Practice · Combine sum and difference of two weights · Sensim Math

Variant 1 easy answer: 2 lb 8 oz (40 oz)

The weight of $4$ bags of beans plus $1$ bag of rice is $23\ \text{oz}$ . The weight of $4$ bags of beans minus $1$ bag of rice is $17\ \text{oz}$ . How much do $8$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

4 bags of beans together with one bag of rice weigh 23 oz, and 4 bags of beans with one bag of rice removed weigh 17 oz. Every bean bag weighs the same. We want the weight of 8 bean bags, written in pounds and ounces.

Givens

4 bags of beans + 1 bag of rice = 23 oz.
4 bags of beans - 1 bag of rice = 17 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 8 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 8 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '4 beans + 1 rice = 23' and '4 beans - 1 rice = 17', adding them makes the +1 rice and -1 rice cancel, leaving 8 bean bags equal to 23 + 17 = 40 oz.

(4B+R)+(4B-R)=23+17 \Rightarrow 8B = 40

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

8 bean bags weigh 40 oz, so one bean bag weighs 40 / 8 = 5 oz.

B = 40 \div 8 = 5\ \text{oz}

Splitting 40 oz equally among 8 bags is a basic within-100 division.

3Find 8 bean bags

#11 Work Backwards 3.OA.A.3

8 bags each weigh 5 oz, so together they weigh 8 x 5 = 40 oz.

8 \times 5 = 40\ \text{oz}

Multiplying the single-bag weight by 8 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 40 oz is 2 full pounds (32 oz) with 8 oz left over: 40 = 16 x 2 + 8.

40 \div 16 = 2\ \text{R}\ 8 \Rightarrow 2\ \text{lb}\ 8\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 8 oz (40 oz)

4 · Reviewdoes it hold up?

Check: with B=5, rice R = 23 - 20 = 3 oz, and 4 beans - rice = 20 - 3 = 17 oz, which matches. 8 bags = 40 oz, and 2 lb 8 oz = 32 + 8 = 40 oz, so the conversion is right.

Another way: Subtract instead: (4B+R) - (4B-R) = 2R = 6, so R = 3 oz; then 4B = 23 - 3 = 20, B = 5 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 40 by 8 and multiplying 5 by 8.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 40 oz into 2 lb 8 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 2 easy answer: 2 lb 10 oz (42 oz)

The weight of $3$ bags of beans plus $1$ bag of rice is $26\ \text{oz}$ . The weight of $3$ bags of beans minus $1$ bag of rice is $10\ \text{oz}$ . How much do $7$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

3 bags of beans together with one bag of rice weigh 26 oz, and 3 bags of beans with one bag of rice removed weigh 10 oz. Every bean bag weighs the same. We want the weight of 7 bean bags, written in pounds and ounces.

Givens

3 bags of beans + 1 bag of rice = 26 oz.
3 bags of beans - 1 bag of rice = 10 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 7 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 7 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '3 beans + 1 rice = 26' and '3 beans - 1 rice = 10', adding them makes the +1 rice and -1 rice cancel, leaving 6 bean bags equal to 26 + 10 = 36 oz.

(3B+R)+(3B-R)=26+10 \Rightarrow 6B = 36

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

6 bean bags weigh 36 oz, so one bean bag weighs 36 / 6 = 6 oz.

B = 36 \div 6 = 6\ \text{oz}

Splitting 36 oz equally among 6 bags is a basic within-100 division.

3Find 7 bean bags

#11 Work Backwards 3.OA.A.3

7 bags each weigh 6 oz, so together they weigh 7 x 6 = 42 oz.

7 \times 6 = 42\ \text{oz}

Multiplying the single-bag weight by 7 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 42 oz is 2 full pounds (32 oz) with 10 oz left over: 42 = 16 x 2 + 10.

42 \div 16 = 2\ \text{R}\ 10 \Rightarrow 2\ \text{lb}\ 10\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 10 oz (42 oz)

4 · Reviewdoes it hold up?

Check: with B=6, rice R = 26 - 18 = 8 oz, and 3 beans - rice = 18 - 8 = 10 oz, which matches. 7 bags = 42 oz, and 2 lb 10 oz = 32 + 10 = 42 oz, so the conversion is right.

Another way: Subtract instead: (3B+R) - (3B-R) = 2R = 16, so R = 8 oz; then 3B = 26 - 8 = 18, B = 6 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 36 by 6 and multiplying 6 by 7.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 42 oz into 2 lb 10 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 3 easy answer: 2 lb (32 oz)

The weight of $3$ bags of beans plus $1$ bag of rice is $30\ \text{oz}$ . The weight of $3$ bags of beans minus $1$ bag of rice is $18\ \text{oz}$ . How much do $4$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

3 bags of beans together with one bag of rice weigh 30 oz, and 3 bags of beans with one bag of rice removed weigh 18 oz. Every bean bag weighs the same. We want the weight of 4 bean bags, written in pounds and ounces.

Givens

3 bags of beans + 1 bag of rice = 30 oz.
3 bags of beans - 1 bag of rice = 18 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 4 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 4 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '3 beans + 1 rice = 30' and '3 beans - 1 rice = 18', adding them makes the +1 rice and -1 rice cancel, leaving 6 bean bags equal to 30 + 18 = 48 oz.

(3B+R)+(3B-R)=30+18 \Rightarrow 6B = 48

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

6 bean bags weigh 48 oz, so one bean bag weighs 48 / 6 = 8 oz.

B = 48 \div 6 = 8\ \text{oz}

Splitting 48 oz equally among 6 bags is a basic within-100 division.

3Find 4 bean bags

#11 Work Backwards 3.OA.A.3

4 bags each weigh 8 oz, so together they weigh 4 x 8 = 32 oz.

4 \times 8 = 32\ \text{oz}

Multiplying the single-bag weight by 4 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 32 oz is 2 full pounds (32 oz) with 0 oz left over: 32 = 16 x 2 + 0.

32 \div 16 = 2\ \text{R}\ 0 \Rightarrow 2\ \text{lb}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb (32 oz)

4 · Reviewdoes it hold up?

Check: with B=8, rice R = 30 - 24 = 6 oz, and 3 beans - rice = 24 - 6 = 18 oz, which matches. 4 bags = 32 oz, and 2 lb = 32 + 0 = 32 oz, so the conversion is right.

Another way: Subtract instead: (3B+R) - (3B-R) = 2R = 12, so R = 6 oz; then 3B = 30 - 6 = 24, B = 8 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 48 by 6 and multiplying 8 by 4.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 32 oz into 2 lb.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 4 easy answer: 2 lb 12 oz (44 oz)

The weight of $2$ bags of beans plus $1$ bag of rice is $31\ \text{oz}$ . The weight of $2$ bags of beans minus $1$ bag of rice is $13\ \text{oz}$ . How much do $4$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

2 bags of beans together with one bag of rice weigh 31 oz, and 2 bags of beans with one bag of rice removed weigh 13 oz. Every bean bag weighs the same. We want the weight of 4 bean bags, written in pounds and ounces.

Givens

2 bags of beans + 1 bag of rice = 31 oz.
2 bags of beans - 1 bag of rice = 13 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 4 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 4 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '2 beans + 1 rice = 31' and '2 beans - 1 rice = 13', adding them makes the +1 rice and -1 rice cancel, leaving 4 bean bags equal to 31 + 13 = 44 oz.

(2B+R)+(2B-R)=31+13 \Rightarrow 4B = 44

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

4 bean bags weigh 44 oz, so one bean bag weighs 44 / 4 = 11 oz.

B = 44 \div 4 = 11\ \text{oz}

Splitting 44 oz equally among 4 bags is a basic within-100 division.

3Find 4 bean bags

#11 Work Backwards 3.OA.A.3

4 bags each weigh 11 oz, so together they weigh 4 x 11 = 44 oz.

4 \times 11 = 44\ \text{oz}

Multiplying the single-bag weight by 4 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 44 oz is 2 full pounds (32 oz) with 12 oz left over: 44 = 16 x 2 + 12.

44 \div 16 = 2\ \text{R}\ 12 \Rightarrow 2\ \text{lb}\ 12\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 12 oz (44 oz)

4 · Reviewdoes it hold up?

Check: with B=11, rice R = 31 - 22 = 9 oz, and 2 beans - rice = 22 - 9 = 13 oz, which matches. 4 bags = 44 oz, and 2 lb 12 oz = 32 + 12 = 44 oz, so the conversion is right.

Another way: Subtract instead: (2B+R) - (2B-R) = 2R = 18, so R = 9 oz; then 2B = 31 - 9 = 22, B = 11 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 44 by 4 and multiplying 11 by 4.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 44 oz into 2 lb 12 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 5 medium answer: 3 lb 6 oz (54 oz)

The weight of $3$ bags of beans plus $1$ bag of rice is $32\ \text{oz}$ . The weight of $3$ bags of beans minus $1$ bag of rice is $22\ \text{oz}$ . How much do $6$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

3 bags of beans together with one bag of rice weigh 32 oz, and 3 bags of beans with one bag of rice removed weigh 22 oz. Every bean bag weighs the same. We want the weight of 6 bean bags, written in pounds and ounces.

Givens

3 bags of beans + 1 bag of rice = 32 oz.
3 bags of beans - 1 bag of rice = 22 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 6 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 6 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '3 beans + 1 rice = 32' and '3 beans - 1 rice = 22', adding them makes the +1 rice and -1 rice cancel, leaving 6 bean bags equal to 32 + 22 = 54 oz.

(3B+R)+(3B-R)=32+22 \Rightarrow 6B = 54

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

6 bean bags weigh 54 oz, so one bean bag weighs 54 / 6 = 9 oz.

B = 54 \div 6 = 9\ \text{oz}

Splitting 54 oz equally among 6 bags is a basic within-100 division.

3Find 6 bean bags

#11 Work Backwards 3.OA.A.3

6 bags each weigh 9 oz, so together they weigh 6 x 9 = 54 oz.

6 \times 9 = 54\ \text{oz}

Multiplying the single-bag weight by 6 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 54 oz is 3 full pounds (48 oz) with 6 oz left over: 54 = 16 x 3 + 6.

54 \div 16 = 3\ \text{R}\ 6 \Rightarrow 3\ \text{lb}\ 6\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 3 lb 6 oz (54 oz)

4 · Reviewdoes it hold up?

Check: with B=9, rice R = 32 - 27 = 5 oz, and 3 beans - rice = 27 - 5 = 22 oz, which matches. 6 bags = 54 oz, and 3 lb 6 oz = 48 + 6 = 54 oz, so the conversion is right.

Another way: Subtract instead: (3B+R) - (3B-R) = 2R = 10, so R = 5 oz; then 3B = 32 - 5 = 27, B = 9 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 54 by 6 and multiplying 9 by 6.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 54 oz into 3 lb 6 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 6 medium answer: 2 lb 3 oz (35 oz)

The weight of $4$ bags of beans plus $1$ bag of rice is $32\ \text{oz}$ . The weight of $4$ bags of beans minus $1$ bag of rice is $24\ \text{oz}$ . How much do $5$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

4 bags of beans together with one bag of rice weigh 32 oz, and 4 bags of beans with one bag of rice removed weigh 24 oz. Every bean bag weighs the same. We want the weight of 5 bean bags, written in pounds and ounces.

Givens

4 bags of beans + 1 bag of rice = 32 oz.
4 bags of beans - 1 bag of rice = 24 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 5 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 5 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '4 beans + 1 rice = 32' and '4 beans - 1 rice = 24', adding them makes the +1 rice and -1 rice cancel, leaving 8 bean bags equal to 32 + 24 = 56 oz.

(4B+R)+(4B-R)=32+24 \Rightarrow 8B = 56

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

8 bean bags weigh 56 oz, so one bean bag weighs 56 / 8 = 7 oz.

B = 56 \div 8 = 7\ \text{oz}

Splitting 56 oz equally among 8 bags is a basic within-100 division.

3Find 5 bean bags

#11 Work Backwards 3.OA.A.3

5 bags each weigh 7 oz, so together they weigh 5 x 7 = 35 oz.

5 \times 7 = 35\ \text{oz}

Multiplying the single-bag weight by 5 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 35 oz is 2 full pounds (32 oz) with 3 oz left over: 35 = 16 x 2 + 3.

35 \div 16 = 2\ \text{R}\ 3 \Rightarrow 2\ \text{lb}\ 3\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 3 oz (35 oz)

4 · Reviewdoes it hold up?

Check: with B=7, rice R = 32 - 28 = 4 oz, and 4 beans - rice = 28 - 4 = 24 oz, which matches. 5 bags = 35 oz, and 2 lb 3 oz = 32 + 3 = 35 oz, so the conversion is right.

Another way: Subtract instead: (4B+R) - (4B-R) = 2R = 8, so R = 4 oz; then 4B = 32 - 4 = 28, B = 7 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 56 by 8 and multiplying 7 by 5.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 35 oz into 2 lb 3 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 7 medium answer: 2 lb 7 oz (39 oz)

The weight of $2$ bags of beans plus $1$ bag of rice is $33\ \text{oz}$ . The weight of $2$ bags of beans minus $1$ bag of rice is $19\ \text{oz}$ . How much do $3$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

2 bags of beans together with one bag of rice weigh 33 oz, and 2 bags of beans with one bag of rice removed weigh 19 oz. Every bean bag weighs the same. We want the weight of 3 bean bags, written in pounds and ounces.

Givens

2 bags of beans + 1 bag of rice = 33 oz.
2 bags of beans - 1 bag of rice = 19 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 3 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 3 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '2 beans + 1 rice = 33' and '2 beans - 1 rice = 19', adding them makes the +1 rice and -1 rice cancel, leaving 4 bean bags equal to 33 + 19 = 52 oz.

(2B+R)+(2B-R)=33+19 \Rightarrow 4B = 52

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

4 bean bags weigh 52 oz, so one bean bag weighs 52 / 4 = 13 oz.

B = 52 \div 4 = 13\ \text{oz}

Splitting 52 oz equally among 4 bags is a basic within-100 division.

3Find 3 bean bags

#11 Work Backwards 3.OA.A.3

3 bags each weigh 13 oz, so together they weigh 3 x 13 = 39 oz.

3 \times 13 = 39\ \text{oz}

Multiplying the single-bag weight by 3 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 39 oz is 2 full pounds (32 oz) with 7 oz left over: 39 = 16 x 2 + 7.

39 \div 16 = 2\ \text{R}\ 7 \Rightarrow 2\ \text{lb}\ 7\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 7 oz (39 oz)

4 · Reviewdoes it hold up?

Check: with B=13, rice R = 33 - 26 = 7 oz, and 2 beans - rice = 26 - 7 = 19 oz, which matches. 3 bags = 39 oz, and 2 lb 7 oz = 32 + 7 = 39 oz, so the conversion is right.

Another way: Subtract instead: (2B+R) - (2B-R) = 2R = 14, so R = 7 oz; then 2B = 33 - 7 = 26, B = 13 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 52 by 4 and multiplying 13 by 3.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 39 oz into 2 lb 7 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 8 medium answer: 2 lb 4 oz (36 oz)

The weight of $2$ bags of beans plus $1$ bag of rice is $34\ \text{oz}$ . The weight of $2$ bags of beans minus $1$ bag of rice is $14\ \text{oz}$ . How much do $3$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

2 bags of beans together with one bag of rice weigh 34 oz, and 2 bags of beans with one bag of rice removed weigh 14 oz. Every bean bag weighs the same. We want the weight of 3 bean bags, written in pounds and ounces.

Givens

2 bags of beans + 1 bag of rice = 34 oz.
2 bags of beans - 1 bag of rice = 14 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 3 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 3 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '2 beans + 1 rice = 34' and '2 beans - 1 rice = 14', adding them makes the +1 rice and -1 rice cancel, leaving 4 bean bags equal to 34 + 14 = 48 oz.

(2B+R)+(2B-R)=34+14 \Rightarrow 4B = 48

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

4 bean bags weigh 48 oz, so one bean bag weighs 48 / 4 = 12 oz.

B = 48 \div 4 = 12\ \text{oz}

Splitting 48 oz equally among 4 bags is a basic within-100 division.

3Find 3 bean bags

#11 Work Backwards 3.OA.A.3

3 bags each weigh 12 oz, so together they weigh 3 x 12 = 36 oz.

3 \times 12 = 36\ \text{oz}

Multiplying the single-bag weight by 3 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 36 oz is 2 full pounds (32 oz) with 4 oz left over: 36 = 16 x 2 + 4.

36 \div 16 = 2\ \text{R}\ 4 \Rightarrow 2\ \text{lb}\ 4\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 4 oz (36 oz)

4 · Reviewdoes it hold up?

Check: with B=12, rice R = 34 - 24 = 10 oz, and 2 beans - rice = 24 - 10 = 14 oz, which matches. 3 bags = 36 oz, and 2 lb 4 oz = 32 + 4 = 36 oz, so the conversion is right.

Another way: Subtract instead: (2B+R) - (2B-R) = 2R = 20, so R = 10 oz; then 2B = 34 - 10 = 24, B = 12 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 48 by 4 and multiplying 12 by 3.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 36 oz into 2 lb 4 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 9 hard answer: 1 lb 14 oz (30 oz)

The weight of $2$ bags of beans plus $1$ bag of rice is $41\ \text{oz}$ . The weight of $2$ bags of beans minus $1$ bag of rice is $19\ \text{oz}$ . How much do $2$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

2 bags of beans together with one bag of rice weigh 41 oz, and 2 bags of beans with one bag of rice removed weigh 19 oz. Every bean bag weighs the same. We want the weight of 2 bean bags, written in pounds and ounces.

Givens

2 bags of beans + 1 bag of rice = 41 oz.
2 bags of beans - 1 bag of rice = 19 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 2 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 2 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '2 beans + 1 rice = 41' and '2 beans - 1 rice = 19', adding them makes the +1 rice and -1 rice cancel, leaving 4 bean bags equal to 41 + 19 = 60 oz.

(2B+R)+(2B-R)=41+19 \Rightarrow 4B = 60

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

4 bean bags weigh 60 oz, so one bean bag weighs 60 / 4 = 15 oz.

B = 60 \div 4 = 15\ \text{oz}

Splitting 60 oz equally among 4 bags is a basic within-100 division.

3Find 2 bean bags

#11 Work Backwards 3.OA.A.3

2 bags each weigh 15 oz, so together they weigh 2 x 15 = 30 oz.

2 \times 15 = 30\ \text{oz}

Multiplying the single-bag weight by 2 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 30 oz is 1 full pound (16 oz) with 14 oz left over: 30 = 16 x 1 + 14.

30 \div 16 = 1\ \text{R}\ 14 \Rightarrow 1\ \text{lb}\ 14\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 1 lb 14 oz (30 oz)

4 · Reviewdoes it hold up?

Check: with B=15, rice R = 41 - 30 = 11 oz, and 2 beans - rice = 30 - 11 = 19 oz, which matches. 2 bags = 30 oz, and 1 lb 14 oz = 16 + 14 = 30 oz, so the conversion is right.

Another way: Subtract instead: (2B+R) - (2B-R) = 2R = 22, so R = 11 oz; then 2B = 41 - 11 = 30, B = 15 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 60 by 4 and multiplying 15 by 2.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 30 oz into 1 lb 14 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 10 hard answer: 2 lb 8 oz (40 oz)

The weight of $3$ bags of beans plus $1$ bag of rice is $42\ \text{oz}$ . The weight of $3$ bags of beans minus $1$ bag of rice is $18\ \text{oz}$ . How much do $4$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

3 bags of beans together with one bag of rice weigh 42 oz, and 3 bags of beans with one bag of rice removed weigh 18 oz. Every bean bag weighs the same. We want the weight of 4 bean bags, written in pounds and ounces.

Givens

3 bags of beans + 1 bag of rice = 42 oz.
3 bags of beans - 1 bag of rice = 18 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 4 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 4 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '3 beans + 1 rice = 42' and '3 beans - 1 rice = 18', adding them makes the +1 rice and -1 rice cancel, leaving 6 bean bags equal to 42 + 18 = 60 oz.

(3B+R)+(3B-R)=42+18 \Rightarrow 6B = 60

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

6 bean bags weigh 60 oz, so one bean bag weighs 60 / 6 = 10 oz.

B = 60 \div 6 = 10\ \text{oz}

Splitting 60 oz equally among 6 bags is a basic within-100 division.

3Find 4 bean bags

#11 Work Backwards 3.OA.A.3

4 bags each weigh 10 oz, so together they weigh 4 x 10 = 40 oz.

4 \times 10 = 40\ \text{oz}

Multiplying the single-bag weight by 4 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 40 oz is 2 full pounds (32 oz) with 8 oz left over: 40 = 16 x 2 + 8.

40 \div 16 = 2\ \text{R}\ 8 \Rightarrow 2\ \text{lb}\ 8\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 8 oz (40 oz)

4 · Reviewdoes it hold up?

Check: with B=10, rice R = 42 - 30 = 12 oz, and 3 beans - rice = 30 - 12 = 18 oz, which matches. 4 bags = 40 oz, and 2 lb 8 oz = 32 + 8 = 40 oz, so the conversion is right.

Another way: Subtract instead: (3B+R) - (3B-R) = 2R = 24, so R = 12 oz; then 3B = 42 - 12 = 30, B = 10 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 60 by 6 and multiplying 10 by 4.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 40 oz into 2 lb 8 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 11 hard answer: 2 lb 10 oz (42 oz)

The weight of $2$ bags of beans plus $1$ bag of rice is $46\ \text{oz}$ . The weight of $2$ bags of beans minus $1$ bag of rice is $10\ \text{oz}$ . How much do $3$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

2 bags of beans together with one bag of rice weigh 46 oz, and 2 bags of beans with one bag of rice removed weigh 10 oz. Every bean bag weighs the same. We want the weight of 3 bean bags, written in pounds and ounces.

Givens

2 bags of beans + 1 bag of rice = 46 oz.
2 bags of beans - 1 bag of rice = 10 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 3 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 3 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '2 beans + 1 rice = 46' and '2 beans - 1 rice = 10', adding them makes the +1 rice and -1 rice cancel, leaving 4 bean bags equal to 46 + 10 = 56 oz.

(2B+R)+(2B-R)=46+10 \Rightarrow 4B = 56

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

4 bean bags weigh 56 oz, so one bean bag weighs 56 / 4 = 14 oz.

B = 56 \div 4 = 14\ \text{oz}

Splitting 56 oz equally among 4 bags is a basic within-100 division.

3Find 3 bean bags

#11 Work Backwards 3.OA.A.3

3 bags each weigh 14 oz, so together they weigh 3 x 14 = 42 oz.

3 \times 14 = 42\ \text{oz}

Multiplying the single-bag weight by 3 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 42 oz is 2 full pounds (32 oz) with 10 oz left over: 42 = 16 x 2 + 10.

42 \div 16 = 2\ \text{R}\ 10 \Rightarrow 2\ \text{lb}\ 10\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 2 lb 10 oz (42 oz)

4 · Reviewdoes it hold up?

Check: with B=14, rice R = 46 - 28 = 18 oz, and 2 beans - rice = 28 - 18 = 10 oz, which matches. 3 bags = 42 oz, and 2 lb 10 oz = 32 + 10 = 42 oz, so the conversion is right.

Another way: Subtract instead: (2B+R) - (2B-R) = 2R = 36, so R = 18 oz; then 2B = 46 - 18 = 28, B = 14 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 56 by 4 and multiplying 14 by 3.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 42 oz into 2 lb 10 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Variant 12 hard answer: 3 lb 2 oz (50 oz)

The weight of $3$ bags of beans plus $1$ bag of rice is $46\ \text{oz}$ . The weight of $3$ bags of beans minus $1$ bag of rice is $14\ \text{oz}$ . How much do $5$ bags of beans weigh, in pounds and ounces? (Every bag of beans weighs the same.)

Show solution

1 · Understandwhat's really being asked

3 bags of beans together with one bag of rice weigh 46 oz, and 3 bags of beans with one bag of rice removed weigh 14 oz. Every bean bag weighs the same. We want the weight of 5 bean bags, written in pounds and ounces.

Givens

3 bags of beans + 1 bag of rice = 46 oz.
3 bags of beans - 1 bag of rice = 14 oz.
All bean bags weigh the same.
1 lb = 16 oz.

Unknowns

The weight of 5 bags of beans, in pounds and ounces.

Constraints

Each bean bag has the same weight.
The rice bag and bean bag have fixed positive weights.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #11 Work Backwards

Adding the sum-relation and the difference-relation cancels the rice bag, leaving only beans - a re-grouping of the given facts. Once one bean bag is known we scale up to 5 and convert to pounds and ounces.

3 · Execute4 carry out the plan

1Add the two statements to remove the rice

#15 Organize Information in More Ways 3.OA.D.8

In '3 beans + 1 rice = 46' and '3 beans - 1 rice = 14', adding them makes the +1 rice and -1 rice cancel, leaving 6 bean bags equal to 46 + 14 = 60 oz.

(3B+R)+(3B-R)=46+14 \Rightarrow 6B = 60

Lining up the two facts and combining them so the rice disappears is the same idea as a two-step word problem.

2Find one bean bag

#15 Organize Information in More Ways 3.OA.A.3

6 bean bags weigh 60 oz, so one bean bag weighs 60 / 6 = 10 oz.

B = 60 \div 6 = 10\ \text{oz}

Splitting 60 oz equally among 6 bags is a basic within-100 division.

3Find 5 bean bags

#11 Work Backwards 3.OA.A.3

5 bags each weigh 10 oz, so together they weigh 5 x 10 = 50 oz.

5 \times 10 = 50\ \text{oz}

Multiplying the single-bag weight by 5 is repeated addition of equal groups.

4Convert ounces to pounds and ounces

#11 Work Backwards 3.MD.A.2

Since 16 oz = 1 lb, 50 oz is 3 full pounds (48 oz) with 2 oz left over: 50 = 16 x 3 + 2.

50 \div 16 = 3\ \text{R}\ 2 \Rightarrow 3\ \text{lb}\ 2\ \text{oz}

Trading every 16 oz for 1 lb is measurement-unit sense built in Grade 3.

Answer: 3 lb 2 oz (50 oz)

4 · Reviewdoes it hold up?

Check: with B=10, rice R = 46 - 30 = 16 oz, and 3 beans - rice = 30 - 16 = 14 oz, which matches. 5 bags = 50 oz, and 3 lb 2 oz = 48 + 2 = 50 oz, so the conversion is right.

Another way: Subtract instead: (3B+R) - (3B-R) = 2R = 32, so R = 16 oz; then 3B = 46 - 16 = 30, B = 10 oz - same answer by Work Backwards.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the sum and difference statements to isolate the bean bags.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 60 by 6 and multiplying 10 by 5.
3.MD.A.2 Measure and estimate liquid volumes and masses of objects — Converting 50 oz into 3 lb 2 oz.

💡Takeaway. Add the two clues so the rice cancels - then it's just Grade 3 dividing and an ounce-to-pound trade!

Combine sum and difference of two weights · 12 practice problems

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