matches the second weighing; and 32 oz + 112 oz = 144 oz matches the first weighing. So 32 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 144 and b + a/2 = 88; subtracting gives a/2 = 56, so a = 112 and b = 32 oz.

Standardsmin 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.

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

Variant 5 medium answer: 16 oz

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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

#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.

2Find the weight of the apples that were eaten

#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.

3Find the weight of all the apples

#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.

4Subtract all apples from the full weight

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 6 medium answer: 32 oz

A basket full of apples is weighed and comes to $10\ \text{lb}\ 0\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $6\ \text{lb}\ 0\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 10 lb 0 oz. After half the apples are eaten, it weighs 6 lb 0 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 10 lb 0 oz.
Basket with half the apples weighs 6 lb 0 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 10 lb 0 oz = 10 x 16 + 0 = 160 oz; half-eaten basket 6 lb 0 oz = 6 x 16 + 0 = 96 oz.

10 \times 16 + 0 = 160,\quad 6 \times 16 + 0 = 96

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 160 oz to 96 oz, a loss of 160 - 96 = 64 oz. That 64 oz is exactly half of all the apples.

160 - 96 = 64\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 64 oz, then all the apples weigh twice that: 64 x 2 = 128 oz.

64 \times 2 = 128\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 160 - 128 = 32 oz.

160 - 128 = 32\ \text{oz}

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

Answer: 32 oz

4 · Reviewdoes it hold up?

Check: basket 32 oz + half apples 64 oz = 96 oz, which matches the second weighing; and 32 oz + 128 oz = 160 oz matches the first weighing. So 32 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 160 and b + a/2 = 96; subtracting gives a/2 = 64, so a = 128 and b = 32 oz.

Standardsmin 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.

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

Variant 7 medium answer: 32 oz

A basket full of apples is weighed and comes to $10\ \text{lb}\ 8\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $6\ \text{lb}\ 4\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 10 lb 8 oz. After half the apples are eaten, it weighs 6 lb 4 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 10 lb 8 oz.
Basket with half the apples weighs 6 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 10 lb 8 oz = 10 x 16 + 8 = 168 oz; half-eaten basket 6 lb 4 oz = 6 x 16 + 4 = 100 oz.

10 \times 16 + 8 = 168,\quad 6 \times 16 + 4 = 100

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 168 oz to 100 oz, a loss of 168 - 100 = 68 oz. That 68 oz is exactly half of all the apples.

168 - 100 = 68\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#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.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 168 - 136 = 32 oz.

168 - 136 = 32\ \text{oz}

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

Answer: 32 oz

4 · Reviewdoes it hold up?

Check: basket 32 oz + half apples 68 oz = 100 oz, which matches the second weighing; and 32 oz + 136 oz = 168 oz matches the first weighing. So 32 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 168 and b + a/2 = 100; subtracting gives a/2 = 68, so a = 136 and b = 32 oz.

Standardsmin 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.

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

Variant 8 hard answer: 32 oz

A basket full of apples is weighed and comes to $11\ \text{lb}\ 0\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $6\ \text{lb}\ 8\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 11 lb 0 oz. After half the apples are eaten, it weighs 6 lb 8 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 11 lb 0 oz.
Basket with half the apples weighs 6 lb 8 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 11 lb 0 oz = 11 x 16 + 0 = 176 oz; half-eaten basket 6 lb 8 oz = 6 x 16 + 8 = 104 oz.

11 \times 16 + 0 = 176,\quad 6 \times 16 + 8 = 104

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 176 oz to 104 oz, a loss of 176 - 104 = 72 oz. That 72 oz is exactly half of all the apples.

176 - 104 = 72\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 72 oz, then all the apples weigh twice that: 72 x 2 = 144 oz.

72 \times 2 = 144\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 176 - 144 = 32 oz.

176 - 144 = 32\ \text{oz}

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

Answer: 32 oz

4 · Reviewdoes it hold up?

Check: basket 32 oz + half apples 72 oz = 104 oz, which matches the second weighing; and 32 oz + 144 oz = 176 oz matches the first weighing. So 32 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 176 and b + a/2 = 104; subtracting gives a/2 = 72, so a = 144 and b = 32 oz.

Standardsmin 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.

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

Variant 9 hard answer: 48 oz

A basket full of apples is weighed and comes to $7\ \text{lb}\ 12\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $5\ \text{lb}\ 6\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 7 lb 12 oz. After half the apples are eaten, it weighs 5 lb 6 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 7 lb 12 oz.
Basket with half the apples weighs 5 lb 6 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 7 lb 12 oz = 7 x 16 + 12 = 124 oz; half-eaten basket 5 lb 6 oz = 5 x 16 + 6 = 86 oz.

7 \times 16 + 12 = 124,\quad 5 \times 16 + 6 = 86

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 124 oz to 86 oz, a loss of 124 - 86 = 38 oz. That 38 oz is exactly half of all the apples.

124 - 86 = 38\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 38 oz, then all the apples weigh twice that: 38 x 2 = 76 oz.

38 \times 2 = 76\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 124 - 76 = 48 oz.

124 - 76 = 48\ \text{oz}

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

Answer: 48 oz

4 · Reviewdoes it hold up?

Check: basket 48 oz + half apples 38 oz = 86 oz, which matches the second weighing; and 48 oz + 76 oz = 124 oz matches the first weighing. So 48 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 124 and b + a/2 = 86; subtracting gives a/2 = 38, so a = 76 and b = 48 oz.

Standardsmin 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.

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

Variant 10 hard answer: 32 oz

A basket full of apples is weighed and comes to $12\ \text{lb}\ 4\ \text{oz}$ . After half of the apples are eaten, the basket is weighed again and comes to $7\ \text{lb}\ 2\ \text{oz}$ . How much does the empty basket weigh, in ounces? (Every apple weighs the same.)

Show solution

1 · Understandwhat's really being asked

A basket of apples weighs 12 lb 4 oz. After half the apples are eaten, it weighs 7 lb 2 oz. All apples weigh the same. Find the weight of the empty basket in ounces.

Givens

Basket with all apples weighs 12 lb 4 oz.
Basket with half the apples weighs 7 lb 2 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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Change both weights to ounces

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

With 1 lb = 16 oz: full basket 12 lb 4 oz = 12 x 16 + 4 = 196 oz; half-eaten basket 7 lb 2 oz = 7 x 16 + 2 = 114 oz.

12 \times 16 + 4 = 196,\quad 7 \times 16 + 2 = 114

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

2Find the weight of the apples that were eaten

#11 Work Backwards 3.MD.A.2

Eating half the apples dropped the weight from 196 oz to 114 oz, a loss of 196 - 114 = 82 oz. That 82 oz is exactly half of all the apples.

196 - 114 = 82\ \text{oz (half the apples)}

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

3Find the weight of all the apples

#11 Work Backwards 3.OA.A.3

If half the apples weigh 82 oz, then all the apples weigh twice that: 82 x 2 = 164 oz.

82 \times 2 = 164\ \text{oz (all apples)}

Doubling half of something gives the whole.

4Subtract all apples from the full weight

#11 Work Backwards 3.MD.A.2

The empty basket weighs the full weight minus all the apples: 196 - 164 = 32 oz.

196 - 164 = 32\ \text{oz}

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

Answer: 32 oz

4 · Reviewdoes it hold up?

Check: basket 32 oz + half apples 82 oz = 114 oz, which matches the second weighing; and 32 oz + 164 oz = 196 oz matches the first weighing. So 32 oz is consistent.

Another way: Convert to algebra (tool 13): let basket = b and all apples = a. Then b + a = 196 and b + a/2 = 114; subtracting gives a/2 = 82, so a = 164 and b = 32 oz.

Standardsmin 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.

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