Practice · Find two lengths from sum and difference · Sensim Math

Variant 1 easy

There are two bars, A and B. The sum of the lengths of A and B is $50\ \text{cm}$ , and their difference is $16\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 50 cm. A is 16 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 50 cm.
The difference between the bars' lengths is 16 cm (A is longer than B).
The figure shows bar A as the full 50 cm length with a 16 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 16 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 16 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 50 cm by 16 cm.

50 - 16 = 34

Taking the extra piece away leaves two equal bars, so the leftover 34 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 34 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

34 \div 2 = 17

Sharing 34 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 16 cm extra piece.

17 + 16 = 33

A is just B with its extra piece restored.

Answer: The longer bar A is 33 cm and the shorter bar B is 17 cm.

4 · Reviewdoes it hold up?

Check both facts: 33 + 17 = 50 cm (the sum) and 33 - 17 = 16 cm (the difference). Both match, and A (33) is longer than B (17), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 33, B = 17; their sum is 50 and difference is 16, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 2 easy

There are two bars, A and B. The sum of the lengths of A and B is $60\ \text{cm}$ , and their difference is $20\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 60 cm. A is 20 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 60 cm.
The difference between the bars' lengths is 20 cm (A is longer than B).
The figure shows bar A as the full 60 cm length with a 20 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 20 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 20 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 60 cm by 20 cm.

60 - 20 = 40

Taking the extra piece away leaves two equal bars, so the leftover 40 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 40 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

40 \div 2 = 20

Sharing 40 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 20 cm extra piece.

20 + 20 = 40

A is just B with its extra piece restored.

Answer: The longer bar A is 40 cm and the shorter bar B is 20 cm.

4 · Reviewdoes it hold up?

Check both facts: 40 + 20 = 60 cm (the sum) and 40 - 20 = 20 cm (the difference). Both match, and A (40) is longer than B (20), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 40, B = 20; their sum is 60 and difference is 20, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 3 medium

There are two bars, A and B. The sum of the lengths of A and B is $70\ \text{cm}$ , and their difference is $10\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 70 cm. A is 10 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 70 cm.
The difference between the bars' lengths is 10 cm (A is longer than B).
The figure shows bar A as the full 70 cm length with a 10 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 10 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 10 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 70 cm by 10 cm.

70 - 10 = 60

Taking the extra piece away leaves two equal bars, so the leftover 60 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 60 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

60 \div 2 = 30

Sharing 60 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 10 cm extra piece.

30 + 10 = 40

A is just B with its extra piece restored.

Answer: The longer bar A is 40 cm and the shorter bar B is 30 cm.

4 · Reviewdoes it hold up?

Check both facts: 40 + 30 = 70 cm (the sum) and 40 - 30 = 10 cm (the difference). Both match, and A (40) is longer than B (30), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 40, B = 30; their sum is 70 and difference is 10, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 4 medium

There are two bars, A and B. The sum of the lengths of A and B is $80\ \text{cm}$ , and their difference is $20\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 80 cm. A is 20 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 80 cm.
The difference between the bars' lengths is 20 cm (A is longer than B).
The figure shows bar A as the full 80 cm length with a 20 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 20 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 20 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 80 cm by 20 cm.

80 - 20 = 60

Taking the extra piece away leaves two equal bars, so the leftover 60 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 60 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

60 \div 2 = 30

Sharing 60 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 20 cm extra piece.

30 + 20 = 50

A is just B with its extra piece restored.

Answer: The longer bar A is 50 cm and the shorter bar B is 30 cm.

4 · Reviewdoes it hold up?

Check both facts: 50 + 30 = 80 cm (the sum) and 50 - 30 = 20 cm (the difference). Both match, and A (50) is longer than B (30), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 50, B = 30; their sum is 80 and difference is 20, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 5 medium

There are two bars, A and B. The sum of the lengths of A and B is $84\ \text{cm}$ , and their difference is $12\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 84 cm. A is 12 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 84 cm.
The difference between the bars' lengths is 12 cm (A is longer than B).
The figure shows bar A as the full 84 cm length with a 12 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 12 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 12 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 84 cm by 12 cm.

84 - 12 = 72

Taking the extra piece away leaves two equal bars, so the leftover 72 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 72 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

72 \div 2 = 36

Sharing 72 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 12 cm extra piece.

36 + 12 = 48

A is just B with its extra piece restored.

Answer: The longer bar A is 48 cm and the shorter bar B is 36 cm.

4 · Reviewdoes it hold up?

Check both facts: 48 + 36 = 84 cm (the sum) and 48 - 36 = 12 cm (the difference). Both match, and A (48) is longer than B (36), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 48, B = 36; their sum is 84 and difference is 12, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 6 medium

There are two bars, A and B. The sum of the lengths of A and B is $90\ \text{cm}$ , and their difference is $10\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 90 cm. A is 10 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 90 cm.
The difference between the bars' lengths is 10 cm (A is longer than B).
The figure shows bar A as the full 90 cm length with a 10 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 10 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 10 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 90 cm by 10 cm.

90 - 10 = 80

Taking the extra piece away leaves two equal bars, so the leftover 80 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 80 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

80 \div 2 = 40

Sharing 80 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 10 cm extra piece.

40 + 10 = 50

A is just B with its extra piece restored.

Answer: The longer bar A is 50 cm and the shorter bar B is 40 cm.

4 · Reviewdoes it hold up?

Check both facts: 50 + 40 = 90 cm (the sum) and 50 - 40 = 10 cm (the difference). Both match, and A (50) is longer than B (40), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 50, B = 40; their sum is 90 and difference is 10, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 7 hard

There are two bars, A and B. The sum of the lengths of A and B is $100\ \text{cm}$ , and their difference is $40\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 100 cm. A is 40 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 100 cm.
The difference between the bars' lengths is 40 cm (A is longer than B).
The figure shows bar A as the full 100 cm length with a 40 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 40 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 40 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 100 cm by 40 cm.

100 - 40 = 60

Taking the extra piece away leaves two equal bars, so the leftover 60 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 60 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

60 \div 2 = 30

Sharing 60 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 40 cm extra piece.

30 + 40 = 70

A is just B with its extra piece restored.

Answer: The longer bar A is 70 cm and the shorter bar B is 30 cm.

4 · Reviewdoes it hold up?

Check both facts: 70 + 30 = 100 cm (the sum) and 70 - 30 = 40 cm (the difference). Both match, and A (70) is longer than B (30), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 70, B = 30; their sum is 100 and difference is 40, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Variant 8 hard

There are two bars, A and B. The sum of the lengths of A and B is $120\ \text{cm}$ , and their difference is $30\ \text{cm}$ . (A is longer than B.)

Find the length, in centimeters, of the longer bar A and the shorter bar B.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two bars, A and B, together measure 120 cm. A is 30 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 120 cm.
The difference between the bars' lengths is 30 cm (A is longer than B).
The figure shows bar A as the full 120 cm length with a 30 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 30 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

3 · Execute3 carry out the plan

1Remove the difference from the sum

#1 Draw a Diagram 2.MD.B.5

If we cut off the 30 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 120 cm by 30 cm.

120 - 30 = 90

Taking the extra piece away leaves two equal bars, so the leftover 90 cm is just B counted twice.

2Split into two equal bars to get B

#7 Identify Subproblems 2.MD.B.5

The 90 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

90 \div 2 = 45

Sharing 90 equally into two groups is a second-grade halving fact.

3Add the difference back to get A

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 30 cm extra piece.

45 + 30 = 75

A is just B with its extra piece restored.

Answer: The longer bar A is 75 cm and the shorter bar B is 45 cm.

4 · Reviewdoes it hold up?

Check both facts: 75 + 45 = 120 cm (the sum) and 75 - 45 = 30 cm (the difference). Both match, and A (75) is longer than B (45), so the answer is sensible.

Another way: Guess and check (tool 6): try A = 75, B = 45; their sum is 120 and difference is 30, confirming the answer on the first reasonable guess.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡Takeaway. Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!

Find two lengths from sum and difference · 8 practice problems

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