Practice · Count the gaps, then add and subtract · Sensim Math

Variant 1 easy

A concert was planned to mix $7$ -minute pieces and $9$ -minute pieces so that the whole performance would take $2$ hours $3$ minutes. At the orchestra's request, the numbers of $7$ -minute pieces and $9$ -minute pieces were swapped, and the performance took $2$ hours $13$ minutes in all. There is a $2$ -minute break between each piece and the next. How many $7$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (7 and 9 minutes) are played with a 2-minute break between neighbors. The planned program lasts 123 minutes; swapping how many of each kind makes it last 133 minutes. Find the planned number of 7-minute pieces.

Givens

Pieces last $7$ minutes or $9$ minutes.
The planned total is $2$ hours $3$ minutes.
With the two counts swapped the total is $2$ hours $13$ minutes.
There is a $2$ -minute break between each piece and the next.

Unknowns

The planned number of $7$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

7

minutes and

y

pieces of

9

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{break minutes} = 2(x + y - 1)

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

2

hours

3

minutes is

123

minutes and

2

hours

13

minutes is

133

minutes. In the swapped program there are

y

seven-minute pieces and

x

nine-minute pieces.

7x + 9y + 2(x + y - 1) = 123,\quad 9x + 11y = 125;\qquad 7y + 9x + 2(x + y - 1) = 133,\quad 11x + 9y = 135

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

20x + 20y = 260,\quad x + y = 13

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

2x - 2y = 10,\quad x - y = 5

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{13 + 5}{2} = 9,\quad y = \frac{13 - 5}{2} = 4

Sum and difference together pin down two numbers.

Answer:

9

pieces

4 · Reviewdoes it hold up?

Planned: $7 \times 9 + 9 \times 4 + 24 = 63 + 36 + 24 = 123$ minutes. Swapped: $7 \times 4 + 9 \times 9 + 24 = 28 + 81 + 24 = 133$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $7$ to $9$ minutes adds $2$ minutes and each that switches from $9$ to $7$ takes away $2$ , so the swap changes the total by $2(x - y) = 133 - 123 = 10$ , giving $x - y = 5$ . With $x + y = 13$ this again gives $x = 9$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 2 easy

A concert was planned to mix $7$ -minute pieces and $10$ -minute pieces so that the whole performance would take $1$ hour $17$ minutes. At the orchestra's request, the numbers of $7$ -minute pieces and $10$ -minute pieces were swapped, and the performance took $1$ hour $32$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $7$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (7 and 10 minutes) are played with a 1-minute break between neighbors. The planned program lasts 77 minutes; swapping how many of each kind makes it last 92 minutes. Find the planned number of 7-minute pieces.

Givens

Pieces last $7$ minutes or $10$ minutes.
The planned total is $1$ hour $17$ minutes.
With the two counts swapped the total is $1$ hour $32$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $7$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

7

minutes and

y

pieces of

10

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

17

minutes is

77

minutes and

1

hour

32

minutes is

92

minutes. In the swapped program there are

y

seven-minute pieces and

x

ten-minute pieces.

7x + 10y + (x + y - 1) = 77,\quad 8x + 11y = 78;\qquad 7y + 10x + (x + y - 1) = 92,\quad 11x + 8y = 93

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

19x + 19y = 171,\quad x + y = 9

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

3x - 3y = 15,\quad x - y = 5

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{9 + 5}{2} = 7,\quad y = \frac{9 - 5}{2} = 2

Sum and difference together pin down two numbers.

Answer:

7

pieces

4 · Reviewdoes it hold up?

Planned: $7 \times 7 + 10 \times 2 + 8 = 49 + 20 + 8 = 77$ minutes. Swapped: $7 \times 2 + 10 \times 7 + 8 = 14 + 70 + 8 = 92$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $7$ to $10$ minutes adds $3$ minutes and each that switches from $10$ to $7$ takes away $3$ , so the swap changes the total by $3(x - y) = 92 - 77 = 15$ , giving $x - y = 5$ . With $x + y = 9$ this again gives $x = 7$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 3 easy

A concert was planned to mix $5$ -minute pieces and $8$ -minute pieces so that the whole performance would take $1$ hour $20$ minutes. At the orchestra's request, the numbers of $5$ -minute pieces and $8$ -minute pieces were swapped, and the performance took $1$ hour $38$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $5$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (5 and 8 minutes) are played with a 1-minute break between neighbors. The planned program lasts 80 minutes; swapping how many of each kind makes it last 98 minutes. Find the planned number of 5-minute pieces.

Givens

Pieces last $5$ minutes or $8$ minutes.
The planned total is $1$ hour $20$ minutes.
With the two counts swapped the total is $1$ hour $38$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $5$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

5

minutes and

y

pieces of

8

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

20

minutes is

80

minutes and

1

hour

38

minutes is

98

minutes. In the swapped program there are

y

five-minute pieces and

x

eight-minute pieces.

5x + 8y + (x + y - 1) = 80,\quad 6x + 9y = 81;\qquad 5y + 8x + (x + y - 1) = 98,\quad 9x + 6y = 99

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

15x + 15y = 180,\quad x + y = 12

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

3x - 3y = 18,\quad x - y = 6

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{12 + 6}{2} = 9,\quad y = \frac{12 - 6}{2} = 3

Sum and difference together pin down two numbers.

Answer:

9

pieces

4 · Reviewdoes it hold up?

Planned: $5 \times 9 + 8 \times 3 + 11 = 45 + 24 + 11 = 80$ minutes. Swapped: $5 \times 3 + 8 \times 9 + 11 = 15 + 72 + 11 = 98$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $5$ to $8$ minutes adds $3$ minutes and each that switches from $8$ to $5$ takes away $3$ , so the swap changes the total by $3(x - y) = 98 - 80 = 18$ , giving $x - y = 6$ . With $x + y = 12$ this again gives $x = 9$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 4 easy

A concert was planned to mix $8$ -minute pieces and $11$ -minute pieces so that the whole performance would take $1$ hour $29$ minutes. At the orchestra's request, the numbers of $8$ -minute pieces and $11$ -minute pieces were swapped, and the performance took $1$ hour $38$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $8$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (8 and 11 minutes) are played with a 1-minute break between neighbors. The planned program lasts 89 minutes; swapping how many of each kind makes it last 98 minutes. Find the planned number of 8-minute pieces.

Givens

Pieces last $8$ minutes or $11$ minutes.
The planned total is $1$ hour $29$ minutes.
With the two counts swapped the total is $1$ hour $38$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $8$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

8

minutes and

y

pieces of

11

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

29

minutes is

89

minutes and

1

hour

38

minutes is

98

minutes. In the swapped program there are

y

eight-minute pieces and

x

eleven-minute pieces.

8x + 11y + (x + y - 1) = 89,\quad 9x + 12y = 90;\qquad 8y + 11x + (x + y - 1) = 98,\quad 12x + 9y = 99

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

21x + 21y = 189,\quad x + y = 9

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

3x - 3y = 9,\quad x - y = 3

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{9 + 3}{2} = 6,\quad y = \frac{9 - 3}{2} = 3

Sum and difference together pin down two numbers.

Answer:

6

pieces

4 · Reviewdoes it hold up?

Planned: $8 \times 6 + 11 \times 3 + 8 = 48 + 33 + 8 = 89$ minutes. Swapped: $8 \times 3 + 11 \times 6 + 8 = 24 + 66 + 8 = 98$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $8$ to $11$ minutes adds $3$ minutes and each that switches from $11$ to $8$ takes away $3$ , so the swap changes the total by $3(x - y) = 98 - 89 = 9$ , giving $x - y = 3$ . With $x + y = 9$ this again gives $x = 6$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 5 medium

A concert was planned to mix $5$ -minute pieces and $9$ -minute pieces so that the whole performance would take $1$ hour $37$ minutes. At the orchestra's request, the numbers of $5$ -minute pieces and $9$ -minute pieces were swapped, and the performance took $1$ hour $49$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $5$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (5 and 9 minutes) are played with a 1-minute break between neighbors. The planned program lasts 97 minutes; swapping how many of each kind makes it last 109 minutes. Find the planned number of 5-minute pieces.

Givens

Pieces last $5$ minutes or $9$ minutes.
The planned total is $1$ hour $37$ minutes.
With the two counts swapped the total is $1$ hour $49$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $5$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

5

minutes and

y

pieces of

9

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

37

minutes is

97

minutes and

1

hour

49

minutes is

109

minutes. In the swapped program there are

y

five-minute pieces and

x

nine-minute pieces.

5x + 9y + (x + y - 1) = 97,\quad 6x + 10y = 98;\qquad 5y + 9x + (x + y - 1) = 109,\quad 10x + 6y = 110

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

16x + 16y = 208,\quad x + y = 13

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

4x - 4y = 12,\quad x - y = 3

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{13 + 3}{2} = 8,\quad y = \frac{13 - 3}{2} = 5

Sum and difference together pin down two numbers.

Answer:

8

pieces

4 · Reviewdoes it hold up?

Planned: $5 \times 8 + 9 \times 5 + 12 = 40 + 45 + 12 = 97$ minutes. Swapped: $5 \times 5 + 9 \times 8 + 12 = 25 + 72 + 12 = 109$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $5$ to $9$ minutes adds $4$ minutes and each that switches from $9$ to $5$ takes away $4$ , so the swap changes the total by $4(x - y) = 109 - 97 = 12$ , giving $x - y = 3$ . With $x + y = 13$ this again gives $x = 8$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 6 medium

A concert was planned to mix $6$ -minute pieces and $9$ -minute pieces so that the whole performance would take $1$ hour $35$ minutes. At the orchestra's request, the numbers of $6$ -minute pieces and $9$ -minute pieces were swapped, and the performance took $1$ hour $50$ minutes in all. There is a $2$ -minute break between each piece and the next. How many $6$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (6 and 9 minutes) are played with a 2-minute break between neighbors. The planned program lasts 95 minutes; swapping how many of each kind makes it last 110 minutes. Find the planned number of 6-minute pieces.

Givens

Pieces last $6$ minutes or $9$ minutes.
The planned total is $1$ hour $35$ minutes.
With the two counts swapped the total is $1$ hour $50$ minutes.
There is a $2$ -minute break between each piece and the next.

Unknowns

The planned number of $6$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

6

minutes and

y

pieces of

9

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{break minutes} = 2(x + y - 1)

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

35

minutes is

95

minutes and

1

hour

50

minutes is

110

minutes. In the swapped program there are

y

six-minute pieces and

x

nine-minute pieces.

6x + 9y + 2(x + y - 1) = 95,\quad 8x + 11y = 97;\qquad 6y + 9x + 2(x + y - 1) = 110,\quad 11x + 8y = 112

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

19x + 19y = 209,\quad x + y = 11

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

3x - 3y = 15,\quad x - y = 5

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{11 + 5}{2} = 8,\quad y = \frac{11 - 5}{2} = 3

Sum and difference together pin down two numbers.

Answer:

8

pieces

4 · Reviewdoes it hold up?

Planned: $6 \times 8 + 9 \times 3 + 20 = 48 + 27 + 20 = 95$ minutes. Swapped: $6 \times 3 + 9 \times 8 + 20 = 18 + 72 + 20 = 110$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $6$ to $9$ minutes adds $3$ minutes and each that switches from $9$ to $6$ takes away $3$ , so the swap changes the total by $3(x - y) = 110 - 95 = 15$ , giving $x - y = 5$ . With $x + y = 11$ this again gives $x = 8$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 7 medium

A concert was planned to mix $4$ -minute pieces and $6$ -minute pieces so that the whole performance would take $1$ hour $51$ minutes. At the orchestra's request, the numbers of $4$ -minute pieces and $6$ -minute pieces were swapped, and the performance took $2$ hours $7$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $4$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (4 and 6 minutes) are played with a 1-minute break between neighbors. The planned program lasts 111 minutes; swapping how many of each kind makes it last 127 minutes. Find the planned number of 4-minute pieces.

Givens

Pieces last $4$ minutes or $6$ minutes.
The planned total is $1$ hour $51$ minutes.
With the two counts swapped the total is $2$ hours $7$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $4$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

4

minutes and

y

pieces of

6

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

51

minutes is

111

minutes and

2

hours

7

minutes is

127

minutes. In the swapped program there are

y

four-minute pieces and

x

six-minute pieces.

4x + 6y + (x + y - 1) = 111,\quad 5x + 7y = 112;\qquad 4y + 6x + (x + y - 1) = 127,\quad 7x + 5y = 128

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

12x + 12y = 240,\quad x + y = 20

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

2x - 2y = 16,\quad x - y = 8

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{20 + 8}{2} = 14,\quad y = \frac{20 - 8}{2} = 6

Sum and difference together pin down two numbers.

Answer:

14

pieces

4 · Reviewdoes it hold up?

Planned: $4 \times 14 + 6 \times 6 + 19 = 56 + 36 + 19 = 111$ minutes. Swapped: $4 \times 6 + 6 \times 14 + 19 = 24 + 84 + 19 = 127$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $4$ to $6$ minutes adds $2$ minutes and each that switches from $6$ to $4$ takes away $2$ , so the swap changes the total by $2(x - y) = 127 - 111 = 16$ , giving $x - y = 8$ . With $x + y = 20$ this again gives $x = 14$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 8 medium

A concert was planned to mix $5$ -minute pieces and $7$ -minute pieces so that the whole performance would take $1$ hour $53$ minutes. At the orchestra's request, the numbers of $5$ -minute pieces and $7$ -minute pieces were swapped, and the performance took $2$ hours $3$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $5$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (5 and 7 minutes) are played with a 1-minute break between neighbors. The planned program lasts 113 minutes; swapping how many of each kind makes it last 123 minutes. Find the planned number of 5-minute pieces.

Givens

Pieces last $5$ minutes or $7$ minutes.
The planned total is $1$ hour $53$ minutes.
With the two counts swapped the total is $2$ hours $3$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $5$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

5

minutes and

y

pieces of

7

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

53

minutes is

113

minutes and

2

hours

3

minutes is

123

minutes. In the swapped program there are

y

five-minute pieces and

x

seven-minute pieces.

5x + 7y + (x + y - 1) = 113,\quad 6x + 8y = 114;\qquad 5y + 7x + (x + y - 1) = 123,\quad 8x + 6y = 124

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

14x + 14y = 238,\quad x + y = 17

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

2x - 2y = 10,\quad x - y = 5

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{17 + 5}{2} = 11,\quad y = \frac{17 - 5}{2} = 6

Sum and difference together pin down two numbers.

Answer:

11

pieces

4 · Reviewdoes it hold up?

Planned: $5 \times 11 + 7 \times 6 + 16 = 55 + 42 + 16 = 113$ minutes. Swapped: $5 \times 6 + 7 \times 11 + 16 = 30 + 77 + 16 = 123$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $5$ to $7$ minutes adds $2$ minutes and each that switches from $7$ to $5$ takes away $2$ , so the swap changes the total by $2(x - y) = 123 - 113 = 10$ , giving $x - y = 5$ . With $x + y = 17$ this again gives $x = 11$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 9 hard

A concert was planned to mix $3$ -minute pieces and $5$ -minute pieces so that the whole performance would take $1$ hour $53$ minutes. At the orchestra's request, the numbers of $3$ -minute pieces and $5$ -minute pieces were swapped, and the performance took $2$ hours $5$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $3$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (3 and 5 minutes) are played with a 1-minute break between neighbors. The planned program lasts 113 minutes; swapping how many of each kind makes it last 125 minutes. Find the planned number of 3-minute pieces.

Givens

Pieces last $3$ minutes or $5$ minutes.
The planned total is $1$ hour $53$ minutes.
With the two counts swapped the total is $2$ hours $5$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $3$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

3

minutes and

y

pieces of

5

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

53

minutes is

113

minutes and

2

hours

5

minutes is

125

minutes. In the swapped program there are

y

three-minute pieces and

x

five-minute pieces.

3x + 5y + (x + y - 1) = 113,\quad 4x + 6y = 114;\qquad 3y + 5x + (x + y - 1) = 125,\quad 6x + 4y = 126

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

10x + 10y = 240,\quad x + y = 24

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

2x - 2y = 12,\quad x - y = 6

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{24 + 6}{2} = 15,\quad y = \frac{24 - 6}{2} = 9

Sum and difference together pin down two numbers.

Answer:

15

pieces

4 · Reviewdoes it hold up?

Planned: $3 \times 15 + 5 \times 9 + 23 = 45 + 45 + 23 = 113$ minutes. Swapped: $3 \times 9 + 5 \times 15 + 23 = 27 + 75 + 23 = 125$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $3$ to $5$ minutes adds $2$ minutes and each that switches from $5$ to $3$ takes away $2$ , so the swap changes the total by $2(x - y) = 125 - 113 = 12$ , giving $x - y = 6$ . With $x + y = 24$ this again gives $x = 15$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 10 hard

A concert was planned to mix $6$ -minute pieces and $10$ -minute pieces so that the whole performance would take $1$ hour $42$ minutes. At the orchestra's request, the numbers of $6$ -minute pieces and $10$ -minute pieces were swapped, and the performance took $1$ hour $54$ minutes in all. There is a $2$ -minute break between each piece and the next. How many $6$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (6 and 10 minutes) are played with a 2-minute break between neighbors. The planned program lasts 102 minutes; swapping how many of each kind makes it last 114 minutes. Find the planned number of 6-minute pieces.

Givens

Pieces last $6$ minutes or $10$ minutes.
The planned total is $1$ hour $42$ minutes.
With the two counts swapped the total is $1$ hour $54$ minutes.
There is a $2$ -minute break between each piece and the next.

Unknowns

The planned number of $6$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

6

minutes and

y

pieces of

10

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{break minutes} = 2(x + y - 1)

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

42

minutes is

102

minutes and

1

hour

54

minutes is

114

minutes. In the swapped program there are

y

six-minute pieces and

x

ten-minute pieces.

6x + 10y + 2(x + y - 1) = 102,\quad 8x + 12y = 104;\qquad 6y + 10x + 2(x + y - 1) = 114,\quad 12x + 8y = 116

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

20x + 20y = 220,\quad x + y = 11

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

4x - 4y = 12,\quad x - y = 3

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{11 + 3}{2} = 7,\quad y = \frac{11 - 3}{2} = 4

Sum and difference together pin down two numbers.

Answer:

7

pieces

4 · Reviewdoes it hold up?

Planned: $6 \times 7 + 10 \times 4 + 20 = 42 + 40 + 20 = 102$ minutes. Swapped: $6 \times 4 + 10 \times 7 + 20 = 24 + 70 + 20 = 114$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $6$ to $10$ minutes adds $4$ minutes and each that switches from $10$ to $6$ takes away $4$ , so the swap changes the total by $4(x - y) = 114 - 102 = 12$ , giving $x - y = 3$ . With $x + y = 11$ this again gives $x = 7$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 11 hard

A concert was planned to mix $4$ -minute pieces and $7$ -minute pieces so that the whole performance would take $1$ hour $31$ minutes. At the orchestra's request, the numbers of $4$ -minute pieces and $7$ -minute pieces were swapped, and the performance took $1$ hour $55$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $4$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (4 and 7 minutes) are played with a 1-minute break between neighbors. The planned program lasts 91 minutes; swapping how many of each kind makes it last 115 minutes. Find the planned number of 4-minute pieces.

Givens

Pieces last $4$ minutes or $7$ minutes.
The planned total is $1$ hour $31$ minutes.
With the two counts swapped the total is $1$ hour $55$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $4$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

4

minutes and

y

pieces of

7

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

31

minutes is

91

minutes and

1

hour

55

minutes is

115

minutes. In the swapped program there are

y

four-minute pieces and

x

seven-minute pieces.

4x + 7y + (x + y - 1) = 91,\quad 5x + 8y = 92;\qquad 4y + 7x + (x + y - 1) = 115,\quad 8x + 5y = 116

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

13x + 13y = 208,\quad x + y = 16

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

3x - 3y = 24,\quad x - y = 8

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{16 + 8}{2} = 12,\quad y = \frac{16 - 8}{2} = 4

Sum and difference together pin down two numbers.

Answer:

12

pieces

4 · Reviewdoes it hold up?

Planned: $4 \times 12 + 7 \times 4 + 15 = 48 + 28 + 15 = 91$ minutes. Swapped: $4 \times 4 + 7 \times 12 + 15 = 16 + 84 + 15 = 115$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $4$ to $7$ minutes adds $3$ minutes and each that switches from $7$ to $4$ takes away $3$ , so the swap changes the total by $3(x - y) = 115 - 91 = 24$ , giving $x - y = 8$ . With $x + y = 16$ this again gives $x = 12$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Variant 12 hard

A concert was planned to mix $6$ -minute pieces and $8$ -minute pieces so that the whole performance would take $1$ hour $45$ minutes. At the orchestra's request, the numbers of $6$ -minute pieces and $8$ -minute pieces were swapped, and the performance took $1$ hour $57$ minutes in all. There is a $1$ -minute break between each piece and the next. How many $6$ -minute pieces were originally planned?

Your answer

Show solution

1 · Understandwhat's really being asked

Two kinds of pieces (6 and 8 minutes) are played with a 1-minute break between neighbors. The planned program lasts 105 minutes; swapping how many of each kind makes it last 117 minutes. Find the planned number of 6-minute pieces.

Givens

Pieces last $6$ minutes or $8$ minutes.
The planned total is $1$ hour $45$ minutes.
With the two counts swapped the total is $1$ hour $57$ minutes.
There is a $1$ -minute break between each piece and the next.

Unknowns

The planned number of $6$ -minute pieces.

Constraints

Counts are whole numbers.
Breaks come only between pieces, not before the first or after the last.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#8 Analyze the Units#15 Organize Information in More Ways

Picture the program as pieces in a row with breaks in the gaps to count the breaks correctly, convert the times to minutes, and write one equation for each program. The two equations are mirror images, so adding and subtracting them is the quickest way through.

3 · Execute5 carry out the plan

1Count the breaks

#1 Draw a Diagram 6.EE.B.6

Plan

x

pieces of

6

minutes and

y

pieces of

8

minutes. In a row of

x + y

pieces there is a gap between each neighboring pair, so there are

x + y - 1

breaks. Swapping the counts does not change the number of pieces, so the breaks stay the same.

\text{breaks} = x + y - 1

Like fence posts and gaps:

n

items in a line have

n - 1

spaces between them.

2Write both programs in minutes

#8 Analyze the Units 8.EE.C.8

1

hour

45

minutes is

105

minutes and

1

hour

57

minutes is

117

minutes. In the swapped program there are

y

six-minute pieces and

x

eight-minute pieces.

6x + 8y + (x + y - 1) = 105,\quad 7x + 9y = 106;\qquad 6y + 8x + (x + y - 1) = 117,\quad 9x + 7y = 118

Put every time in the same unit before comparing.

3Add the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Adding gives the same coefficient on both letters.

16x + 16y = 224,\quad x + y = 14

Mirror-image equations add to a statement about the total.

4Subtract the mirror equations

#15 Organize Information in More Ways 8.EE.C.8

Subtracting the first from the second leaves the difference.

2x - 2y = 12,\quad x - y = 6

...and subtract to a statement about the difference.

5Recover the counts

#13 Convert to Algebra 8.EE.C.8

Knowing the sum and the difference gives each count.

x = \frac{14 + 6}{2} = 10,\quad y = \frac{14 - 6}{2} = 4

Sum and difference together pin down two numbers.

Answer:

10

pieces

4 · Reviewdoes it hold up?

Planned: $6 \times 10 + 8 \times 4 + 13 = 60 + 32 + 13 = 105$ minutes. Swapped: $6 \times 4 + 8 \times 10 + 13 = 24 + 80 + 13 = 117$ minutes. Both match.

Another way: Look only at the change. Each piece that switches from $6$ to $8$ minutes adds $2$ minutes and each that switches from $8$ to $6$ takes away $2$ , so the swap changes the total by $2(x - y) = 117 - 105 = 12$ , giving $x - y = 6$ . With $x + y = 14$ this again gives $x = 10$ .

Standardsmin grade 8

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations — Writing and solving the two total-time equations.
6.EE.B.6 Use variables to represent numbers and write expressions — Expressing the number of breaks as $x + y - 1$.

💡Takeaway. Count the gaps carefully, then let the mirror-image equations do the work: add for the sum, subtract for the difference.

Count the gaps, then add and subtract · 12 practice problems

Generated variants — 12

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