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Percentages of different groups compare only through counts · 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: 120 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 150150 boys in all, and 66 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (38%) Games (24%) Reading (20%) Other (18%) Share of girls by hobby Music (35%) Reading (30%) Sports (25%) Other (10%)
Show solution
1 · Understandwhat's really being asked

150 boys are split by hobby, 20 percent of them reading. The girls' graph gives reading 30 percent, and 6 more girls than boys read. We want the number of girls.

Givens
  • There are 150 boys.
  • Reading is 20 percent of the boys and 30 percent of the girls.
  • 6 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

20 percent and 30 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
20 percent of the 150 boys.
150×0.2=30150 \times 0.2 = 30
30 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
6 more than the boys.
30+6=3630 + 6 = 36
36 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 36 girls are 30 percent of all the girls.
girls×0.3=36\text{girls} \times 0.3 = 36
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
36÷0.3=12036 \div 0.3 = 120
There are 120 girls.
Answer: 120 girls
4 · Reviewdoes it hold up?

30 percent of 120 girls is 36 readers, which is 6 more than the boys' 30 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read more, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 2 easy answer: 260 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 150150 boys in all, and 2020 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (32%) Games (22%) Reading (30%) Other (16%) Share of girls by hobby Music (15%) Reading (25%) Sports (15%) Other (45%)
Show solution
1 · Understandwhat's really being asked

150 boys are split by hobby, 30 percent of them reading. The girls' graph gives reading 25 percent, and 20 more girls than boys read. We want the number of girls.

Givens
  • There are 150 boys.
  • Reading is 30 percent of the boys and 25 percent of the girls.
  • 20 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

30 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
30 percent of the 150 boys.
150×0.3=45150 \times 0.3 = 45
45 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
20 more than the boys.
45+20=6545 + 20 = 65
65 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 65 girls are 25 percent of all the girls.
girls×0.25=65\text{girls} \times 0.25 = 65
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
65÷0.25=26065 \div 0.25 = 260
There are 260 girls.
Answer: 260 girls
4 · Reviewdoes it hold up?

25 percent of 260 girls is 65 readers, which is 20 more than the boys' 45 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 3 easy answer: 208 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 160160 boys in all, and 2020 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (36%) Games (36%) Reading (20%) Other (8%) Share of girls by hobby Music (15%) Reading (25%) Sports (30%) Other (30%)
Show solution
1 · Understandwhat's really being asked

160 boys are split by hobby, 20 percent of them reading. The girls' graph gives reading 25 percent, and 20 more girls than boys read. We want the number of girls.

Givens
  • There are 160 boys.
  • Reading is 20 percent of the boys and 25 percent of the girls.
  • 20 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

20 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
20 percent of the 160 boys.
160×0.2=32160 \times 0.2 = 32
32 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
20 more than the boys.
32+20=5232 + 20 = 52
52 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 52 girls are 25 percent of all the girls.
girls×0.25=52\text{girls} \times 0.25 = 52
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
52÷0.25=20852 \div 0.25 = 208
There are 208 girls.
Answer: 208 girls
4 · Reviewdoes it hold up?

25 percent of 208 girls is 52 readers, which is 20 more than the boys' 32 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read more, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 4 easy answer: 305 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 200200 boys in all, and 55 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (28%) Games (24%) Reading (28%) Other (20%) Share of girls by hobby Music (35%) Reading (20%) Sports (35%) Other (10%)
Show solution
1 · Understandwhat's really being asked

200 boys are split by hobby, 28 percent of them reading. The girls' graph gives reading 20 percent, and 5 more girls than boys read. We want the number of girls.

Givens
  • There are 200 boys.
  • Reading is 28 percent of the boys and 20 percent of the girls.
  • 5 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

28 percent and 20 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
28 percent of the 200 boys.
200×0.28=56200 \times 0.28 = 56
56 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
5 more than the boys.
56+5=6156 + 5 = 61
61 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 61 girls are 20 percent of all the girls.
girls×0.2=61\text{girls} \times 0.2 = 61
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
61÷0.2=30561 \div 0.2 = 305
There are 305 girls.
Answer: 305 girls
4 · Reviewdoes it hold up?

20 percent of 305 girls is 61 readers, which is 5 more than the boys' 56 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 5 medium answer: 360 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 200200 boys in all, and 66 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (28%) Games (28%) Reading (24%) Other (20%) Share of girls by hobby Music (15%) Reading (15%) Sports (25%) Other (45%)
Show solution
1 · Understandwhat's really being asked

200 boys are split by hobby, 24 percent of them reading. The girls' graph gives reading 15 percent, and 6 more girls than boys read. We want the number of girls.

Givens
  • There are 200 boys.
  • Reading is 24 percent of the boys and 15 percent of the girls.
  • 6 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

24 percent and 15 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
24 percent of the 200 boys.
200×0.24=48200 \times 0.24 = 48
48 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
6 more than the boys.
48+6=5448 + 6 = 54
54 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 54 girls are 15 percent of all the girls.
girls×0.15=54\text{girls} \times 0.15 = 54
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
54÷0.15=36054 \div 0.15 = 360
There are 360 girls.
Answer: 360 girls
4 · Reviewdoes it hold up?

15 percent of 360 girls is 54 readers, which is 6 more than the boys' 48 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 6 medium answer: 248 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 200200 boys in all, and 66 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (34%) Games (20%) Reading (28%) Other (18%) Share of girls by hobby Music (20%) Reading (25%) Sports (30%) Other (25%)
Show solution
1 · Understandwhat's really being asked

200 boys are split by hobby, 28 percent of them reading. The girls' graph gives reading 25 percent, and 6 more girls than boys read. We want the number of girls.

Givens
  • There are 200 boys.
  • Reading is 28 percent of the boys and 25 percent of the girls.
  • 6 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

28 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
28 percent of the 200 boys.
200×0.28=56200 \times 0.28 = 56
56 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
6 more than the boys.
56+6=6256 + 6 = 62
62 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 62 girls are 25 percent of all the girls.
girls×0.25=62\text{girls} \times 0.25 = 62
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
62÷0.25=24862 \div 0.25 = 248
There are 248 girls.
Answer: 248 girls
4 · Reviewdoes it hold up?

25 percent of 248 girls is 62 readers, which is 6 more than the boys' 56 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 7 medium answer: 304 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 250250 boys in all, and 66 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (26%) Games (30%) Reading (28%) Other (16%) Share of girls by hobby Music (15%) Reading (25%) Sports (35%) Other (25%)
Show solution
1 · Understandwhat's really being asked

250 boys are split by hobby, 28 percent of them reading. The girls' graph gives reading 25 percent, and 6 more girls than boys read. We want the number of girls.

Givens
  • There are 250 boys.
  • Reading is 28 percent of the boys and 25 percent of the girls.
  • 6 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

28 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
28 percent of the 250 boys.
250×0.28=70250 \times 0.28 = 70
70 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
6 more than the boys.
70+6=7670 + 6 = 76
76 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 76 girls are 25 percent of all the girls.
girls×0.25=76\text{girls} \times 0.25 = 76
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
76÷0.25=30476 \div 0.25 = 304
There are 304 girls.
Answer: 304 girls
4 · Reviewdoes it hold up?

25 percent of 304 girls is 76 readers, which is 6 more than the boys' 70 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 8 medium answer: 248 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 250250 boys in all, and 1212 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (34%) Games (32%) Reading (20%) Other (14%) Share of girls by hobby Music (25%) Reading (25%) Sports (20%) Other (30%)
Show solution
1 · Understandwhat's really being asked

250 boys are split by hobby, 20 percent of them reading. The girls' graph gives reading 25 percent, and 12 more girls than boys read. We want the number of girls.

Givens
  • There are 250 boys.
  • Reading is 20 percent of the boys and 25 percent of the girls.
  • 12 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

20 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
20 percent of the 250 boys.
250×0.2=50250 \times 0.2 = 50
50 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
12 more than the boys.
50+12=6250 + 12 = 62
62 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 62 girls are 25 percent of all the girls.
girls×0.25=62\text{girls} \times 0.25 = 62
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
62÷0.25=24862 \div 0.25 = 248
There are 248 girls.
Answer: 248 girls
4 · Reviewdoes it hold up?

25 percent of 248 girls is 62 readers, which is 12 more than the boys' 50 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read more, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 9 hard answer: 484 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 300300 boys in all, and 2525 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (26%) Games (22%) Reading (32%) Other (20%) Share of girls by hobby Music (30%) Reading (25%) Sports (20%) Other (25%)
Show solution
1 · Understandwhat's really being asked

300 boys are split by hobby, 32 percent of them reading. The girls' graph gives reading 25 percent, and 25 more girls than boys read. We want the number of girls.

Givens
  • There are 300 boys.
  • Reading is 32 percent of the boys and 25 percent of the girls.
  • 25 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

32 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
32 percent of the 300 boys.
300×0.32=96300 \times 0.32 = 96
96 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
25 more than the boys.
96+25=12196 + 25 = 121
121 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 121 girls are 25 percent of all the girls.
girls×0.25=121\text{girls} \times 0.25 = 121
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
121÷0.25=484121 \div 0.25 = 484
There are 484 girls.
Answer: 484 girls
4 · Reviewdoes it hold up?

25 percent of 484 girls is 121 readers, which is 25 more than the boys' 96 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 10 hard answer: 530 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 400400 boys in all, and 77 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (22%) Games (34%) Reading (38%) Other (6%) Share of girls by hobby Music (30%) Reading (30%) Sports (30%) Other (10%)
Show solution
1 · Understandwhat's really being asked

400 boys are split by hobby, 38 percent of them reading. The girls' graph gives reading 30 percent, and 7 more girls than boys read. We want the number of girls.

Givens
  • There are 400 boys.
  • Reading is 38 percent of the boys and 30 percent of the girls.
  • 7 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

38 percent and 30 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
38 percent of the 400 boys.
400×0.38=152400 \times 0.38 = 152
152 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
7 more than the boys.
152+7=159152 + 7 = 159
159 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 159 girls are 30 percent of all the girls.
girls×0.3=159\text{girls} \times 0.3 = 159
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
159÷0.3=530159 \div 0.3 = 530
There are 530 girls.
Answer: 530 girls
4 · Reviewdoes it hold up?

30 percent of 530 girls is 159 readers, which is 7 more than the boys' 152 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 11 hard answer: 528 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 500500 boys in all, and 1212 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (36%) Games (38%) Reading (24%) Other (2%) Share of girls by hobby Music (20%) Reading (25%) Sports (20%) Other (35%)
Show solution
1 · Understandwhat's really being asked

500 boys are split by hobby, 24 percent of them reading. The girls' graph gives reading 25 percent, and 12 more girls than boys read. We want the number of girls.

Givens
  • There are 500 boys.
  • Reading is 24 percent of the boys and 25 percent of the girls.
  • 12 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

24 percent and 25 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
24 percent of the 500 boys.
500×0.24=120500 \times 0.24 = 120
120 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
12 more than the boys.
120+12=132120 + 12 = 132
132 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 132 girls are 25 percent of all the girls.
girls×0.25=132\text{girls} \times 0.25 = 132
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
132÷0.25=528132 \div 0.25 = 528
There are 528 girls.
Answer: 528 girls
4 · Reviewdoes it hold up?

25 percent of 528 girls is 132 readers, which is 12 more than the boys' 120 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read more, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.
Variant 12 hard answer: 580 girls

The circle graphs show the hobbies of the sixth-grade boys and girls at Jiwoo's school. There are 500500 boys in all, and 66 more girls than boys give reading as their hobby. How many girls are there in all?

Share of boys by hobby Sports (28%) Games (24%) Reading (22%) Other (26%) Share of girls by hobby Music (30%) Reading (20%) Sports (15%) Other (35%)
Show solution
1 · Understandwhat's really being asked

500 boys are split by hobby, 22 percent of them reading. The girls' graph gives reading 20 percent, and 6 more girls than boys read. We want the number of girls.

Givens
  • There are 500 boys.
  • Reading is 22 percent of the boys and 20 percent of the girls.
  • 6 more girls than boys give reading as their hobby.
Unknowns
  • The total number of girls.
Constraints
  • The two graphs describe different groups, so their percentages are not on the same base.
2 · Planchoose the strategy

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

22 percent and 20 percent sit on different totals and cannot be compared. Counts can. Turn the boys' percentage into a count, add the gap to get the girls' count, and let that count name the girls' total.

3 · Execute4 carry out the plan

1Count the boys who read

#8 Analyze the Units 7.RP.A.3
22 percent of the 500 boys.
500×0.22=110500 \times 0.22 = 110
110 boys read.

2Count the girls who read

#15 Organize Information in More Ways 7.RP.A.3
6 more than the boys.
110+6=116110 + 6 = 116
116 girls read.

3Use that as a share of the girls

#13 Convert to Algebra 6.EE.B.7
Those 116 girls are 20 percent of all the girls.
girls×0.2=116\text{girls} \times 0.2 = 116
One equation, one unknown.

4Solve for the total

#13 Convert to Algebra 6.EE.B.7
Divide by the share.
116÷0.2=580116 \div 0.2 = 580
There are 580 girls.
Answer: 580 girls
4 · Reviewdoes it hold up?

20 percent of 580 girls is 116 readers, which is 6 more than the boys' 110 -- both conditions hold.

Another way: Comparing the percentages directly would say girls read less, but a percentage is not a count until the group size is known.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each group's percentage into a count.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the unknown total from a known share.
💡Takeaway. Two percentages of two different groups say nothing to each other. Counts do.