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The second graph's whole is the first graph's slice · 12 practice problems

7.RP.A.36.SP.B.5

Generated variants — 12

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

Variant 1 easy answer: 336,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 11 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (20%) Share of votes by candidate A B 27% C 18% D 8% Other 5%
Show solution
1 · Understandwhat's really being asked

1,000,000 people could vote and 20 percent did not. Of the votes cast, B took 27 percent, C 18, D 8 and other 5; A is unlabelled. We want A's votes.

Givens
  • There are 1,000,000 eligible voters.
  • 20 percent did not vote.
  • B 27 percent, C 18 percent, D 8 percent, other 5 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 20 percent stayed home, 80 percent voted.
100−20=80100 - 20 = 80
80 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
1,000,000×0.8=800,0001{,}000{,}000 \times 0.8 = 800{,}000
800,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−27−18−8−5=42100 - 27 - 18 - 8 - 5 = 42
A took 42 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
800,000×0.42=336,000800{,}000 \times 0.42 = 336{,}000
A received 336,000 votes.
Answer: 336,000 votes
4 · Reviewdoes it hold up?

A's 336,000 votes out of the 800,000 cast is 42 percent, and out of the 1,000,000 electorate only 33.6 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 42 percent of the 1,000,000 electorate gives 420,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 2 easy answer: 2,362,500 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 77 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (25%) Share of votes by candidate A B 29% C 16% D 6% Other 4%
Show solution
1 · Understandwhat's really being asked

7,000,000 people could vote and 25 percent did not. Of the votes cast, B took 29 percent, C 16, D 6 and other 4; A is unlabelled. We want A's votes.

Givens
  • There are 7,000,000 eligible voters.
  • 25 percent did not vote.
  • B 29 percent, C 16 percent, D 6 percent, other 4 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 25 percent stayed home, 75 percent voted.
100−25=75100 - 25 = 75
75 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
7,000,000×0.75=5,250,0007{,}000{,}000 \times 0.75 = 5{,}250{,}000
5,250,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−29−16−6−4=45100 - 29 - 16 - 6 - 4 = 45
A took 45 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
5,250,000×0.45=2,362,5005{,}250{,}000 \times 0.45 = 2{,}362{,}500
A received 2,362,500 votes.
Answer: 2,362,500 votes
4 · Reviewdoes it hold up?

A's 2,362,500 votes out of the 5,250,000 cast is 45 percent, and out of the 7,000,000 electorate only 33.75 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 45 percent of the 7,000,000 electorate gives 3,150,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 3 easy answer: 1,410,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 44 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (25%) Share of votes by candidate A B 30% C 12% D 6% Other 5%
Show solution
1 · Understandwhat's really being asked

4,000,000 people could vote and 25 percent did not. Of the votes cast, B took 30 percent, C 12, D 6 and other 5; A is unlabelled. We want A's votes.

Givens
  • There are 4,000,000 eligible voters.
  • 25 percent did not vote.
  • B 30 percent, C 12 percent, D 6 percent, other 5 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 25 percent stayed home, 75 percent voted.
100−25=75100 - 25 = 75
75 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
4,000,000×0.75=3,000,0004{,}000{,}000 \times 0.75 = 3{,}000{,}000
3,000,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−30−12−6−5=47100 - 30 - 12 - 6 - 5 = 47
A took 47 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
3,000,000×0.47=1,410,0003{,}000{,}000 \times 0.47 = 1{,}410{,}000
A received 1,410,000 votes.
Answer: 1,410,000 votes
4 · Reviewdoes it hold up?

A's 1,410,000 votes out of the 3,000,000 cast is 47 percent, and out of the 4,000,000 electorate only 35.25 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 47 percent of the 4,000,000 electorate gives 1,880,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 4 easy answer: 945,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 33 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (30%) Share of votes by candidate A B 28% C 15% D 7% Other 5%
Show solution
1 · Understandwhat's really being asked

3,000,000 people could vote and 30 percent did not. Of the votes cast, B took 28 percent, C 15, D 7 and other 5; A is unlabelled. We want A's votes.

Givens
  • There are 3,000,000 eligible voters.
  • 30 percent did not vote.
  • B 28 percent, C 15 percent, D 7 percent, other 5 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 30 percent stayed home, 70 percent voted.
100−30=70100 - 30 = 70
70 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
3,000,000×0.7=2,100,0003{,}000{,}000 \times 0.7 = 2{,}100{,}000
2,100,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−28−15−7−5=45100 - 28 - 15 - 7 - 5 = 45
A took 45 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
2,100,000×0.45=945,0002{,}100{,}000 \times 0.45 = 945{,}000
A received 945,000 votes.
Answer: 945,000 votes
4 · Reviewdoes it hold up?

A's 945,000 votes out of the 2,100,000 cast is 45 percent, and out of the 3,000,000 electorate only 31.5 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 45 percent of the 3,000,000 electorate gives 1,350,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 5 medium answer: 782,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 22 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (15%) Share of votes by candidate A B 30% C 10% D 8% Other 6%
Show solution
1 · Understandwhat's really being asked

2,000,000 people could vote and 15 percent did not. Of the votes cast, B took 30 percent, C 10, D 8 and other 6; A is unlabelled. We want A's votes.

Givens
  • There are 2,000,000 eligible voters.
  • 15 percent did not vote.
  • B 30 percent, C 10 percent, D 8 percent, other 6 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 15 percent stayed home, 85 percent voted.
100−15=85100 - 15 = 85
85 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
2,000,000×0.85=1,700,0002{,}000{,}000 \times 0.85 = 1{,}700{,}000
1,700,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−30−10−8−6=46100 - 30 - 10 - 8 - 6 = 46
A took 46 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
1,700,000×0.46=782,0001{,}700{,}000 \times 0.46 = 782{,}000
A received 782,000 votes.
Answer: 782,000 votes
4 · Reviewdoes it hold up?

A's 782,000 votes out of the 1,700,000 cast is 46 percent, and out of the 2,000,000 electorate only 39.1 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 46 percent of the 2,000,000 electorate gives 920,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 6 medium answer: 3,483,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 99 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (10%) Share of votes by candidate A B 31% C 14% D 7% Other 5%
Show solution
1 · Understandwhat's really being asked

9,000,000 people could vote and 10 percent did not. Of the votes cast, B took 31 percent, C 14, D 7 and other 5; A is unlabelled. We want A's votes.

Givens
  • There are 9,000,000 eligible voters.
  • 10 percent did not vote.
  • B 31 percent, C 14 percent, D 7 percent, other 5 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 10 percent stayed home, 90 percent voted.
100−10=90100 - 10 = 90
90 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
9,000,000×0.9=8,100,0009{,}000{,}000 \times 0.9 = 8{,}100{,}000
8,100,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−31−14−7−5=43100 - 31 - 14 - 7 - 5 = 43
A took 43 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
8,100,000×0.43=3,483,0008{,}100{,}000 \times 0.43 = 3{,}483{,}000
A received 3,483,000 votes.
Answer: 3,483,000 votes
4 · Reviewdoes it hold up?

A's 3,483,000 votes out of the 8,100,000 cast is 43 percent, and out of the 9,000,000 electorate only 38.7 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 43 percent of the 9,000,000 electorate gives 3,870,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 7 medium answer: 2,268,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 66 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (10%) Share of votes by candidate A B 32% C 14% D 8% Other 4%
Show solution
1 · Understandwhat's really being asked

6,000,000 people could vote and 10 percent did not. Of the votes cast, B took 32 percent, C 14, D 8 and other 4; A is unlabelled. We want A's votes.

Givens
  • There are 6,000,000 eligible voters.
  • 10 percent did not vote.
  • B 32 percent, C 14 percent, D 8 percent, other 4 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 10 percent stayed home, 90 percent voted.
100−10=90100 - 10 = 90
90 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
6,000,000×0.9=5,400,0006{,}000{,}000 \times 0.9 = 5{,}400{,}000
5,400,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−32−14−8−4=42100 - 32 - 14 - 8 - 4 = 42
A took 42 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
5,400,000×0.42=2,268,0005{,}400{,}000 \times 0.42 = 2{,}268{,}000
A received 2,268,000 votes.
Answer: 2,268,000 votes
4 · Reviewdoes it hold up?

A's 2,268,000 votes out of the 5,400,000 cast is 42 percent, and out of the 6,000,000 electorate only 37.8 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 42 percent of the 6,000,000 electorate gives 2,520,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 8 medium answer: 5,160,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 1515 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (20%) Share of votes by candidate A B 32% C 12% D 8% Other 5%
Show solution
1 · Understandwhat's really being asked

15,000,000 people could vote and 20 percent did not. Of the votes cast, B took 32 percent, C 12, D 8 and other 5; A is unlabelled. We want A's votes.

Givens
  • There are 15,000,000 eligible voters.
  • 20 percent did not vote.
  • B 32 percent, C 12 percent, D 8 percent, other 5 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 20 percent stayed home, 80 percent voted.
100−20=80100 - 20 = 80
80 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
15,000,000×0.8=12,000,00015{,}000{,}000 \times 0.8 = 12{,}000{,}000
12,000,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−32−12−8−5=43100 - 32 - 12 - 8 - 5 = 43
A took 43 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
12,000,000×0.43=5,160,00012{,}000{,}000 \times 0.43 = 5{,}160{,}000
A received 5,160,000 votes.
Answer: 5,160,000 votes
4 · Reviewdoes it hold up?

A's 5,160,000 votes out of the 12,000,000 cast is 43 percent, and out of the 15,000,000 electorate only 34.4 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 43 percent of the 15,000,000 electorate gives 6,450,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 9 hard answer: 2,880,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 88 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (20%) Share of votes by candidate A B 34% C 12% D 6% Other 3%
Show solution
1 · Understandwhat's really being asked

8,000,000 people could vote and 20 percent did not. Of the votes cast, B took 34 percent, C 12, D 6 and other 3; A is unlabelled. We want A's votes.

Givens
  • There are 8,000,000 eligible voters.
  • 20 percent did not vote.
  • B 34 percent, C 12 percent, D 6 percent, other 3 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 20 percent stayed home, 80 percent voted.
100−20=80100 - 20 = 80
80 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
8,000,000×0.8=6,400,0008{,}000{,}000 \times 0.8 = 6{,}400{,}000
6,400,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−34−12−6−3=45100 - 34 - 12 - 6 - 3 = 45
A took 45 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
6,400,000×0.45=2,880,0006{,}400{,}000 \times 0.45 = 2{,}880{,}000
A received 2,880,000 votes.
Answer: 2,880,000 votes
4 · Reviewdoes it hold up?

A's 2,880,000 votes out of the 6,400,000 cast is 45 percent, and out of the 8,000,000 electorate only 36 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 45 percent of the 8,000,000 electorate gives 3,600,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 10 hard answer: 3,510,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 1212 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (35%) Share of votes by candidate A B 30% C 15% D 6% Other 4%
Show solution
1 · Understandwhat's really being asked

12,000,000 people could vote and 35 percent did not. Of the votes cast, B took 30 percent, C 15, D 6 and other 4; A is unlabelled. We want A's votes.

Givens
  • There are 12,000,000 eligible voters.
  • 35 percent did not vote.
  • B 30 percent, C 15 percent, D 6 percent, other 4 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 35 percent stayed home, 65 percent voted.
100−35=65100 - 35 = 65
65 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
12,000,000×0.65=7,800,00012{,}000{,}000 \times 0.65 = 7{,}800{,}000
7,800,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−30−15−6−4=45100 - 30 - 15 - 6 - 4 = 45
A took 45 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
7,800,000×0.45=3,510,0007{,}800{,}000 \times 0.45 = 3{,}510{,}000
A received 3,510,000 votes.
Answer: 3,510,000 votes
4 · Reviewdoes it hold up?

A's 3,510,000 votes out of the 7,800,000 cast is 45 percent, and out of the 12,000,000 electorate only 29.25 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 45 percent of the 12,000,000 electorate gives 5,400,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 11 hard answer: 1,800,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 55 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (20%) Share of votes by candidate A B 35% C 11% D 5% Other 4%
Show solution
1 · Understandwhat's really being asked

5,000,000 people could vote and 20 percent did not. Of the votes cast, B took 35 percent, C 11, D 5 and other 4; A is unlabelled. We want A's votes.

Givens
  • There are 5,000,000 eligible voters.
  • 20 percent did not vote.
  • B 35 percent, C 11 percent, D 5 percent, other 4 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 20 percent stayed home, 80 percent voted.
100−20=80100 - 20 = 80
80 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
5,000,000×0.8=4,000,0005{,}000{,}000 \times 0.8 = 4{,}000{,}000
4,000,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−35−11−5−4=45100 - 35 - 11 - 5 - 4 = 45
A took 45 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
4,000,000×0.45=1,800,0004{,}000{,}000 \times 0.45 = 1{,}800{,}000
A received 1,800,000 votes.
Answer: 1,800,000 votes
4 · Reviewdoes it hold up?

A's 1,800,000 votes out of the 4,000,000 cast is 45 percent, and out of the 5,000,000 electorate only 36 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 45 percent of the 5,000,000 electorate gives 2,250,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.
Variant 12 hard answer: 2,700,000 votes

The graphs show the turnout of an election and the share of votes each candidate received. If there are 1010 million eligible voters in all, how many votes did candidate A receive?

Turnout Voted Did not vote (40%) Share of votes by candidate A B 33% C 13% D 5% Other 4%
Show solution
1 · Understandwhat's really being asked

10,000,000 people could vote and 40 percent did not. Of the votes cast, B took 33 percent, C 13, D 5 and other 4; A is unlabelled. We want A's votes.

Givens
  • There are 10,000,000 eligible voters.
  • 40 percent did not vote.
  • B 33 percent, C 13 percent, D 5 percent, other 4 percent of the votes cast.
Unknowns
  • How many votes candidate A received.
Constraints
  • The second graph divides the votes cast, not the electorate.
2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #8 Analyze the Units#16 Count the Complement

The two graphs do not share a whole: the bar graph's 100 percent is the circle graph's voting slice. Spend the first graph to find how many votes were cast, then let the second graph divide that.

3 · Execute4 carry out the plan

1Find the turnout

#16 Count the Complement 6.SP.B.5
If 40 percent stayed home, 60 percent voted.
100−40=60100 - 40 = 60
60 percent of voters turned out.

2Count the votes cast

#8 Analyze the Units 7.RP.A.3
That share of the whole electorate.
10,000,000×0.6=6,000,00010{,}000{,}000 \times 0.6 = 6{,}000{,}000
6,000,000 votes were cast.

3Find A's share of the votes cast

#16 Count the Complement 6.SP.B.5
The five sections make the whole ballot, so A is what the others leave.
100−33−13−5−4=45100 - 33 - 13 - 5 - 4 = 45
A took 45 percent.

4Take that share of the votes cast

#8 Analyze the Units 7.RP.A.3
Not of the electorate -- the bar graph never divided that.
6,000,000×0.45=2,700,0006{,}000{,}000 \times 0.45 = 2{,}700{,}000
A received 2,700,000 votes.
Answer: 2,700,000 votes
4 · Reviewdoes it hold up?

A's 2,700,000 votes out of the 6,000,000 cast is 45 percent, and out of the 10,000,000 electorate only 27 percent -- lower, as it has to be when a fifth of voters stayed home.

Another way: Taking 45 percent of the 10,000,000 electorate gives 4,500,000, which counts votes from people who did not vote.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Applying two percentages, each to its own whole.
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading each unlabelled section as what the others leave.
💡Takeaway. When one graph divides up a slice of another, the second graph's whole hundred percent is only part of the first.