← Sharing in a ratio cuts the whole into as many parts as the terms add to · Proportion and Proportional Division

Sharing in a ratio cuts the whole into as many parts as the terms add to · 12 practice problems

7.RP.A.36.RP.A.3

Generated variants — 12

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

Variant 1 easy answer: 240,000 won

Eunji invested 800,000800{,}000 won and Hyunsu invested 1,200,0001{,}200{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 600,000600{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 800,000 and 1,200,000 won and share a profit of 600,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 800,000 won and Hyunsu 1,200,000 won.
  • The profit is 600,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 800,000 to 1,200,000 is 2 to 3.
800,000:1,200,000=2:3800{,}000 : 1{,}200{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
600,000÷5=120,000600{,}000 \div 5 = 120{,}000
Each piece is 120,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
120,000×2=240,000120{,}000 \times 2 = 240{,}000
Eunji gets 240,000 won.
Answer: 240,000 won
4 · Reviewdoes it hold up?

The two shares, 240,000 and 360,000 won, add to 600,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 240,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 2 easy answer: 150,000 won

Eunji invested 900,000900{,}000 won and Hyunsu invested 1,500,0001{,}500{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 400,000400{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 900,000 and 1,500,000 won and share a profit of 400,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 900,000 won and Hyunsu 1,500,000 won.
  • The profit is 400,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 900,000 to 1,500,000 is 3 to 5.
900,000:1,500,000=3:5900{,}000 : 1{,}500{,}000 = 3 : 5
A ratio of 3 to 5.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
3+5=83 + 5 = 8
8 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
400,000÷8=50,000400{,}000 \div 8 = 50{,}000
Each piece is 50,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 3 of the 8 pieces.
50,000×3=150,00050{,}000 \times 3 = 150{,}000
Eunji gets 150,000 won.
Answer: 150,000 won
4 · Reviewdoes it hold up?

The two shares, 150,000 and 250,000 won, add to 400,000 won -- the whole profit -- and stand in the ratio 3 to 5.

Another way: Taking 3 over 8 of the profit directly gives the same 150,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 3 easy answer: 300,000 won

Eunji invested 1,500,0001{,}500{,}000 won and Hyunsu invested 2,000,0002{,}000{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 700,000700{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 1,500,000 and 2,000,000 won and share a profit of 700,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 1,500,000 won and Hyunsu 2,000,000 won.
  • The profit is 700,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 1,500,000 to 2,000,000 is 3 to 4.
1,500,000:2,000,000=3:41{,}500{,}000 : 2{,}000{,}000 = 3 : 4
A ratio of 3 to 4.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
3+4=73 + 4 = 7
7 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
700,000÷7=100,000700{,}000 \div 7 = 100{,}000
Each piece is 100,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 3 of the 7 pieces.
100,000×3=300,000100{,}000 \times 3 = 300{,}000
Eunji gets 300,000 won.
Answer: 300,000 won
4 · Reviewdoes it hold up?

The two shares, 300,000 and 400,000 won, add to 700,000 won -- the whole profit -- and stand in the ratio 3 to 4.

Another way: Taking 3 over 7 of the profit directly gives the same 300,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 4 easy answer: 300,000 won

Eunji invested 3,000,0003{,}000{,}000 won and Hyunsu invested 4,500,0004{,}500{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 750,000750{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 3,000,000 and 4,500,000 won and share a profit of 750,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 3,000,000 won and Hyunsu 4,500,000 won.
  • The profit is 750,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 3,000,000 to 4,500,000 is 2 to 3.
3,000,000:4,500,000=2:33{,}000{,}000 : 4{,}500{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
750,000÷5=150,000750{,}000 \div 5 = 150{,}000
Each piece is 150,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
150,000×2=300,000150{,}000 \times 2 = 300{,}000
Eunji gets 300,000 won.
Answer: 300,000 won
4 · Reviewdoes it hold up?

The two shares, 300,000 and 450,000 won, add to 750,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 300,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 5 medium answer: 500,000 won

Eunji invested 2,500,0002{,}500{,}000 won and Hyunsu invested 1,500,0001{,}500{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 800,000800{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 2,500,000 and 1,500,000 won and share a profit of 800,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 2,500,000 won and Hyunsu 1,500,000 won.
  • The profit is 800,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 2,500,000 to 1,500,000 is 5 to 3.
2,500,000:1,500,000=5:32{,}500{,}000 : 1{,}500{,}000 = 5 : 3
A ratio of 5 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
5+3=85 + 3 = 8
8 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
800,000÷8=100,000800{,}000 \div 8 = 100{,}000
Each piece is 100,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 5 of the 8 pieces.
100,000×5=500,000100{,}000 \times 5 = 500{,}000
Eunji gets 500,000 won.
Answer: 500,000 won
4 · Reviewdoes it hold up?

The two shares, 500,000 and 300,000 won, add to 800,000 won -- the whole profit -- and stand in the ratio 5 to 3.

Another way: Taking 5 over 8 of the profit directly gives the same 500,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 6 medium answer: 300,000 won

Eunji invested 2,100,0002{,}100{,}000 won and Hyunsu invested 2,800,0002{,}800{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 700,000700{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 2,100,000 and 2,800,000 won and share a profit of 700,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 2,100,000 won and Hyunsu 2,800,000 won.
  • The profit is 700,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 2,100,000 to 2,800,000 is 3 to 4.
2,100,000:2,800,000=3:42{,}100{,}000 : 2{,}800{,}000 = 3 : 4
A ratio of 3 to 4.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
3+4=73 + 4 = 7
7 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
700,000÷7=100,000700{,}000 \div 7 = 100{,}000
Each piece is 100,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 3 of the 7 pieces.
100,000×3=300,000100{,}000 \times 3 = 300{,}000
Eunji gets 300,000 won.
Answer: 300,000 won
4 · Reviewdoes it hold up?

The two shares, 300,000 and 400,000 won, add to 700,000 won -- the whole profit -- and stand in the ratio 3 to 4.

Another way: Taking 3 over 7 of the profit directly gives the same 300,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 7 medium answer: 200,000 won

Eunji invested 1,200,0001{,}200{,}000 won and Hyunsu invested 1,800,0001{,}800{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 500,000500{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 1,200,000 and 1,800,000 won and share a profit of 500,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 1,200,000 won and Hyunsu 1,800,000 won.
  • The profit is 500,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 1,200,000 to 1,800,000 is 2 to 3.
1,200,000:1,800,000=2:31{,}200{,}000 : 1{,}800{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
500,000÷5=100,000500{,}000 \div 5 = 100{,}000
Each piece is 100,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
100,000×2=200,000100{,}000 \times 2 = 200{,}000
Eunji gets 200,000 won.
Answer: 200,000 won
4 · Reviewdoes it hold up?

The two shares, 200,000 and 300,000 won, add to 500,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 200,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 8 medium answer: 1,000,000 won

Eunji invested 4,000,0004{,}000{,}000 won and Hyunsu invested 6,000,0006{,}000{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 2,500,0002{,}500{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 4,000,000 and 6,000,000 won and share a profit of 2,500,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 4,000,000 won and Hyunsu 6,000,000 won.
  • The profit is 2,500,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 4,000,000 to 6,000,000 is 2 to 3.
4,000,000:6,000,000=2:34{,}000{,}000 : 6{,}000{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
2,500,000÷5=500,0002{,}500{,}000 \div 5 = 500{,}000
Each piece is 500,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
500,000×2=1,000,000500{,}000 \times 2 = 1{,}000{,}000
Eunji gets 1,000,000 won.
Answer: 1,000,000 won
4 · Reviewdoes it hold up?

The two shares, 1,000,000 and 1,500,000 won, add to 2,500,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 1,000,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 9 hard answer: 360,000 won

Eunji invested 2,400,0002{,}400{,}000 won and Hyunsu invested 3,600,0003{,}600{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 900,000900{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 2,400,000 and 3,600,000 won and share a profit of 900,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 2,400,000 won and Hyunsu 3,600,000 won.
  • The profit is 900,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 2,400,000 to 3,600,000 is 2 to 3.
2,400,000:3,600,000=2:32{,}400{,}000 : 3{,}600{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
900,000÷5=180,000900{,}000 \div 5 = 180{,}000
Each piece is 180,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
180,000×2=360,000180{,}000 \times 2 = 360{,}000
Eunji gets 360,000 won.
Answer: 360,000 won
4 · Reviewdoes it hold up?

The two shares, 360,000 and 540,000 won, add to 900,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 360,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 10 hard answer: 1,000,000 won

Eunji invested 5,000,0005{,}000{,}000 won and Hyunsu invested 3,000,0003{,}000{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 1,600,0001{,}600{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 5,000,000 and 3,000,000 won and share a profit of 1,600,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 5,000,000 won and Hyunsu 3,000,000 won.
  • The profit is 1,600,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 5,000,000 to 3,000,000 is 5 to 3.
5,000,000:3,000,000=5:35{,}000{,}000 : 3{,}000{,}000 = 5 : 3
A ratio of 5 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
5+3=85 + 3 = 8
8 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
1,600,000÷8=200,0001{,}600{,}000 \div 8 = 200{,}000
Each piece is 200,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 5 of the 8 pieces.
200,000×5=1,000,000200{,}000 \times 5 = 1{,}000{,}000
Eunji gets 1,000,000 won.
Answer: 1,000,000 won
4 · Reviewdoes it hold up?

The two shares, 1,000,000 and 600,000 won, add to 1,600,000 won -- the whole profit -- and stand in the ratio 5 to 3.

Another way: Taking 5 over 8 of the profit directly gives the same 1,000,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 11 hard answer: 400,000 won

Eunji invested 1,600,0001{,}600{,}000 won and Hyunsu invested 2,400,0002{,}400{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 1,000,0001{,}000{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 1,600,000 and 2,400,000 won and share a profit of 1,000,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 1,600,000 won and Hyunsu 2,400,000 won.
  • The profit is 1,000,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 1,600,000 to 2,400,000 is 2 to 3.
1,600,000:2,400,000=2:31{,}600{,}000 : 2{,}400{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
1,000,000÷5=200,0001{,}000{,}000 \div 5 = 200{,}000
Each piece is 200,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
200,000×2=400,000200{,}000 \times 2 = 400{,}000
Eunji gets 400,000 won.
Answer: 400,000 won
4 · Reviewdoes it hold up?

The two shares, 400,000 and 600,000 won, add to 1,000,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 400,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.
Variant 12 hard answer: 600,000 won

Eunji invested 1,800,0001{,}800{,}000 won and Hyunsu invested 2,700,0002{,}700{,}000 won, and they will split the profit in the ratio of what they invested. If the total profit is 1,500,0001{,}500{,}000 won, how much does Eunji get?

Show solution
1 · Understandwhat's really being asked

Two people put in 1,800,000 and 2,700,000 won and share a profit of 1,500,000 won in that ratio. We want the first one's share.

Givens
  • Eunji invested 1,800,000 won and Hyunsu 2,700,000 won.
  • The profit is 1,500,000 won.
  • The profit is split in the ratio of the investments.
Unknowns
  • Eunji's share of the profit.
Constraints
  • The two shares together are the whole profit.
2 · Planchoose the strategy

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

Simplify the ratio before doing anything with it, then add its terms: that sum is how many equal pieces the profit is cut into, which is the number the division needs.

3 · Execute4 carry out the plan

1Simplify the ratio of the investments

#7 Identify Subproblems 6.RP.A.3
Both amounts share a factor, so 1,800,000 to 2,700,000 is 2 to 3.
1,800,000:2,700,000=2:31{,}800{,}000 : 2{,}700{,}000 = 2 : 3
A ratio of 2 to 3.

2Count the pieces

#8 Analyze the Units 7.RP.A.3
Adding the two terms says how many equal parts the profit is cut into.
2+3=52 + 3 = 5
5 pieces in all.

3Find one piece

#13 Convert to Algebra 7.RP.A.3
The profit shared equally between those pieces.
1,500,000÷5=300,0001{,}500{,}000 \div 5 = 300{,}000
Each piece is 300,000 won.

4Give Eunji her share

#13 Convert to Algebra 7.RP.A.3
She takes 2 of the 5 pieces.
300,000×2=600,000300{,}000 \times 2 = 600{,}000
Eunji gets 600,000 won.
Answer: 600,000 won
4 · Reviewdoes it hold up?

The two shares, 600,000 and 900,000 won, add to 1,500,000 won -- the whole profit -- and stand in the ratio 2 to 3.

Another way: Taking 2 over 5 of the profit directly gives the same 600,000 won, with the piece never named.

Standardsmin grade 7
  • 7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Dividing a total in a given ratio.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Simplifying the ratio of the two investments.
💡Takeaway. Add the terms of a ratio and you know how many pieces to cut into. Then it is one division and one multiplication.