← A difference in counts and a difference in shares fix the whole · Read and Scale a Data Graph

A difference in counts and a difference in shares fix the whole · 12 practice problems

7.RP.A.36.EE.B.7

Generated variants — 12

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

Variant 1 easy answer: 20 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 2424 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (35%) Chinese (20%) Japanese (14%) German (5%) Other (26%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 35, Chinese 20, Japanese 14, German 5 and other 26 percent -- and 24 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 35 percent, Chinese 20 percent, Japanese 14 percent, German 5 percent, other 26 percent.
  • 24 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 20 percent and Japanese 14, so the extra students are the difference.
20−14=6%20 - 14 = 6\%
Those 24 students are 6 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
24 students are 6 percent of the total.
24÷0.06=40024 \div 0.06 = 400
400 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
5 percent of the whole school.
400×0.05=20400 \times 0.05 = 20
20 students want German.
Answer: 20 students
4 · Reviewdoes it hold up?

Checking the gap: 20 percent of 400 is 80 and 14 percent is 56, a difference of 24 exactly.

Another way: German's 5 percent stands to the gap's 6 percent as 5:65 : 6, so scaling the 24 extra students in that ratio gives 20 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 2 easy answer: 12 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 2020 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (36%) Chinese (26%) Japanese (16%) German (6%) Other (16%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 36, Chinese 26, Japanese 16, German 6 and other 16 percent -- and 20 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 36 percent, Chinese 26 percent, Japanese 16 percent, German 6 percent, other 16 percent.
  • 20 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 26 percent and Japanese 16, so the extra students are the difference.
26−16=10%26 - 16 = 10\%
Those 20 students are 10 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
20 students are 10 percent of the total.
20÷0.1=20020 \div 0.1 = 200
200 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
6 percent of the whole school.
200×0.06=12200 \times 0.06 = 12
12 students want German.
Answer: 12 students
4 · Reviewdoes it hold up?

Checking the gap: 26 percent of 200 is 52 and 16 percent is 32, a difference of 20 exactly.

Another way: German's 6 percent stands to the gap's 10 percent as 6:106 : 10, so scaling the 20 extra students in that ratio gives 12 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 3 easy answer: 4 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 88 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (40%) Chinese (30%) Japanese (20%) German (5%) Other (5%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 40, Chinese 30, Japanese 20, German 5 and other 5 percent -- and 8 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 40 percent, Chinese 30 percent, Japanese 20 percent, German 5 percent, other 5 percent.
  • 8 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 30 percent and Japanese 20, so the extra students are the difference.
30−20=10%30 - 20 = 10\%
Those 8 students are 10 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
8 students are 10 percent of the total.
8÷0.1=808 \div 0.1 = 80
80 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
5 percent of the whole school.
80×0.05=480 \times 0.05 = 4
4 students want German.
Answer: 4 students
4 · Reviewdoes it hold up?

Checking the gap: 30 percent of 80 is 24 and 20 percent is 16, a difference of 8 exactly.

Another way: German's 5 percent stands to the gap's 10 percent as 5:105 : 10, so scaling the 8 extra students in that ratio gives 4 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 4 easy answer: 6 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 1212 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (41%) Chinese (30%) Japanese (24%) German (3%) Other (2%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 41, Chinese 30, Japanese 24, German 3 and other 2 percent -- and 12 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 41 percent, Chinese 30 percent, Japanese 24 percent, German 3 percent, other 2 percent.
  • 12 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 30 percent and Japanese 24, so the extra students are the difference.
30−24=6%30 - 24 = 6\%
Those 12 students are 6 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
12 students are 6 percent of the total.
12÷0.06=20012 \div 0.06 = 200
200 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
3 percent of the whole school.
200×0.03=6200 \times 0.03 = 6
6 students want German.
Answer: 6 students
4 · Reviewdoes it hold up?

Checking the gap: 30 percent of 200 is 60 and 24 percent is 48, a difference of 12 exactly.

Another way: German's 3 percent stands to the gap's 6 percent as 3:63 : 6, so scaling the 12 extra students in that ratio gives 6 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 5 medium answer: 45 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 2525 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (42%) Chinese (24%) Japanese (19%) German (9%) Other (6%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 42, Chinese 24, Japanese 19, German 9 and other 6 percent -- and 25 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 42 percent, Chinese 24 percent, Japanese 19 percent, German 9 percent, other 6 percent.
  • 25 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 24 percent and Japanese 19, so the extra students are the difference.
24−19=5%24 - 19 = 5\%
Those 25 students are 5 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
25 students are 5 percent of the total.
25÷0.05=50025 \div 0.05 = 500
500 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
9 percent of the whole school.
500×0.09=45500 \times 0.09 = 45
45 students want German.
Answer: 45 students
4 · Reviewdoes it hold up?

Checking the gap: 24 percent of 500 is 120 and 19 percent is 95, a difference of 25 exactly.

Another way: German's 9 percent stands to the gap's 5 percent as 9:59 : 5, so scaling the 25 extra students in that ratio gives 45 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 6 medium answer: 30 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 2424 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (43%) Chinese (21%) Japanese (17%) German (5%) Other (14%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 43, Chinese 21, Japanese 17, German 5 and other 14 percent -- and 24 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 43 percent, Chinese 21 percent, Japanese 17 percent, German 5 percent, other 14 percent.
  • 24 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 21 percent and Japanese 17, so the extra students are the difference.
21−17=4%21 - 17 = 4\%
Those 24 students are 4 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
24 students are 4 percent of the total.
24÷0.04=60024 \div 0.04 = 600
600 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
5 percent of the whole school.
600×0.05=30600 \times 0.05 = 30
30 students want German.
Answer: 30 students
4 · Reviewdoes it hold up?

Checking the gap: 21 percent of 600 is 126 and 17 percent is 102, a difference of 24 exactly.

Another way: German's 5 percent stands to the gap's 4 percent as 5:45 : 4, so scaling the 24 extra students in that ratio gives 30 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 7 medium answer: 48 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 1212 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (48%) Chinese (24%) Japanese (23%) German (4%) Other (1%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 48, Chinese 24, Japanese 23, German 4 and other 1 percent -- and 12 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 48 percent, Chinese 24 percent, Japanese 23 percent, German 4 percent, other 1 percent.
  • 12 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 24 percent and Japanese 23, so the extra students are the difference.
24−23=1%24 - 23 = 1\%
Those 12 students are 1 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
12 students are 1 percent of the total.
12÷0.01=120012 \div 0.01 = 1200
1200 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
4 percent of the whole school.
1200×0.04=481200 \times 0.04 = 48
48 students want German.
Answer: 48 students
4 · Reviewdoes it hold up?

Checking the gap: 24 percent of 1200 is 288 and 23 percent is 276, a difference of 12 exactly.

Another way: German's 4 percent stands to the gap's 1 percent as 4:14 : 1, so scaling the 12 extra students in that ratio gives 48 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 8 medium answer: 18 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 3030 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (49%) Chinese (26%) Japanese (21%) German (3%) Other (1%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 49, Chinese 26, Japanese 21, German 3 and other 1 percent -- and 30 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 49 percent, Chinese 26 percent, Japanese 21 percent, German 3 percent, other 1 percent.
  • 30 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 26 percent and Japanese 21, so the extra students are the difference.
26−21=5%26 - 21 = 5\%
Those 30 students are 5 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
30 students are 5 percent of the total.
30÷0.05=60030 \div 0.05 = 600
600 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
3 percent of the whole school.
600×0.03=18600 \times 0.03 = 18
18 students want German.
Answer: 18 students
4 · Reviewdoes it hold up?

Checking the gap: 26 percent of 600 is 156 and 21 percent is 126, a difference of 30 exactly.

Another way: German's 3 percent stands to the gap's 5 percent as 3:53 : 5, so scaling the 30 extra students in that ratio gives 18 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 9 hard answer: 16 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 1212 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (50%) Chinese (21%) Japanese (15%) German (8%) Other (6%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 50, Chinese 21, Japanese 15, German 8 and other 6 percent -- and 12 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 50 percent, Chinese 21 percent, Japanese 15 percent, German 8 percent, other 6 percent.
  • 12 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 21 percent and Japanese 15, so the extra students are the difference.
21−15=6%21 - 15 = 6\%
Those 12 students are 6 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
12 students are 6 percent of the total.
12÷0.06=20012 \div 0.06 = 200
200 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
8 percent of the whole school.
200×0.08=16200 \times 0.08 = 16
16 students want German.
Answer: 16 students
4 · Reviewdoes it hold up?

Checking the gap: 21 percent of 200 is 42 and 15 percent is 30, a difference of 12 exactly.

Another way: German's 8 percent stands to the gap's 6 percent as 8:68 : 6, so scaling the 12 extra students in that ratio gives 16 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 10 hard answer: 6 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 2020 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (50%) Chinese (23%) Japanese (13%) German (3%) Other (11%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 50, Chinese 23, Japanese 13, German 3 and other 11 percent -- and 20 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 50 percent, Chinese 23 percent, Japanese 13 percent, German 3 percent, other 11 percent.
  • 20 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 23 percent and Japanese 13, so the extra students are the difference.
23−13=10%23 - 13 = 10\%
Those 20 students are 10 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
20 students are 10 percent of the total.
20÷0.1=20020 \div 0.1 = 200
200 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
3 percent of the whole school.
200×0.03=6200 \times 0.03 = 6
6 students want German.
Answer: 6 students
4 · Reviewdoes it hold up?

Checking the gap: 23 percent of 200 is 46 and 13 percent is 26, a difference of 20 exactly.

Another way: German's 3 percent stands to the gap's 10 percent as 3:103 : 10, so scaling the 20 extra students in that ratio gives 6 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 11 hard answer: 21 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 2424 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (52%) Chinese (22%) Japanese (14%) German (7%) Other (5%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 52, Chinese 22, Japanese 14, German 7 and other 5 percent -- and 24 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 52 percent, Chinese 22 percent, Japanese 14 percent, German 7 percent, other 5 percent.
  • 24 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 22 percent and Japanese 14, so the extra students are the difference.
22−14=8%22 - 14 = 8\%
Those 24 students are 8 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
24 students are 8 percent of the total.
24÷0.08=30024 \div 0.08 = 300
300 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
7 percent of the whole school.
300×0.07=21300 \times 0.07 = 21
21 students want German.
Answer: 21 students
4 · Reviewdoes it hold up?

Checking the gap: 22 percent of 300 is 66 and 14 percent is 42, a difference of 24 exactly.

Another way: German's 7 percent stands to the gap's 8 percent as 7:87 : 8, so scaling the 24 extra students in that ratio gives 21 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.
Variant 12 hard answer: 21 students

The circle graph at the right shows which foreign language students at Yeona's school want to learn. There are 3030 more students who want Chinese than students who want Japanese. How many students want to learn German?

Share of students by the foreign language they want to learn English (57%) Chinese (21%) Japanese (11%) German (7%) Other (4%)
Show solution
1 · Understandwhat's really being asked

A circle graph gives every language's share -- English 57, Chinese 21, Japanese 11, German 7 and other 4 percent -- and 30 more students want Chinese than Japanese. We want the number wanting German.

Givens
  • English 57 percent, Chinese 21 percent, Japanese 11 percent, German 7 percent, other 4 percent.
  • 30 more students want Chinese than Japanese.
Unknowns
  • How many students want to learn German.
Constraints
  • The graph gives no counts at all, only shares.
2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #8 Analyze the Units#7 Identify Subproblems

The graph has shares and no counts; the sentence has a count and no share. They meet at the Chinese-Japanese difference: subtracting the two percentages says what share those extra students are, and that fixes the size of the school.

3 · Execute3 carry out the plan

1Turn the gap into a share

#8 Analyze the Units 7.RP.A.3
Chinese is 21 percent and Japanese 11, so the extra students are the difference.
21−11=10%21 - 11 = 10\%
Those 30 students are 10 percent of the school.

2Find the whole school

#13 Convert to Algebra 6.EE.B.7
30 students are 10 percent of the total.
30÷0.1=30030 \div 0.1 = 300
300 students in all.

3Take German's share of it

#7 Identify Subproblems 7.RP.A.3
7 percent of the whole school.
300×0.07=21300 \times 0.07 = 21
21 students want German.
Answer: 21 students
4 · Reviewdoes it hold up?

Checking the gap: 21 percent of 300 is 63 and 11 percent is 33, a difference of 30 exactly.

Another way: German's 7 percent stands to the gap's 10 percent as 7:107 : 10, so scaling the 30 extra students in that ratio gives 21 straight away -- the same reasoning without naming the school's size.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Moving between counts and percentages of the same whole.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the school's size from one known count.
💡Takeaway. One real count is all a graph of percentages needs. Find where the count and the shares describe the same thing.