← Cutting the same from both leaves the difference untouched · Find Two Unknowns from Sum and Difference

Cutting the same from both leaves the difference untouched · 12 practice problems

7.EE.B.46.EE.B.6

Generated variants — 12

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

Variant 1 easy answer: 21 cm

String A is $24\ \text{cm}$ long and string B is $26\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $3 : 5$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 24 cm and 26 cm, each lose the same length, and what is left is in the ratio 3 to 5. We want the length cut off.

Givens

String A is 24 cm and string B is 26 cm.
The same length is cut from each.
What remains is in the ratio 3 to 5.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

26 - 24 = 2

Still 2 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 3 to 5, the gap is 2 parts.

5 - 3 = 2

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

2 parts come to 2 cm.

2 \div 2 = 1

Each part is 1 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 3 parts, so the rest of its 24 cm was cut off.

24 - 1 \times 3 = 21

21 cm was cut from each.

Answer: 21 cm

4 · Reviewdoes it hold up?

After cutting 21 cm the strings are 3 cm and 5 cm -- still 2 cm apart, and in the ratio 3 to 5.

Another way: Following string B instead gives 26 minus 21, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 2 easy answer: 15 cm

String A is $27\ \text{cm}$ long and string B is $30\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $4 : 5$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 27 cm and 30 cm, each lose the same length, and what is left is in the ratio 4 to 5. We want the length cut off.

Givens

String A is 27 cm and string B is 30 cm.
The same length is cut from each.
What remains is in the ratio 4 to 5.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

30 - 27 = 3

Still 3 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 4 to 5, the gap is 1 part.

5 - 4 = 1

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

1 part come to 3 cm.

3 \div 1 = 3

Each part is 3 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 4 parts, so the rest of its 27 cm was cut off.

27 - 3 \times 4 = 15

15 cm was cut from each.

Answer: 15 cm

4 · Reviewdoes it hold up?

After cutting 15 cm the strings are 12 cm and 15 cm -- still 3 cm apart, and in the ratio 4 to 5.

Another way: Following string B instead gives 30 minus 15, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 3 easy answer: 16 cm

String A is $23\ \text{cm}$ long and string B is $37\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $1 : 3$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 23 cm and 37 cm, each lose the same length, and what is left is in the ratio 1 to 3. We want the length cut off.

Givens

String A is 23 cm and string B is 37 cm.
The same length is cut from each.
What remains is in the ratio 1 to 3.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

37 - 23 = 14

Still 14 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 1 to 3, the gap is 2 parts.

3 - 1 = 2

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

2 parts come to 14 cm.

14 \div 2 = 7

Each part is 7 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 1 part, so the rest of its 23 cm was cut off.

23 - 7 \times 1 = 16

16 cm was cut from each.

Answer: 16 cm

4 · Reviewdoes it hold up?

After cutting 16 cm the strings are 7 cm and 21 cm -- still 14 cm apart, and in the ratio 1 to 3.

Another way: Following string B instead gives 37 minus 16, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 4 easy answer: 22 cm

String A is $29\ \text{cm}$ long and string B is $43\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $1 : 3$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 29 cm and 43 cm, each lose the same length, and what is left is in the ratio 1 to 3. We want the length cut off.

Givens

String A is 29 cm and string B is 43 cm.
The same length is cut from each.
What remains is in the ratio 1 to 3.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

43 - 29 = 14

Still 14 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 1 to 3, the gap is 2 parts.

3 - 1 = 2

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

2 parts come to 14 cm.

14 \div 2 = 7

Each part is 7 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 1 part, so the rest of its 29 cm was cut off.

29 - 7 \times 1 = 22

22 cm was cut from each.

Answer: 22 cm

4 · Reviewdoes it hold up?

After cutting 22 cm the strings are 7 cm and 21 cm -- still 14 cm apart, and in the ratio 1 to 3.

Another way: Following string B instead gives 43 minus 22, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 5 medium answer: 2 cm

String A is $35\ \text{cm}$ long and string B is $46\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $3 : 4$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 35 cm and 46 cm, each lose the same length, and what is left is in the ratio 3 to 4. We want the length cut off.

Givens

String A is 35 cm and string B is 46 cm.
The same length is cut from each.
What remains is in the ratio 3 to 4.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

46 - 35 = 11

Still 11 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 3 to 4, the gap is 1 part.

4 - 3 = 1

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

1 part come to 11 cm.

11 \div 1 = 11

Each part is 11 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 3 parts, so the rest of its 35 cm was cut off.

35 - 11 \times 3 = 2

2 cm was cut from each.

Answer: 2 cm

4 · Reviewdoes it hold up?

After cutting 2 cm the strings are 33 cm and 44 cm -- still 11 cm apart, and in the ratio 3 to 4.

Another way: Following string B instead gives 46 minus 2, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 6 medium answer: 20 cm

String A is $30\ \text{cm}$ long and string B is $50\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $1 : 3$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 30 cm and 50 cm, each lose the same length, and what is left is in the ratio 1 to 3. We want the length cut off.

Givens

String A is 30 cm and string B is 50 cm.
The same length is cut from each.
What remains is in the ratio 1 to 3.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

50 - 30 = 20

Still 20 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 1 to 3, the gap is 2 parts.

3 - 1 = 2

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

2 parts come to 20 cm.

20 \div 2 = 10

Each part is 10 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 1 part, so the rest of its 30 cm was cut off.

30 - 10 \times 1 = 20

20 cm was cut from each.

Answer: 20 cm

4 · Reviewdoes it hold up?

After cutting 20 cm the strings are 10 cm and 30 cm -- still 20 cm apart, and in the ratio 1 to 3.

Another way: Following string B instead gives 50 minus 20, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 7 medium answer: 40 cm

String A is $49\ \text{cm}$ long and string B is $52\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $3 : 4$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 49 cm and 52 cm, each lose the same length, and what is left is in the ratio 3 to 4. We want the length cut off.

Givens

String A is 49 cm and string B is 52 cm.
The same length is cut from each.
What remains is in the ratio 3 to 4.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

52 - 49 = 3

Still 3 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 3 to 4, the gap is 1 part.

4 - 3 = 1

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

1 part come to 3 cm.

3 \div 1 = 3

Each part is 3 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 3 parts, so the rest of its 49 cm was cut off.

49 - 3 \times 3 = 40

40 cm was cut from each.

Answer: 40 cm

4 · Reviewdoes it hold up?

After cutting 40 cm the strings are 9 cm and 12 cm -- still 3 cm apart, and in the ratio 3 to 4.

Another way: Following string B instead gives 52 minus 40, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 8 medium answer: 16 cm

String A is $44\ \text{cm}$ long and string B is $58\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $2 : 3$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 44 cm and 58 cm, each lose the same length, and what is left is in the ratio 2 to 3. We want the length cut off.

Givens

String A is 44 cm and string B is 58 cm.
The same length is cut from each.
What remains is in the ratio 2 to 3.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

58 - 44 = 14

Still 14 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 2 to 3, the gap is 1 part.

3 - 2 = 1

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

1 part come to 14 cm.

14 \div 1 = 14

Each part is 14 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 2 parts, so the rest of its 44 cm was cut off.

44 - 14 \times 2 = 16

16 cm was cut from each.

Answer: 16 cm

4 · Reviewdoes it hold up?

After cutting 16 cm the strings are 28 cm and 42 cm -- still 14 cm apart, and in the ratio 2 to 3.

Another way: Following string B instead gives 58 minus 16, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 9 hard answer: 1 cm

String A is $17\ \text{cm}$ long and string B is $65\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $1 : 4$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 17 cm and 65 cm, each lose the same length, and what is left is in the ratio 1 to 4. We want the length cut off.

Givens

String A is 17 cm and string B is 65 cm.
The same length is cut from each.
What remains is in the ratio 1 to 4.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

65 - 17 = 48

Still 48 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 1 to 4, the gap is 3 parts.

4 - 1 = 3

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

3 parts come to 48 cm.

48 \div 3 = 16

Each part is 16 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 1 part, so the rest of its 17 cm was cut off.

17 - 16 \times 1 = 1

1 cm was cut from each.

Answer: 1 cm

4 · Reviewdoes it hold up?

After cutting 1 cm the strings are 16 cm and 64 cm -- still 48 cm apart, and in the ratio 1 to 4.

Another way: Following string B instead gives 65 minus 1, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 10 hard answer: 32 cm

String A is $46\ \text{cm}$ long and string B is $67\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $2 : 5$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 46 cm and 67 cm, each lose the same length, and what is left is in the ratio 2 to 5. We want the length cut off.

Givens

String A is 46 cm and string B is 67 cm.
The same length is cut from each.
What remains is in the ratio 2 to 5.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

67 - 46 = 21

Still 21 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 2 to 5, the gap is 3 parts.

5 - 2 = 3

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

3 parts come to 21 cm.

21 \div 3 = 7

Each part is 7 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 2 parts, so the rest of its 46 cm was cut off.

46 - 7 \times 2 = 32

32 cm was cut from each.

Answer: 32 cm

4 · Reviewdoes it hold up?

After cutting 32 cm the strings are 14 cm and 35 cm -- still 21 cm apart, and in the ratio 2 to 5.

Another way: Following string B instead gives 67 minus 32, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 11 hard answer: 61 cm

String A is $69\ \text{cm}$ long and string B is $75\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $4 : 7$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 69 cm and 75 cm, each lose the same length, and what is left is in the ratio 4 to 7. We want the length cut off.

Givens

String A is 69 cm and string B is 75 cm.
The same length is cut from each.
What remains is in the ratio 4 to 7.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

75 - 69 = 6

Still 6 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 4 to 7, the gap is 3 parts.

7 - 4 = 3

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

3 parts come to 6 cm.

6 \div 3 = 2

Each part is 2 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 4 parts, so the rest of its 69 cm was cut off.

69 - 2 \times 4 = 61

61 cm was cut from each.

Answer: 61 cm

4 · Reviewdoes it hold up?

After cutting 61 cm the strings are 8 cm and 14 cm -- still 6 cm apart, and in the ratio 4 to 7.

Another way: Following string B instead gives 75 minus 61, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.

Variant 12 hard answer: 53 cm

String A is $61\ \text{cm}$ long and string B is $93\ \text{cm}$ long. The same length is cut from each, after which the lengths remaining are in the ratio $1 : 5$ . How many centimeters were cut from string A?

Show solution

1 · Understandwhat's really being asked

Two strings, 61 cm and 93 cm, each lose the same length, and what is left is in the ratio 1 to 5. We want the length cut off.

Givens

String A is 61 cm and string B is 93 cm.
The same length is cut from each.
What remains is in the ratio 1 to 5.

Unknowns

The length cut from each string.

Constraints

Both strings lose the same amount, so their difference is unchanged.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#10 Create a Physical Representation

Both lengths change, so neither can be followed directly. Their difference does not change, and the new ratio says how many parts that difference is -- which fixes the size of a part.

3 · Execute4 carry out the plan

1Notice what the cut leaves alone

#16 Count the Complement 6.EE.B.6

Taking the same amount off both keeps the gap between them the same.

93 - 61 = 32

Still 32 cm apart.

2Count the parts in that gap

#10 Create a Physical Representation 7.EE.B.4

With the remainders in the ratio 1 to 5, the gap is 4 parts.

5 - 1 = 4

The gap is measured in parts.

3Find one part

#13 Convert to Algebra 7.EE.B.4

4 parts come to 32 cm.

32 \div 4 = 8

Each part is 8 cm.

4Work back to the cut

#13 Convert to Algebra 7.EE.B.4

String A keeps 1 part, so the rest of its 61 cm was cut off.

61 - 8 \times 1 = 53

53 cm was cut from each.

Answer: 53 cm

4 · Reviewdoes it hold up?

After cutting 53 cm the strings are 8 cm and 40 cm -- still 32 cm apart, and in the ratio 1 to 5.

Another way: Following string B instead gives 93 minus 53, the same cut reached from the longer string.

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving for the part size and then the cut.
6.EE.B.6 Use variables to represent numbers and write expressions — Naming the cut as one unknown taken from both strings.

💡Takeaway. Take the same amount from two lengths and the gap between them never moves. Work with the gap.