← Moving things between two people leaves the total alone · Track a Quantity Through Changes

Moving things between two people leaves the total alone · 12 practice problems

6.EE.B.76.RP.A.3

Generated variants — 12

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

Variant 1 easy answer: 2 candies

Jia and Minho each had 1818 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 4:54 : 5. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 18 candies each. After one gives some to the other, their counts are in the ratio 4 to 5. We want how many were given.

Givens
  • Each started with 18 candies.
  • Afterwards the ratio is 4 to 5.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
18+18=3618 + 18 = 36
Still 36 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
4 to 5 means 9 equal parts of the 36.
36÷9=436 \div 9 = 4
Each part is 4 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 4 of the 9 parts.
4×4=164 \times 4 = 16
Jia has 16 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
18−16=218 - 16 = 2
She gave away 2.
Answer: 2 candies
4 · Reviewdoes it hold up?

After the gift Jia has 16 and Minho 20: they add to 36 and stand in the ratio 4 to 5.

Another way: Following Minho instead -- he gains what Jia loses, so 18 plus the gift is 20 -- gives the same 2 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 2 easy answer: 10 candies

Jia and Minho each had 2020 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 1:31 : 3. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 20 candies each. After one gives some to the other, their counts are in the ratio 1 to 3. We want how many were given.

Givens
  • Each started with 20 candies.
  • Afterwards the ratio is 1 to 3.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
20+20=4020 + 20 = 40
Still 40 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
1 to 3 means 4 equal parts of the 40.
40÷4=1040 \div 4 = 10
Each part is 10 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 1 of the 4 parts.
10×1=1010 \times 1 = 10
Jia has 10 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
20−10=1020 - 10 = 10
She gave away 10.
Answer: 10 candies
4 · Reviewdoes it hold up?

After the gift Jia has 10 and Minho 30: they add to 40 and stand in the ratio 1 to 3.

Another way: Following Minho instead -- he gains what Jia loses, so 20 plus the gift is 30 -- gives the same 10 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 3 easy answer: 8 candies

Jia and Minho each had 2424 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 1:21 : 2. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 24 candies each. After one gives some to the other, their counts are in the ratio 1 to 2. We want how many were given.

Givens
  • Each started with 24 candies.
  • Afterwards the ratio is 1 to 2.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
24+24=4824 + 24 = 48
Still 48 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
1 to 2 means 3 equal parts of the 48.
48÷3=1648 \div 3 = 16
Each part is 16 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 1 of the 3 parts.
16×1=1616 \times 1 = 16
Jia has 16 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
24−16=824 - 16 = 8
She gave away 8.
Answer: 8 candies
4 · Reviewdoes it hold up?

After the gift Jia has 16 and Minho 32: they add to 48 and stand in the ratio 1 to 2.

Another way: Following Minho instead -- he gains what Jia loses, so 24 plus the gift is 32 -- gives the same 8 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 4 easy answer: 4 candies

Jia and Minho each had 2828 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 3:43 : 4. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 28 candies each. After one gives some to the other, their counts are in the ratio 3 to 4. We want how many were given.

Givens
  • Each started with 28 candies.
  • Afterwards the ratio is 3 to 4.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
28+28=5628 + 28 = 56
Still 56 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
3 to 4 means 7 equal parts of the 56.
56÷7=856 \div 7 = 8
Each part is 8 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 3 of the 7 parts.
8×3=248 \times 3 = 24
Jia has 24 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
28−24=428 - 24 = 4
She gave away 4.
Answer: 4 candies
4 · Reviewdoes it hold up?

After the gift Jia has 24 and Minho 32: they add to 56 and stand in the ratio 3 to 4.

Another way: Following Minho instead -- he gains what Jia loses, so 28 plus the gift is 32 -- gives the same 4 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 5 medium answer: 6 candies

Jia and Minho each had 3030 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 2:32 : 3. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 30 candies each. After one gives some to the other, their counts are in the ratio 2 to 3. We want how many were given.

Givens
  • Each started with 30 candies.
  • Afterwards the ratio is 2 to 3.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
30+30=6030 + 30 = 60
Still 60 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
2 to 3 means 5 equal parts of the 60.
60÷5=1260 \div 5 = 12
Each part is 12 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 2 of the 5 parts.
12×2=2412 \times 2 = 24
Jia has 24 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
30−24=630 - 24 = 6
She gave away 6.
Answer: 6 candies
4 · Reviewdoes it hold up?

After the gift Jia has 24 and Minho 36: they add to 60 and stand in the ratio 2 to 3.

Another way: Following Minho instead -- he gains what Jia loses, so 30 plus the gift is 36 -- gives the same 6 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 6 medium answer: 3 candies

Jia and Minho each had 3333 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 5:65 : 6. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 33 candies each. After one gives some to the other, their counts are in the ratio 5 to 6. We want how many were given.

Givens
  • Each started with 33 candies.
  • Afterwards the ratio is 5 to 6.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
33+33=6633 + 33 = 66
Still 66 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
5 to 6 means 11 equal parts of the 66.
66÷11=666 \div 11 = 6
Each part is 6 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 5 of the 11 parts.
6×5=306 \times 5 = 30
Jia has 30 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
33−30=333 - 30 = 3
She gave away 3.
Answer: 3 candies
4 · Reviewdoes it hold up?

After the gift Jia has 30 and Minho 36: they add to 66 and stand in the ratio 5 to 6.

Another way: Following Minho instead -- he gains what Jia loses, so 33 plus the gift is 36 -- gives the same 3 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 7 medium answer: 24 candies

Jia and Minho each had 3636 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 1:51 : 5. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 36 candies each. After one gives some to the other, their counts are in the ratio 1 to 5. We want how many were given.

Givens
  • Each started with 36 candies.
  • Afterwards the ratio is 1 to 5.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
36+36=7236 + 36 = 72
Still 72 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
1 to 5 means 6 equal parts of the 72.
72÷6=1272 \div 6 = 12
Each part is 12 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 1 of the 6 parts.
12×1=1212 \times 1 = 12
Jia has 12 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
36−12=2436 - 12 = 24
She gave away 24.
Answer: 24 candies
4 · Reviewdoes it hold up?

After the gift Jia has 12 and Minho 60: they add to 72 and stand in the ratio 1 to 5.

Another way: Following Minho instead -- he gains what Jia loses, so 36 plus the gift is 60 -- gives the same 24 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 8 medium answer: 10 candies

Jia and Minho each had 4040 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 3:53 : 5. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 40 candies each. After one gives some to the other, their counts are in the ratio 3 to 5. We want how many were given.

Givens
  • Each started with 40 candies.
  • Afterwards the ratio is 3 to 5.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
40+40=8040 + 40 = 80
Still 80 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
3 to 5 means 8 equal parts of the 80.
80÷8=1080 \div 8 = 10
Each part is 10 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 3 of the 8 parts.
10×3=3010 \times 3 = 30
Jia has 30 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
40−30=1040 - 30 = 10
She gave away 10.
Answer: 10 candies
4 · Reviewdoes it hold up?

After the gift Jia has 30 and Minho 50: they add to 80 and stand in the ratio 3 to 5.

Another way: Following Minho instead -- he gains what Jia loses, so 40 plus the gift is 50 -- gives the same 10 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 9 hard answer: 7 candies

Jia and Minho each had 4242 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 5:75 : 7. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 42 candies each. After one gives some to the other, their counts are in the ratio 5 to 7. We want how many were given.

Givens
  • Each started with 42 candies.
  • Afterwards the ratio is 5 to 7.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
42+42=8442 + 42 = 84
Still 84 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
5 to 7 means 12 equal parts of the 84.
84÷12=784 \div 12 = 7
Each part is 7 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 5 of the 12 parts.
7×5=357 \times 5 = 35
Jia has 35 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
42−35=742 - 35 = 7
She gave away 7.
Answer: 7 candies
4 · Reviewdoes it hold up?

After the gift Jia has 35 and Minho 49: they add to 84 and stand in the ratio 5 to 7.

Another way: Following Minho instead -- he gains what Jia loses, so 42 plus the gift is 49 -- gives the same 7 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 10 hard answer: 25 candies

Jia and Minho each had 4545 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 2:72 : 7. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 45 candies each. After one gives some to the other, their counts are in the ratio 2 to 7. We want how many were given.

Givens
  • Each started with 45 candies.
  • Afterwards the ratio is 2 to 7.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
45+45=9045 + 45 = 90
Still 90 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
2 to 7 means 9 equal parts of the 90.
90÷9=1090 \div 9 = 10
Each part is 10 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 2 of the 9 parts.
10×2=2010 \times 2 = 20
Jia has 20 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
45−20=2545 - 20 = 25
She gave away 25.
Answer: 25 candies
4 · Reviewdoes it hold up?

After the gift Jia has 20 and Minho 70: they add to 90 and stand in the ratio 2 to 7.

Another way: Following Minho instead -- he gains what Jia loses, so 45 plus the gift is 70 -- gives the same 25 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 11 hard answer: 30 candies

Jia and Minho each had 5050 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 1:41 : 4. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 50 candies each. After one gives some to the other, their counts are in the ratio 1 to 4. We want how many were given.

Givens
  • Each started with 50 candies.
  • Afterwards the ratio is 1 to 4.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
50+50=10050 + 50 = 100
Still 100 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
1 to 4 means 5 equal parts of the 100.
100÷5=20100 \div 5 = 20
Each part is 20 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 1 of the 5 parts.
20×1=2020 \times 1 = 20
Jia has 20 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
50−20=3050 - 20 = 30
She gave away 30.
Answer: 30 candies
4 · Reviewdoes it hold up?

After the gift Jia has 20 and Minho 80: they add to 100 and stand in the ratio 1 to 4.

Another way: Following Minho instead -- he gains what Jia loses, so 50 plus the gift is 80 -- gives the same 30 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.
Variant 12 hard answer: 24 candies

Jia and Minho each had 6060 candies. After Jia gave some candies to Minho, the ratio of Jia's candies to Minho's became 3:73 : 7. How many candies did Jia give away?

Show solution
1 · Understandwhat's really being asked

Two people start with 60 candies each. After one gives some to the other, their counts are in the ratio 3 to 7. We want how many were given.

Givens
  • Each started with 60 candies.
  • Afterwards the ratio is 3 to 7.
Unknowns
  • How many candies Jia gave away.
Constraints
  • The candies only moved between the two: none were eaten or added.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #13 Convert to Algebra#7 Identify Subproblems

Both counts changed, so neither can be tracked directly. The total did not change, and that is what the new ratio divides -- work with the total and the two counts fall out.

3 · Execute4 carry out the plan

1Notice what did not change

#16 Count the Complement 6.RP.A.3
Candies moved from one person to the other, so the two together are exactly as many as before.
60+60=12060 + 60 = 120
Still 120 candies.

2Cut that total in the new ratio

#7 Identify Subproblems 6.RP.A.3
3 to 7 means 10 equal parts of the 120.
120÷10=12120 \div 10 = 12
Each part is 12 candies.

3Find what Jia has left

#13 Convert to Algebra 6.EE.B.7
She keeps 3 of the 10 parts.
12×3=3612 \times 3 = 36
Jia has 36 left.

4Compare with what she started with

#13 Convert to Algebra 6.EE.B.7
The question asks what she gave, not what she has.
60−36=2460 - 36 = 24
She gave away 24.
Answer: 24 candies
4 · Reviewdoes it hold up?

After the gift Jia has 36 and Minho 84: they add to 120 and stand in the ratio 3 to 7.

Another way: Following Minho instead -- he gains what Jia loses, so 60 plus the gift is 84 -- gives the same 24 from the other side.

Standardsmin grade 6
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each count and then for the amount moved.
  • 6.RP.A.3 Use ratio and rate reasoning to solve problems — Dividing the unchanged total in the new ratio.
💡Takeaway. When something moves between two people, look for what stayed the same. It is usually the total.