Practice · Split buyers into only-coffee, only-cake, both · Sensim Math

Variant 1 easy

At a bazaar snack corner, a cup of coffee costs $160$ won and a slice of cake costs $140$ won, and anyone who buys both at once gets $20\%$ off the combined price. On the first day, $70$ people bought something at the snack corner, and total sales were $12300$ won. $45$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $20\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $160$ won; cake: $140$ won.
Buying both: $20\%$ off the combined price.
$70$ buyers in total, $45$ of whom bought coffee.
Total sales: $12300$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $70 - 45$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 70 - 45 = 25

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

300

won, less

20\%

.

300 \times 0.8 = 240

A

20\%

discount leaves

80\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

45 - x

bought coffee only.

160(45 - x) + 140 \times 25 + 240x = 12300

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

7200 - 160x + 3500 + 240x = 12300 \;\Rightarrow\; 80x = 1600 \;\Rightarrow\; x = 20

Each combo buyer brings in

80

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

25 + 20 = 45

Read the whole cake circle off the diagram.

Answer: 45 people

4 · Reviewdoes it hold up?

Coffee only $25$ pay $25 \times 160 = 4000$ , cake only pay $25 \times 140 = 3500$ , and both pay $20 \times 240 = 4800$ . The total is $4000 + 3500 + 4800 = 12300$ won, and $25 + 25 + 20 = 70$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 25$ bought both and $70 - x$ bought coffee only, so $160(70 - x) + 140 \times 25 + 240(x - 25) = 12300$ , giving $80x = 3600$ and $x = 45$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $20\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 2 easy

At a bazaar snack corner, a cup of coffee costs $180$ won and a slice of cake costs $120$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $80$ people bought something at the snack corner, and total sales were $14220$ won. $50$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $180$ won; cake: $120$ won.
Buying both: $10\%$ off the combined price.
$80$ buyers in total, $50$ of whom bought coffee.
Total sales: $14220$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $80 - 50$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 80 - 50 = 30

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

300

won, less

10\%

.

300 \times 0.9 = 270

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

50 - x

bought coffee only.

180(50 - x) + 120 \times 30 + 270x = 14220

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

9000 - 180x + 3600 + 270x = 14220 \;\Rightarrow\; 90x = 1620 \;\Rightarrow\; x = 18

Each combo buyer brings in

90

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

30 + 18 = 48

Read the whole cake circle off the diagram.

Answer: 48 people

4 · Reviewdoes it hold up?

Coffee only $32$ pay $32 \times 180 = 5760$ , cake only pay $30 \times 120 = 3600$ , and both pay $18 \times 270 = 4860$ . The total is $5760 + 3600 + 4860 = 14220$ won, and $32 + 30 + 18 = 80$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 30$ bought both and $80 - x$ bought coffee only, so $180(80 - x) + 120 \times 30 + 270(x - 30) = 14220$ , giving $90x = 4320$ and $x = 48$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 3 easy

At a bazaar snack corner, a cup of coffee costs $150$ won and a slice of cake costs $100$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $100$ people bought something at the snack corner, and total sales were $15375$ won. $70$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $150$ won; cake: $100$ won.
Buying both: $10\%$ off the combined price.
$100$ buyers in total, $70$ of whom bought coffee.
Total sales: $15375$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $100 - 70$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 100 - 70 = 30

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

250

won, less

10\%

.

250 \times 0.9 = 225

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

70 - x

bought coffee only.

150(70 - x) + 100 \times 30 + 225x = 15375

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

10500 - 150x + 3000 + 225x = 15375 \;\Rightarrow\; 75x = 1875 \;\Rightarrow\; x = 25

Each combo buyer brings in

75

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

30 + 25 = 55

Read the whole cake circle off the diagram.

Answer: 55 people

4 · Reviewdoes it hold up?

Coffee only $45$ pay $45 \times 150 = 6750$ , cake only pay $30 \times 100 = 3000$ , and both pay $25 \times 225 = 5625$ . The total is $6750 + 3000 + 5625 = 15375$ won, and $45 + 30 + 25 = 100$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 30$ bought both and $100 - x$ bought coffee only, so $150(100 - x) + 100 \times 30 + 225(x - 30) = 15375$ , giving $75x = 4125$ and $x = 55$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 4 easy

At a bazaar snack corner, a cup of coffee costs $120$ won and a slice of cake costs $100$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $120$ people bought something at the snack corner, and total sales were $15904$ won. $86$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $120$ won; cake: $100$ won.
Buying both: $10\%$ off the combined price.
$120$ buyers in total, $86$ of whom bought coffee.
Total sales: $15904$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $120 - 86$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 120 - 86 = 34

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

220

won, less

10\%

.

220 \times 0.9 = 198

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

86 - x

bought coffee only.

120(86 - x) + 100 \times 34 + 198x = 15904

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

10320 - 120x + 3400 + 198x = 15904 \;\Rightarrow\; 78x = 2184 \;\Rightarrow\; x = 28

Each combo buyer brings in

78

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

34 + 28 = 62

Read the whole cake circle off the diagram.

Answer: 62 people

4 · Reviewdoes it hold up?

Coffee only $58$ pay $58 \times 120 = 6960$ , cake only pay $34 \times 100 = 3400$ , and both pay $28 \times 198 = 5544$ . The total is $6960 + 3400 + 5544 = 15904$ won, and $58 + 34 + 28 = 120$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 34$ bought both and $120 - x$ bought coffee only, so $120(120 - x) + 100 \times 34 + 198(x - 34) = 15904$ , giving $78x = 4836$ and $x = 62$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 5 medium

At a bazaar snack corner, a cup of coffee costs $200$ won and a slice of cake costs $150$ won, and anyone who buys both at once gets $20\%$ off the combined price. On the first day, $90$ people bought something at the snack corner, and total sales were $19300$ won. $60$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $20\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $200$ won; cake: $150$ won.
Buying both: $20\%$ off the combined price.
$90$ buyers in total, $60$ of whom bought coffee.
Total sales: $19300$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $90 - 60$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 90 - 60 = 30

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

350

won, less

20\%

.

350 \times 0.8 = 280

A

20\%

discount leaves

80\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

60 - x

bought coffee only.

200(60 - x) + 150 \times 30 + 280x = 19300

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

12000 - 200x + 4500 + 280x = 19300 \;\Rightarrow\; 80x = 2800 \;\Rightarrow\; x = 35

Each combo buyer brings in

80

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

30 + 35 = 65

Read the whole cake circle off the diagram.

Answer: 65 people

4 · Reviewdoes it hold up?

Coffee only $25$ pay $25 \times 200 = 5000$ , cake only pay $30 \times 150 = 4500$ , and both pay $35 \times 280 = 9800$ . The total is $5000 + 4500 + 9800 = 19300$ won, and $25 + 30 + 35 = 90$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 30$ bought both and $90 - x$ bought coffee only, so $200(90 - x) + 150 \times 30 + 280(x - 30) = 19300$ , giving $80x = 5200$ and $x = 65$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $20\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 6 medium

At a bazaar snack corner, a cup of coffee costs $140$ won and a slice of cake costs $100$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $130$ people bought something at the snack corner, and total sales were $20552$ won. $90$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $140$ won; cake: $100$ won.
Buying both: $10\%$ off the combined price.
$130$ buyers in total, $90$ of whom bought coffee.
Total sales: $20552$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $130 - 90$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 130 - 90 = 40

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

240

won, less

10\%

.

240 \times 0.9 = 216

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

90 - x

bought coffee only.

140(90 - x) + 100 \times 40 + 216x = 20552

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

12600 - 140x + 4000 + 216x = 20552 \;\Rightarrow\; 76x = 3952 \;\Rightarrow\; x = 52

Each combo buyer brings in

76

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

40 + 52 = 92

Read the whole cake circle off the diagram.

Answer: 92 people

4 · Reviewdoes it hold up?

Coffee only $38$ pay $38 \times 140 = 5320$ , cake only pay $40 \times 100 = 4000$ , and both pay $52 \times 216 = 11232$ . The total is $5320 + 4000 + 11232 = 20552$ won, and $38 + 40 + 52 = 130$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 40$ bought both and $130 - x$ bought coffee only, so $140(130 - x) + 100 \times 40 + 216(x - 40) = 20552$ , giving $76x = 6992$ and $x = 92$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 7 medium

At a bazaar snack corner, a cup of coffee costs $130$ won and a slice of cake costs $110$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $150$ people bought something at the snack corner, and total sales were $21840$ won. $95$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $130$ won; cake: $110$ won.
Buying both: $10\%$ off the combined price.
$150$ buyers in total, $95$ of whom bought coffee.
Total sales: $21840$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $150 - 95$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 150 - 95 = 55

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

240

won, less

10\%

.

240 \times 0.9 = 216

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

95 - x

bought coffee only.

130(95 - x) + 110 \times 55 + 216x = 21840

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

12350 - 130x + 6050 + 216x = 21840 \;\Rightarrow\; 86x = 3440 \;\Rightarrow\; x = 40

Each combo buyer brings in

86

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

55 + 40 = 95

Read the whole cake circle off the diagram.

Answer: 95 people

4 · Reviewdoes it hold up?

Coffee only $55$ pay $55 \times 130 = 7150$ , cake only pay $55 \times 110 = 6050$ , and both pay $40 \times 216 = 8640$ . The total is $7150 + 6050 + 8640 = 21840$ won, and $55 + 55 + 40 = 150$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 55$ bought both and $150 - x$ bought coffee only, so $130(150 - x) + 110 \times 55 + 216(x - 55) = 21840$ , giving $86x = 8170$ and $x = 95$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 8 medium

At a bazaar snack corner, a cup of coffee costs $110$ won and a slice of cake costs $90$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $160$ people bought something at the snack corner, and total sales were $22050$ won. $120$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $110$ won; cake: $90$ won.
Buying both: $10\%$ off the combined price.
$160$ buyers in total, $120$ of whom bought coffee.
Total sales: $22050$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $160 - 120$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 160 - 120 = 40

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

200

won, less

10\%

.

200 \times 0.9 = 180

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

120 - x

bought coffee only.

110(120 - x) + 90 \times 40 + 180x = 22050

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

13200 - 110x + 3600 + 180x = 22050 \;\Rightarrow\; 70x = 5250 \;\Rightarrow\; x = 75

Each combo buyer brings in

70

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

40 + 75 = 115

Read the whole cake circle off the diagram.

Answer: 115 people

4 · Reviewdoes it hold up?

Coffee only $45$ pay $45 \times 110 = 4950$ , cake only pay $40 \times 90 = 3600$ , and both pay $75 \times 180 = 13500$ . The total is $4950 + 3600 + 13500 = 22050$ won, and $45 + 40 + 75 = 160$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 40$ bought both and $160 - x$ bought coffee only, so $110(160 - x) + 90 \times 40 + 180(x - 40) = 22050$ , giving $70x = 8050$ and $x = 115$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 9 hard

At a bazaar snack corner, a cup of coffee costs $220$ won and a slice of cake costs $180$ won, and anyone who buys both at once gets $20\%$ off the combined price. On the first day, $100$ people bought something at the snack corner, and total sales were $23260$ won. $64$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $20\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $220$ won; cake: $180$ won.
Buying both: $20\%$ off the combined price.
$100$ buyers in total, $64$ of whom bought coffee.
Total sales: $23260$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $100 - 64$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 100 - 64 = 36

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

400

won, less

20\%

.

400 \times 0.8 = 320

A

20\%

discount leaves

80\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

64 - x

bought coffee only.

220(64 - x) + 180 \times 36 + 320x = 23260

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

14080 - 220x + 6480 + 320x = 23260 \;\Rightarrow\; 100x = 2700 \;\Rightarrow\; x = 27

Each combo buyer brings in

100

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

36 + 27 = 63

Read the whole cake circle off the diagram.

Answer: 63 people

4 · Reviewdoes it hold up?

Coffee only $37$ pay $37 \times 220 = 8140$ , cake only pay $36 \times 180 = 6480$ , and both pay $27 \times 320 = 8640$ . The total is $8140 + 6480 + 8640 = 23260$ won, and $37 + 36 + 27 = 100$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 36$ bought both and $100 - x$ bought coffee only, so $220(100 - x) + 180 \times 36 + 320(x - 36) = 23260$ , giving $100x = 6300$ and $x = 63$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $20\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 10 hard

At a bazaar snack corner, a cup of coffee costs $250$ won and a slice of cake costs $150$ won, and anyone who buys both at once gets $20\%$ off the combined price. On the first day, $110$ people bought something at the snack corner, and total sales were $26100$ won. $75$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $20\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $250$ won; cake: $150$ won.
Buying both: $20\%$ off the combined price.
$110$ buyers in total, $75$ of whom bought coffee.
Total sales: $26100$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $110 - 75$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 110 - 75 = 35

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

400

won, less

20\%

.

400 \times 0.8 = 320

A

20\%

discount leaves

80\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

75 - x

bought coffee only.

250(75 - x) + 150 \times 35 + 320x = 26100

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

18750 - 250x + 5250 + 320x = 26100 \;\Rightarrow\; 70x = 2100 \;\Rightarrow\; x = 30

Each combo buyer brings in

70

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

35 + 30 = 65

Read the whole cake circle off the diagram.

Answer: 65 people

4 · Reviewdoes it hold up?

Coffee only $45$ pay $45 \times 250 = 11250$ , cake only pay $35 \times 150 = 5250$ , and both pay $30 \times 320 = 9600$ . The total is $11250 + 5250 + 9600 = 26100$ won, and $45 + 35 + 30 = 110$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 35$ bought both and $110 - x$ bought coffee only, so $250(110 - x) + 150 \times 35 + 320(x - 35) = 26100$ , giving $70x = 4550$ and $x = 65$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $20\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 11 hard

At a bazaar snack corner, a cup of coffee costs $170$ won and a slice of cake costs $130$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $200$ people bought something at the snack corner, and total sales were $40800$ won. $150$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $170$ won; cake: $130$ won.
Buying both: $10\%$ off the combined price.
$200$ buyers in total, $150$ of whom bought coffee.
Total sales: $40800$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $200 - 150$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 200 - 150 = 50

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

300

won, less

10\%

.

300 \times 0.9 = 270

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

150 - x

bought coffee only.

170(150 - x) + 130 \times 50 + 270x = 40800

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

25500 - 170x + 6500 + 270x = 40800 \;\Rightarrow\; 100x = 8800 \;\Rightarrow\; x = 88

Each combo buyer brings in

100

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

50 + 88 = 138

Read the whole cake circle off the diagram.

Answer: 138 people

4 · Reviewdoes it hold up?

Coffee only $62$ pay $62 \times 170 = 10540$ , cake only pay $50 \times 130 = 6500$ , and both pay $88 \times 270 = 23760$ . The total is $10540 + 6500 + 23760 = 40800$ won, and $62 + 50 + 88 = 200$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 50$ bought both and $200 - x$ bought coffee only, so $170(200 - x) + 130 \times 50 + 270(x - 50) = 40800$ , giving $100x = 13800$ and $x = 138$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Variant 12 hard

At a bazaar snack corner, a cup of coffee costs $300$ won and a slice of cake costs $200$ won, and anyone who buys both at once gets $10\%$ off the combined price. On the first day, $140$ people bought something at the snack corner, and total sales were $47150$ won. $100$ people bought coffee, and no one bought more than one of the same item. How many people bought cake?

Your answer

Show solution

1 · Understandwhat's really being asked

Every customer bought coffee, cake, or both (at a $10\%$ combo discount), one of each at most. From the head count, the number of coffee buyers, and the day's sales, find how many bought cake.

Givens

Coffee: $300$ won; cake: $200$ won.
Buying both: $10\%$ off the combined price.
$140$ buyers in total, $100$ of whom bought coffee.
Total sales: $47150$ won.
No one bought two of the same item.

Unknowns

The number of people who bought cake.

Constraints

Each buyer is in exactly one group: coffee only, cake only, or both.

2 · Planchoose the strategy

#12 Draw a Venn Diagram · also uses: #13 Convert to Algebra#7 Identify Subproblems

Draw the two overlapping groups. The $140 - 100$ buyers outside the coffee circle can only be cake-only buyers. Then one unknown, the size of the overlap, is enough to write the day's sales as an equation.

3 · Execute5 carry out the plan

1Sort the buyers

#12 Draw a Venn Diagram 7.EE.B.3

Everyone bought something, so the buyers who did not buy coffee bought only cake.

\text{cake only} = 140 - 100 = 40

In the Venn diagram, the part of the cake circle outside the coffee circle is fixed at once.

2Price of the combo

#7 Identify Subproblems 7.RP.A.3

Coffee and cake together cost

500

won, less

10\%

.

500 \times 0.9 = 450

A

10\%

discount leaves

90\%

of the price.

3Write the sales equation

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

Let

x

people buy both. Then

100 - x

bought coffee only.

300(100 - x) + 200 \times 40 + 450x = 47150

Each group pays its own price, and the three groups together make the day's sales.

4Solve

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

Expand and collect the

x

terms.

30000 - 300x + 8000 + 450x = 47150 \;\Rightarrow\; 150x = 9150 \;\Rightarrow\; x = 61

Each combo buyer brings in

150

won more than a coffee-only buyer.

5Count the cake buyers

#12 Draw a Venn Diagram 7.EE.B.3

Cake buyers are the cake-only group plus the both group.

40 + 61 = 101

Read the whole cake circle off the diagram.

Answer: 101 people

4 · Reviewdoes it hold up?

Coffee only $39$ pay $39 \times 300 = 11700$ , cake only pay $40 \times 200 = 8000$ , and both pay $61 \times 450 = 27450$ . The total is $11700 + 8000 + 27450 = 47150$ won, and $39 + 40 + 61 = 140$ buyers.

Another way: Let $x$ be the number of cake buyers instead: then $x - 40$ bought both and $140 - x$ bought coffee only, so $300(140 - x) + 200 \times 40 + 450(x - 40) = 47150$ , giving $150x = 15150$ and $x = 101$ directly.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying the $10\%$ discount to the combo price.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing total sales as an equation in the number of combo buyers.
7.EE.B.3 Solve multi-step real-world problems posed with positive and negative rational numbers — Counting the groups and checking the total.

💡Takeaway. Sort people into non-overlapping groups first, and one unknown is often enough.

Split buyers into only-coffee, only-cake, both · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?