Combine sum and difference of two weights

3.MD.A.23.OA.D.8 · adapt · grade 3

Archetype: Find Two Unknowns from Sum and Difference · step in a 8-type progression

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.)

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Understand

Three bags of beans together with one bag of rice weigh 46 oz, and three 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.

Plan

#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.

Execute

#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.

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

Six 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.

#11 Work Backwards 3.OA.A.3

Five 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.

#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)

Review

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.

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.

Standards · min 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.

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