← Take some away and every remaining share goes up · Read and Scale a Data Graph

Take some away and every remaining share goes up · 12 practice problems

6.SP.B.56.RP.A.3

Generated variants — 12

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

Variant 1 easy answer: 42.5%

The circle graph at the right shows Junseo's marbles sorted by color. There are 1010 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (34%) Blue (29%) Yellow Purple (17%)
Show solution
1 · Understandwhat's really being asked

Marbles are 34 percent red, 29 percent blue, 17 percent purple and the rest yellow. There are 10 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 34 percent, blue 29 percent, purple 17 percent.
  • There are 10 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−34−29−17=20100 - 34 - 29 - 17 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
10 marbles are 20 percent of it.
10÷0.2=5010 \div 0.2 = 50
50 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
50×0.34=17,50−10=4050 \times 0.34 = 17, \quad 50 - 10 = 40
17 reds out of 40.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
17÷40×100=42.517 \div 40 \times 100 = 42.5
Red is now 42.5 percent.
Answer: 42.5%
4 · Reviewdoes it hold up?

The reds never changed -- still 17 of them -- but their share rose from 34 to 42.5 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 34 out of that -- the same 42.5 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 2 easy answer: 52%

The circle graph at the right shows Junseo's marbles sorted by color. There are 2525 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (39%) Blue (22%) Yellow Purple (14%)
Show solution
1 · Understandwhat's really being asked

Marbles are 39 percent red, 22 percent blue, 14 percent purple and the rest yellow. There are 25 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 39 percent, blue 22 percent, purple 14 percent.
  • There are 25 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−39−22−14=25100 - 39 - 22 - 14 = 25
Yellow is 25 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
25 marbles are 25 percent of it.
25÷0.25=10025 \div 0.25 = 100
100 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
100×0.39=39,100−25=75100 \times 0.39 = 39, \quad 100 - 25 = 75
39 reds out of 75.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
39÷75×100=5239 \div 75 \times 100 = 52
Red is now 52 percent.
Answer: 52%
4 · Reviewdoes it hold up?

The reds never changed -- still 39 of them -- but their share rose from 39 to 52 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 75 percent of the old total, so red's new share is 39 out of that -- the same 52 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 3 easy answer: 50%

The circle graph at the right shows Junseo's marbles sorted by color. There are 2020 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (40%) Blue (25%) Yellow Purple (15%)
Show solution
1 · Understandwhat's really being asked

Marbles are 40 percent red, 25 percent blue, 15 percent purple and the rest yellow. There are 20 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 40 percent, blue 25 percent, purple 15 percent.
  • There are 20 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−40−25−15=20100 - 40 - 25 - 15 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
20 marbles are 20 percent of it.
20÷0.2=10020 \div 0.2 = 100
100 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
100×0.4=40,100−20=80100 \times 0.4 = 40, \quad 100 - 20 = 80
40 reds out of 80.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
40÷80×100=5040 \div 80 \times 100 = 50
Red is now 50 percent.
Answer: 50%
4 · Reviewdoes it hold up?

The reds never changed -- still 40 of them -- but their share rose from 40 to 50 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 40 out of that -- the same 50 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 4 easy answer: 50%

The circle graph at the right shows Junseo's marbles sorted by color. There are 1010 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (40%) Blue (30%) Yellow Purple (10%)
Show solution
1 · Understandwhat's really being asked

Marbles are 40 percent red, 30 percent blue, 10 percent purple and the rest yellow. There are 10 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 40 percent, blue 30 percent, purple 10 percent.
  • There are 10 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−40−30−10=20100 - 40 - 30 - 10 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
10 marbles are 20 percent of it.
10÷0.2=5010 \div 0.2 = 50
50 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
50×0.4=20,50−10=4050 \times 0.4 = 20, \quad 50 - 10 = 40
20 reds out of 40.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
20÷40×100=5020 \div 40 \times 100 = 50
Red is now 50 percent.
Answer: 50%
4 · Reviewdoes it hold up?

The reds never changed -- still 20 of them -- but their share rose from 40 to 50 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 40 out of that -- the same 50 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 5 medium answer: 52.5%

The circle graph at the right shows Junseo's marbles sorted by color. There are 3030 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (42%) Blue (26%) Yellow Purple (12%)
Show solution
1 · Understandwhat's really being asked

Marbles are 42 percent red, 26 percent blue, 12 percent purple and the rest yellow. There are 30 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 42 percent, blue 26 percent, purple 12 percent.
  • There are 30 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−42−26−12=20100 - 42 - 26 - 12 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
30 marbles are 20 percent of it.
30÷0.2=15030 \div 0.2 = 150
150 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
150×0.42=63,150−30=120150 \times 0.42 = 63, \quad 150 - 30 = 120
63 reds out of 120.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
63÷120×100=52.563 \div 120 \times 100 = 52.5
Red is now 52.5 percent.
Answer: 52.5%
4 · Reviewdoes it hold up?

The reds never changed -- still 63 of them -- but their share rose from 42 to 52.5 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 42 out of that -- the same 52.5 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 6 medium answer: 56%

The circle graph at the right shows Junseo's marbles sorted by color. There are 2525 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (42%) Blue (24%) Yellow Purple (9%)
Show solution
1 · Understandwhat's really being asked

Marbles are 42 percent red, 24 percent blue, 9 percent purple and the rest yellow. There are 25 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 42 percent, blue 24 percent, purple 9 percent.
  • There are 25 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−42−24−9=25100 - 42 - 24 - 9 = 25
Yellow is 25 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
25 marbles are 25 percent of it.
25÷0.25=10025 \div 0.25 = 100
100 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
100×0.42=42,100−25=75100 \times 0.42 = 42, \quad 100 - 25 = 75
42 reds out of 75.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
42÷75×100=5642 \div 75 \times 100 = 56
Red is now 56 percent.
Answer: 56%
4 · Reviewdoes it hold up?

The reds never changed -- still 42 of them -- but their share rose from 42 to 56 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 75 percent of the old total, so red's new share is 42 out of that -- the same 56 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 7 medium answer: 52.5%

The circle graph at the right shows Junseo's marbles sorted by color. There are 3030 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (42%) Blue (27%) Yellow Purple (11%)
Show solution
1 · Understandwhat's really being asked

Marbles are 42 percent red, 27 percent blue, 11 percent purple and the rest yellow. There are 30 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 42 percent, blue 27 percent, purple 11 percent.
  • There are 30 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−42−27−11=20100 - 42 - 27 - 11 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
30 marbles are 20 percent of it.
30÷0.2=15030 \div 0.2 = 150
150 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
150×0.42=63,150−30=120150 \times 0.42 = 63, \quad 150 - 30 = 120
63 reds out of 120.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
63÷120×100=52.563 \div 120 \times 100 = 52.5
Red is now 52.5 percent.
Answer: 52.5%
4 · Reviewdoes it hold up?

The reds never changed -- still 63 of them -- but their share rose from 42 to 52.5 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 42 out of that -- the same 52.5 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 8 medium answer: 52.5%

The circle graph at the right shows Junseo's marbles sorted by color. There are 3030 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (42%) Blue (32%) Yellow Purple (6%)
Show solution
1 · Understandwhat's really being asked

Marbles are 42 percent red, 32 percent blue, 6 percent purple and the rest yellow. There are 30 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 42 percent, blue 32 percent, purple 6 percent.
  • There are 30 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−42−32−6=20100 - 42 - 32 - 6 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
30 marbles are 20 percent of it.
30÷0.2=15030 \div 0.2 = 150
150 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
150×0.42=63,150−30=120150 \times 0.42 = 63, \quad 150 - 30 = 120
63 reds out of 120.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
63÷120×100=52.563 \div 120 \times 100 = 52.5
Red is now 52.5 percent.
Answer: 52.5%
4 · Reviewdoes it hold up?

The reds never changed -- still 63 of them -- but their share rose from 42 to 52.5 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 42 out of that -- the same 52.5 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 9 hard answer: 55%

The circle graph at the right shows Junseo's marbles sorted by color. There are 2020 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (44%) Blue (31%) Yellow Purple (5%)
Show solution
1 · Understandwhat's really being asked

Marbles are 44 percent red, 31 percent blue, 5 percent purple and the rest yellow. There are 20 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 44 percent, blue 31 percent, purple 5 percent.
  • There are 20 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−44−31−5=20100 - 44 - 31 - 5 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
20 marbles are 20 percent of it.
20÷0.2=10020 \div 0.2 = 100
100 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
100×0.44=44,100−20=80100 \times 0.44 = 44, \quad 100 - 20 = 80
44 reds out of 80.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
44÷80×100=5544 \div 80 \times 100 = 55
Red is now 55 percent.
Answer: 55%
4 · Reviewdoes it hold up?

The reds never changed -- still 44 of them -- but their share rose from 44 to 55 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 44 out of that -- the same 55 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 10 hard answer: 57.5%

The circle graph at the right shows Junseo's marbles sorted by color. There are 2020 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (46%) Blue (15%) Yellow Purple (19%)
Show solution
1 · Understandwhat's really being asked

Marbles are 46 percent red, 15 percent blue, 19 percent purple and the rest yellow. There are 20 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 46 percent, blue 15 percent, purple 19 percent.
  • There are 20 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−46−15−19=20100 - 46 - 15 - 19 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
20 marbles are 20 percent of it.
20÷0.2=10020 \div 0.2 = 100
100 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
100×0.46=46,100−20=80100 \times 0.46 = 46, \quad 100 - 20 = 80
46 reds out of 80.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
46÷80×100=57.546 \div 80 \times 100 = 57.5
Red is now 57.5 percent.
Answer: 57.5%
4 · Reviewdoes it hold up?

The reds never changed -- still 46 of them -- but their share rose from 46 to 57.5 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 46 out of that -- the same 57.5 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 11 hard answer: 60%

The circle graph at the right shows Junseo's marbles sorted by color. There are 1010 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (48%) Blue (19%) Yellow Purple (13%)
Show solution
1 · Understandwhat's really being asked

Marbles are 48 percent red, 19 percent blue, 13 percent purple and the rest yellow. There are 10 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 48 percent, blue 19 percent, purple 13 percent.
  • There are 10 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−48−19−13=20100 - 48 - 19 - 13 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
10 marbles are 20 percent of it.
10÷0.2=5010 \div 0.2 = 50
50 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
50×0.48=24,50−10=4050 \times 0.48 = 24, \quad 50 - 10 = 40
24 reds out of 40.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
24÷40×100=6024 \div 40 \times 100 = 60
Red is now 60 percent.
Answer: 60%
4 · Reviewdoes it hold up?

The reds never changed -- still 24 of them -- but their share rose from 48 to 60 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 48 out of that -- the same 60 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.
Variant 12 hard answer: 55%

The circle graph at the right shows Junseo's marbles sorted by color. There are 5050 yellow marbles. If Junseo gives every yellow marble away, what percent of his remaining marbles are red?

Share of marbles by color Red (44%) Blue (18%) Yellow Purple (18%)
Show solution
1 · Understandwhat's really being asked

Marbles are 44 percent red, 18 percent blue, 18 percent purple and the rest yellow. There are 50 yellow ones, and they are all given away. We want the red share of what is left.

Givens
  • Red 44 percent, blue 18 percent, purple 18 percent.
  • There are 50 yellow marbles.
  • Every yellow marble is given away.
Unknowns
  • The percent of the remaining marbles that are red.
Constraints
  • Only yellow marbles leave; the red ones are untouched.
2 · Planchoose the strategy

#8 Analyze the Units · also uses: #16 Count the Complement#7 Identify Subproblems

Percentages cannot be edited when the whole changes. Turn them into counts, take the yellow marbles out of the count, and work out a fresh percentage from what is left.

3 · Execute4 carry out the plan

1Find the yellow share

#16 Count the Complement 6.SP.B.5
The four colours make the whole collection.
100−44−18−18=20100 - 44 - 18 - 18 = 20
Yellow is 20 percent.

2Get the whole collection

#7 Identify Subproblems 6.RP.A.3
50 marbles are 20 percent of it.
50÷0.2=25050 \div 0.2 = 250
250 marbles in all.

3Count the reds and what is left

#7 Identify Subproblems 6.RP.A.3
The reds do not move, but the total shrinks by the yellows.
250×0.44=110,250−50=200250 \times 0.44 = 110, \quad 250 - 50 = 200
110 reds out of 200.

4Take the new percentage

#8 Analyze the Units 6.RP.A.3
Red count over the new total.
110÷200×100=55110 \div 200 \times 100 = 55
Red is now 55 percent.
Answer: 55%
4 · Reviewdoes it hold up?

The reds never changed -- still 110 of them -- but their share rose from 44 to 55 percent, which is what taking marbles away must do.

Another way: Rescaling the surviving percentages works too: red, blue and purple make 80 percent of the old total, so red's new share is 44 out of that -- the same 55 percent.

Standardsmin grade 6
  • 6.SP.B.5 Summarize numerical data sets in relation to context — Reading the unlabelled slice and the counts behind the graph.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Recomputing a percentage after the whole changes.
💡Takeaway. A percentage can change without anything happening to the thing it counts. What changed was everything else.