← When you can only add, aim for the multiple above · Divisibility and Remainder Reasoning

When you can only add, aim for the multiple above · 12 practice problems

6.NS.B.3

Generated variants — 12

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

Variant 1 easy answer: 0.73 kg

There are 38.15 kg38.15\ \text{kg} of walnuts to be packed into sacks holding 1.62 kg1.62\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

38.15 kg of walnuts go into 1.62 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 38.15 kg of walnuts.
  • Each sack holds 1.62 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
23 sacks take 37.26 kg, and the rest is not enough for another.
1.62×23=37.261.62 \times 23 = 37.26
23 full sacks, with 0.89 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
1.62×24=38.881.62 \times 24 = 38.88
24 sacks would need 38.88 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
38.8838.15=0.7338.88 - 38.15 = 0.73
0.73 kg more.
Answer: 0.73 kg
4 · Reviewdoes it hold up?

38.15 plus 0.73 is 38.88 kg, which is exactly 24 sacks of 1.62 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 0.89 kg removed, which is more than 0.73 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 2 easy answer: 0.18 kg

There are 41.26 kg41.26\ \text{kg} of walnuts to be packed into sacks holding 1.48 kg1.48\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

41.26 kg of walnuts go into 1.48 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 41.26 kg of walnuts.
  • Each sack holds 1.48 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
27 sacks take 39.96 kg, and the rest is not enough for another.
1.48×27=39.961.48 \times 27 = 39.96
27 full sacks, with 1.3 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
1.48×28=41.441.48 \times 28 = 41.44
28 sacks would need 41.44 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
41.4441.26=0.1841.44 - 41.26 = 0.18
0.18 kg more.
Answer: 0.18 kg
4 · Reviewdoes it hold up?

41.26 plus 0.18 is 41.44 kg, which is exactly 28 sacks of 1.48 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.3 kg removed, which is more than 0.18 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 3 easy answer: 1.1 kg

There are 49.3 kg49.3\ \text{kg} of walnuts to be packed into sacks holding 1.26 kg1.26\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

49.3 kg of walnuts go into 1.26 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 49.3 kg of walnuts.
  • Each sack holds 1.26 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
39 sacks take 49.14 kg, and the rest is not enough for another.
1.26×39=49.141.26 \times 39 = 49.14
39 full sacks, with 0.16 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
1.26×40=50.41.26 \times 40 = 50.4
40 sacks would need 50.4 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
50.449.3=1.150.4 - 49.3 = 1.1
1.1 kg more.
Answer: 1.1 kg
4 · Reviewdoes it hold up?

49.3 plus 1.1 is 50.4 kg, which is exactly 40 sacks of 1.26 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 0.16 kg removed, which is more than 1.1 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 4 easy answer: 0.1 kg

There are 52.4 kg52.4\ \text{kg} of walnuts to be packed into sacks holding 1.75 kg1.75\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

52.4 kg of walnuts go into 1.75 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 52.4 kg of walnuts.
  • Each sack holds 1.75 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
29 sacks take 50.75 kg, and the rest is not enough for another.
1.75×29=50.751.75 \times 29 = 50.75
29 full sacks, with 1.65 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
1.75×30=52.51.75 \times 30 = 52.5
30 sacks would need 52.5 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
52.552.4=0.152.5 - 52.4 = 0.1
0.1 kg more.
Answer: 0.1 kg
4 · Reviewdoes it hold up?

52.4 plus 0.1 is 52.5 kg, which is exactly 30 sacks of 1.75 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.65 kg removed, which is more than 0.1 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 5 medium answer: 1.17 kg

There are 56.73 kg56.73\ \text{kg} of walnuts to be packed into sacks holding 1.93 kg1.93\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

56.73 kg of walnuts go into 1.93 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 56.73 kg of walnuts.
  • Each sack holds 1.93 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
29 sacks take 55.97 kg, and the rest is not enough for another.
1.93×29=55.971.93 \times 29 = 55.97
29 full sacks, with 0.76 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
1.93×30=57.91.93 \times 30 = 57.9
30 sacks would need 57.9 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
57.956.73=1.1757.9 - 56.73 = 1.17
1.17 kg more.
Answer: 1.17 kg
4 · Reviewdoes it hold up?

56.73 plus 1.17 is 57.9 kg, which is exactly 30 sacks of 1.93 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 0.76 kg removed, which is more than 1.17 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 6 medium answer: 1.02 kg

There are 63.18 kg63.18\ \text{kg} of walnuts to be packed into sacks holding 2.14 kg2.14\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

63.18 kg of walnuts go into 2.14 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 63.18 kg of walnuts.
  • Each sack holds 2.14 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
29 sacks take 62.06 kg, and the rest is not enough for another.
2.14×29=62.062.14 \times 29 = 62.06
29 full sacks, with 1.12 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
2.14×30=64.22.14 \times 30 = 64.2
30 sacks would need 64.2 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
64.263.18=1.0264.2 - 63.18 = 1.02
1.02 kg more.
Answer: 1.02 kg
4 · Reviewdoes it hold up?

63.18 plus 1.02 is 64.2 kg, which is exactly 30 sacks of 2.14 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.12 kg removed, which is more than 1.02 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 7 medium answer: 0.25 kg

There are 69.05 kg69.05\ \text{kg} of walnuts to be packed into sacks holding 3.15 kg3.15\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

69.05 kg of walnuts go into 3.15 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 69.05 kg of walnuts.
  • Each sack holds 3.15 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
21 sacks take 66.15 kg, and the rest is not enough for another.
3.15×21=66.153.15 \times 21 = 66.15
21 full sacks, with 2.9 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
3.15×22=69.33.15 \times 22 = 69.3
22 sacks would need 69.3 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
69.369.05=0.2569.3 - 69.05 = 0.25
0.25 kg more.
Answer: 0.25 kg
4 · Reviewdoes it hold up?

69.05 plus 0.25 is 69.3 kg, which is exactly 22 sacks of 3.15 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 2.9 kg removed, which is more than 0.25 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 8 medium answer: 1.83 kg

There are 71.25 kg71.25\ \text{kg} of walnuts to be packed into sacks holding 3.48 kg3.48\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

71.25 kg of walnuts go into 3.48 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 71.25 kg of walnuts.
  • Each sack holds 3.48 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
20 sacks take 69.6 kg, and the rest is not enough for another.
3.48×20=69.63.48 \times 20 = 69.6
20 full sacks, with 1.65 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
3.48×21=73.083.48 \times 21 = 73.08
21 sacks would need 73.08 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
73.0871.25=1.8373.08 - 71.25 = 1.83
1.83 kg more.
Answer: 1.83 kg
4 · Reviewdoes it hold up?

71.25 plus 1.83 is 73.08 kg, which is exactly 21 sacks of 3.48 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.65 kg removed, which is more than 1.83 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 9 hard answer: 0.72 kg

There are 74.8 kg74.8\ \text{kg} of walnuts to be packed into sacks holding 2.36 kg2.36\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

74.8 kg of walnuts go into 2.36 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 74.8 kg of walnuts.
  • Each sack holds 2.36 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
31 sacks take 73.16 kg, and the rest is not enough for another.
2.36×31=73.162.36 \times 31 = 73.16
31 full sacks, with 1.64 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
2.36×32=75.522.36 \times 32 = 75.52
32 sacks would need 75.52 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
75.5274.8=0.7275.52 - 74.8 = 0.72
0.72 kg more.
Answer: 0.72 kg
4 · Reviewdoes it hold up?

74.8 plus 0.72 is 75.52 kg, which is exactly 32 sacks of 2.36 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.64 kg removed, which is more than 0.72 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 10 hard answer: 2.5 kg

There are 82.5 kg82.5\ \text{kg} of walnuts to be packed into sacks holding 4.25 kg4.25\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

82.5 kg of walnuts go into 4.25 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 82.5 kg of walnuts.
  • Each sack holds 4.25 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
19 sacks take 80.75 kg, and the rest is not enough for another.
4.25×19=80.754.25 \times 19 = 80.75
19 full sacks, with 1.75 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
4.25×20=854.25 \times 20 = 85
20 sacks would need 85 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
8582.5=2.585 - 82.5 = 2.5
2.5 kg more.
Answer: 2.5 kg
4 · Reviewdoes it hold up?

82.5 plus 2.5 is 85 kg, which is exactly 20 sacks of 4.25 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.75 kg removed, which is more than 2.5 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 11 hard answer: 0.1 kg

There are 84.06 kg84.06\ \text{kg} of walnuts to be packed into sacks holding 2.63 kg2.63\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

84.06 kg of walnuts go into 2.63 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 84.06 kg of walnuts.
  • Each sack holds 2.63 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
31 sacks take 81.53 kg, and the rest is not enough for another.
2.63×31=81.532.63 \times 31 = 81.53
31 full sacks, with 2.53 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
2.63×32=84.162.63 \times 32 = 84.16
32 sacks would need 84.16 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
84.1684.06=0.184.16 - 84.06 = 0.1
0.1 kg more.
Answer: 0.1 kg
4 · Reviewdoes it hold up?

84.06 plus 0.1 is 84.16 kg, which is exactly 32 sacks of 2.63 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 2.53 kg removed, which is more than 0.1 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.
Variant 12 hard answer: 1.16 kg

There are 96.42 kg96.42\ \text{kg} of walnuts to be packed into sacks holding 2.87 kg2.87\ \text{kg} each. To fill every sack completely with none left over, at least how many more kilograms of walnuts are needed?

Show solution
1 · Understandwhat's really being asked

96.42 kg of walnuts go into 2.87 kg sacks. We want the smallest extra amount that leaves no sack part-full and nothing loose.

Givens
  • There are 96.42 kg of walnuts.
  • Each sack holds 2.87 kg.
  • Walnuts may be added, not taken away.
Unknowns
  • The least extra weight of walnuts needed.
Constraints
  • Every sack must end up completely full.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #8 Analyze the Units#7 Identify Subproblems

The division will not come out, and the direction we may move decides what to do about it. Since walnuts can only be added, the target is the next multiple of a sack, not the one below.

3 · Execute3 carry out the plan

1See how many sacks fill now

#7 Identify Subproblems 6.NS.B.3
33 sacks take 94.71 kg, and the rest is not enough for another.
2.87×33=94.712.87 \times 33 = 94.71
33 full sacks, with 1.71 kg loose.

2Aim for one more sack

#11 Work Backwards 6.NS.B.3
Nothing may be taken away, so the only way to leave no loose walnuts is to finish the next sack.
2.87×34=97.582.87 \times 34 = 97.58
34 sacks would need 97.58 kg.

3Find the gap

#8 Analyze the Units 6.NS.B.3
Subtract what is there from what is needed.
97.5896.42=1.1697.58 - 96.42 = 1.16
1.16 kg more.
Answer: 1.16 kg
4 · Reviewdoes it hold up?

96.42 plus 1.16 is 97.58 kg, which is exactly 34 sacks of 2.87 kg -- nothing over and nothing short.

Another way: Taking walnuts away instead would need 1.71 kg removed, which is more than 1.16 here -- and is not allowed anyway, since the question only offers to add.

Standardsmin grade 6
  • 6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing, multiplying and subtracting the decimal weights.
💡Takeaway. A division that does not come out has two nearest answers. Which one you want depends on which way you are allowed to move.