← Reduce relations to a single-unknown equation · Find Two Unknowns from Sum and Difference

Reduce relations to a single-unknown equation · 10 practice problems

3.OA.D.83.OA.A.3

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Generated variants — 10

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

Variant 1 easy

Gwen and Faye are twin sisters, and Hugo and Reid are twin brothers. The four of them are $40$ years old in all, and Faye is $8$ years older than Hugo. If Reid's age is $3$ times the age of Reid's younger brother, how old is Reid's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Gwen and Faye) are the same age, and two brothers (Hugo and Reid) are the same age. The four ages add to 40. Faye is 8 years older than Hugo. Also, Reid is 3 times as old as his younger brother. We must find the younger brother's age.

Givens

Gwen and Faye are twins, so they have the same age
Hugo and Reid are twins, so they have the same age
The four ages together total 40
Faye is 8 years older than Hugo
Reid's age is 3 times his younger brother's age

Unknowns

The age of Reid's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Faye is 8 more than Hugo'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Reid is 3 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Gwen and Faye are the same age; Hugo and Reid are the same age. So the total is twice Faye's age plus twice Hugo's age, and that total is 40.

2 \times (\text{Faye}) + 2 \times (\text{Hugo}) = 40

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 8-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Faye is 8 older than Hugo, so replace Faye with Hugo + 8. The equation becomes 2(Hugo + 8) + 2(Hugo) = 40, which simplifies to 4 times Hugo plus 16 equals 40.

2(\text{Hugo}+8) + 2\,\text{Hugo} = 40 \;\Rightarrow\; 4\,\text{Hugo} + 16 = 40

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Hugo's (and Reid's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 16 from 40 to get 24, then divide by 4. Hugo is 6, and since Reid is his twin, Reid is also 6.

4\,\text{Hugo} = 40 - 16 = 24 \;\Rightarrow\; \text{Hugo} = 6,\quad \text{Reid} = 6

Undoing 'add 16' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Reid is 6, and Reid's age is 3 times his younger brother's age. So the younger brother's age is 6 divided by 3, which is 2.

\text{younger brother} = 6 \div 3 = 2

'3 times as old' means the younger child is one-3th of Reid's age, found by dividing.

Answer: 2 years old

4 · Reviewdoes it hold up?

Faye = 14, Gwen = 14, Hugo = 6, Reid = 6 add to 40, and Faye is indeed 8 more than Hugo. The younger brother is 2, and 3 times 2 is 6 = Reid's age. Everything checks.

Another way: Guess and check (tool 6): try Hugo = 6, then Faye = 14, total = 2(14) + 2(6) = 40, which matches; then 6 / 3 = 2.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 6 by 3 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 8-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Hugo plus 16 equals 40 for Hugo's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 2 easy

Anna and Iris are twin sisters, and Seth and Paul are twin brothers. The four of them are $40$ years old in all, and Iris is $4$ years older than Seth. If Paul's age is $4$ times the age of Paul's younger brother, how old is Paul's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Anna and Iris) are the same age, and two brothers (Seth and Paul) are the same age. The four ages add to 40. Iris is 4 years older than Seth. Also, Paul is 4 times as old as his younger brother. We must find the younger brother's age.

Givens

Anna and Iris are twins, so they have the same age
Seth and Paul are twins, so they have the same age
The four ages together total 40
Iris is 4 years older than Seth
Paul's age is 4 times his younger brother's age

Unknowns

The age of Paul's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Iris is 4 more than Seth'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Paul is 4 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Anna and Iris are the same age; Seth and Paul are the same age. So the total is twice Iris's age plus twice Seth's age, and that total is 40.

2 \times (\text{Iris}) + 2 \times (\text{Seth}) = 40

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 4-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Iris is 4 older than Seth, so replace Iris with Seth + 4. The equation becomes 2(Seth + 4) + 2(Seth) = 40, which simplifies to 4 times Seth plus 8 equals 40.

2(\text{Seth}+4) + 2\,\text{Seth} = 40 \;\Rightarrow\; 4\,\text{Seth} + 8 = 40

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Seth's (and Paul's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 8 from 40 to get 32, then divide by 4. Seth is 8, and since Paul is his twin, Paul is also 8.

4\,\text{Seth} = 40 - 8 = 32 \;\Rightarrow\; \text{Seth} = 8,\quad \text{Paul} = 8

Undoing 'add 8' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Paul is 8, and Paul's age is 4 times his younger brother's age. So the younger brother's age is 8 divided by 4, which is 2.

\text{younger brother} = 8 \div 4 = 2

'4 times as old' means the younger child is one-4th of Paul's age, found by dividing.

Answer: 2 years old

4 · Reviewdoes it hold up?

Iris = 12, Anna = 12, Seth = 8, Paul = 8 add to 40, and Iris is indeed 4 more than Seth. The younger brother is 2, and 4 times 2 is 8 = Paul's age. Everything checks.

Another way: Guess and check (tool 6): try Seth = 8, then Iris = 12, total = 2(12) + 2(8) = 40, which matches; then 8 / 4 = 2.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 8 by 4 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 4-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Seth plus 8 equals 40 for Seth's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 3 easy

Amy and Eva are twin sisters, and Jack and Finn are twin brothers. The four of them are $48$ years old in all, and Eva is $4$ years older than Jack. If Finn's age is $2$ times the age of Finn's younger brother, how old is Finn's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Amy and Eva) are the same age, and two brothers (Jack and Finn) are the same age. The four ages add to 48. Eva is 4 years older than Jack. Also, Finn is 2 times as old as his younger brother. We must find the younger brother's age.

Givens

Amy and Eva are twins, so they have the same age
Jack and Finn are twins, so they have the same age
The four ages together total 48
Eva is 4 years older than Jack
Finn's age is 2 times his younger brother's age

Unknowns

The age of Finn's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Eva is 4 more than Jack'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Finn is 2 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Amy and Eva are the same age; Jack and Finn are the same age. So the total is twice Eva's age plus twice Jack's age, and that total is 48.

2 \times (\text{Eva}) + 2 \times (\text{Jack}) = 48

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 4-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Eva is 4 older than Jack, so replace Eva with Jack + 4. The equation becomes 2(Jack + 4) + 2(Jack) = 48, which simplifies to 4 times Jack plus 8 equals 48.

2(\text{Jack}+4) + 2\,\text{Jack} = 48 \;\Rightarrow\; 4\,\text{Jack} + 8 = 48

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Jack's (and Finn's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 8 from 48 to get 40, then divide by 4. Jack is 10, and since Finn is his twin, Finn is also 10.

4\,\text{Jack} = 48 - 8 = 40 \;\Rightarrow\; \text{Jack} = 10,\quad \text{Finn} = 10

Undoing 'add 8' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Finn is 10, and Finn's age is 2 times his younger brother's age. So the younger brother's age is 10 divided by 2, which is 5.

\text{younger brother} = 10 \div 2 = 5

'2 times as old' means the younger child is one-2th of Finn's age, found by dividing.

Answer: 5 years old

4 · Reviewdoes it hold up?

Eva = 14, Amy = 14, Jack = 10, Finn = 10 add to 48, and Eva is indeed 4 more than Jack. The younger brother is 5, and 2 times 5 is 10 = Finn's age. Everything checks.

Another way: Guess and check (tool 6): try Jack = 10, then Eva = 14, total = 2(14) + 2(10) = 48, which matches; then 10 / 2 = 5.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 10 by 2 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 4-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Jack plus 8 equals 48 for Jack's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 4 medium

Dana and Cora are twin sisters, and Neil and Gage are twin brothers. The four of them are $56$ years old in all, and Cora is $8$ years older than Neil. If Gage's age is $2$ times the age of Gage's younger brother, how old is Gage's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Dana and Cora) are the same age, and two brothers (Neil and Gage) are the same age. The four ages add to 56. Cora is 8 years older than Neil. Also, Gage is 2 times as old as his younger brother. We must find the younger brother's age.

Givens

Dana and Cora are twins, so they have the same age
Neil and Gage are twins, so they have the same age
The four ages together total 56
Cora is 8 years older than Neil
Gage's age is 2 times his younger brother's age

Unknowns

The age of Gage's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Cora is 8 more than Neil'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Gage is 2 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Dana and Cora are the same age; Neil and Gage are the same age. So the total is twice Cora's age plus twice Neil's age, and that total is 56.

2 \times (\text{Cora}) + 2 \times (\text{Neil}) = 56

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 8-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Cora is 8 older than Neil, so replace Cora with Neil + 8. The equation becomes 2(Neil + 8) + 2(Neil) = 56, which simplifies to 4 times Neil plus 16 equals 56.

2(\text{Neil}+8) + 2\,\text{Neil} = 56 \;\Rightarrow\; 4\,\text{Neil} + 16 = 56

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Neil's (and Gage's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 16 from 56 to get 40, then divide by 4. Neil is 10, and since Gage is his twin, Gage is also 10.

4\,\text{Neil} = 56 - 16 = 40 \;\Rightarrow\; \text{Neil} = 10,\quad \text{Gage} = 10

Undoing 'add 16' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Gage is 10, and Gage's age is 2 times his younger brother's age. So the younger brother's age is 10 divided by 2, which is 5.

\text{younger brother} = 10 \div 2 = 5

'2 times as old' means the younger child is one-2th of Gage's age, found by dividing.

Answer: 5 years old

4 · Reviewdoes it hold up?

Cora = 18, Dana = 18, Neil = 10, Gage = 10 add to 56, and Cora is indeed 8 more than Neil. The younger brother is 5, and 2 times 5 is 10 = Gage's age. Everything checks.

Another way: Guess and check (tool 6): try Neil = 10, then Cora = 18, total = 2(18) + 2(10) = 56, which matches; then 10 / 2 = 5.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 10 by 2 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 8-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Neil plus 16 equals 56 for Neil's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 5 medium

Sara and Kate are twin sisters, and Ryan and Dean are twin brothers. The four of them are $56$ years old in all, and Kate is $4$ years older than Ryan. If Dean's age is $3$ times the age of Dean's younger brother, how old is Dean's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Sara and Kate) are the same age, and two brothers (Ryan and Dean) are the same age. The four ages add to 56. Kate is 4 years older than Ryan. Also, Dean is 3 times as old as his younger brother. We must find the younger brother's age.

Givens

Sara and Kate are twins, so they have the same age
Ryan and Dean are twins, so they have the same age
The four ages together total 56
Kate is 4 years older than Ryan
Dean's age is 3 times his younger brother's age

Unknowns

The age of Dean's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Kate is 4 more than Ryan'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Dean is 3 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Sara and Kate are the same age; Ryan and Dean are the same age. So the total is twice Kate's age plus twice Ryan's age, and that total is 56.

2 \times (\text{Kate}) + 2 \times (\text{Ryan}) = 56

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 4-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Kate is 4 older than Ryan, so replace Kate with Ryan + 4. The equation becomes 2(Ryan + 4) + 2(Ryan) = 56, which simplifies to 4 times Ryan plus 8 equals 56.

2(\text{Ryan}+4) + 2\,\text{Ryan} = 56 \;\Rightarrow\; 4\,\text{Ryan} + 8 = 56

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Ryan's (and Dean's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 8 from 56 to get 48, then divide by 4. Ryan is 12, and since Dean is his twin, Dean is also 12.

4\,\text{Ryan} = 56 - 8 = 48 \;\Rightarrow\; \text{Ryan} = 12,\quad \text{Dean} = 12

Undoing 'add 8' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Dean is 12, and Dean's age is 3 times his younger brother's age. So the younger brother's age is 12 divided by 3, which is 4.

\text{younger brother} = 12 \div 3 = 4

'3 times as old' means the younger child is one-3th of Dean's age, found by dividing.

Answer: 4 years old

4 · Reviewdoes it hold up?

Kate = 16, Sara = 16, Ryan = 12, Dean = 12 add to 56, and Kate is indeed 4 more than Ryan. The younger brother is 4, and 3 times 4 is 12 = Dean's age. Everything checks.

Another way: Guess and check (tool 6): try Ryan = 12, then Kate = 16, total = 2(16) + 2(12) = 56, which matches; then 12 / 3 = 4.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 12 by 3 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 4-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Ryan plus 8 equals 56 for Ryan's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 6 medium

Nora and Beth are twin sisters, and Luke and Mark are twin brothers. The four of them are $60$ years old in all, and Beth is $6$ years older than Luke. If Mark's age is $2$ times the age of Mark's younger brother, how old is Mark's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Nora and Beth) are the same age, and two brothers (Luke and Mark) are the same age. The four ages add to 60. Beth is 6 years older than Luke. Also, Mark is 2 times as old as his younger brother. We must find the younger brother's age.

Givens

Nora and Beth are twins, so they have the same age
Luke and Mark are twins, so they have the same age
The four ages together total 60
Beth is 6 years older than Luke
Mark's age is 2 times his younger brother's age

Unknowns

The age of Mark's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Beth is 6 more than Luke'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Mark is 2 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Nora and Beth are the same age; Luke and Mark are the same age. So the total is twice Beth's age plus twice Luke's age, and that total is 60.

2 \times (\text{Beth}) + 2 \times (\text{Luke}) = 60

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 6-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Beth is 6 older than Luke, so replace Beth with Luke + 6. The equation becomes 2(Luke + 6) + 2(Luke) = 60, which simplifies to 4 times Luke plus 12 equals 60.

2(\text{Luke}+6) + 2\,\text{Luke} = 60 \;\Rightarrow\; 4\,\text{Luke} + 12 = 60

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Luke's (and Mark's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 12 from 60 to get 48, then divide by 4. Luke is 12, and since Mark is his twin, Mark is also 12.

4\,\text{Luke} = 60 - 12 = 48 \;\Rightarrow\; \text{Luke} = 12,\quad \text{Mark} = 12

Undoing 'add 12' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Mark is 12, and Mark's age is 2 times his younger brother's age. So the younger brother's age is 12 divided by 2, which is 6.

\text{younger brother} = 12 \div 2 = 6

'2 times as old' means the younger child is one-2th of Mark's age, found by dividing.

Answer: 6 years old

4 · Reviewdoes it hold up?

Beth = 18, Nora = 18, Luke = 12, Mark = 12 add to 60, and Beth is indeed 6 more than Luke. The younger brother is 6, and 2 times 6 is 12 = Mark's age. Everything checks.

Another way: Guess and check (tool 6): try Luke = 12, then Beth = 18, total = 2(18) + 2(12) = 60, which matches; then 12 / 2 = 6.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 12 by 2 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 6-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Luke plus 12 equals 60 for Luke's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 7 medium

Mia and Ella are twin sisters, and Liam and Noah are twin brothers. The four of them are $64$ years old in all, and Ella is $8$ years older than Liam. If Noah's age is $3$ times the age of Noah's younger brother, how old is Noah's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Mia and Ella) are the same age, and two brothers (Liam and Noah) are the same age. The four ages add to 64. Ella is 8 years older than Liam. Also, Noah is 3 times as old as his younger brother. We must find the younger brother's age.

Givens

Mia and Ella are twins, so they have the same age
Liam and Noah are twins, so they have the same age
The four ages together total 64
Ella is 8 years older than Liam
Noah's age is 3 times his younger brother's age

Unknowns

The age of Noah's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Ella is 8 more than Liam'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Noah is 3 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Mia and Ella are the same age; Liam and Noah are the same age. So the total is twice Ella's age plus twice Liam's age, and that total is 64.

2 \times (\text{Ella}) + 2 \times (\text{Liam}) = 64

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 8-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Ella is 8 older than Liam, so replace Ella with Liam + 8. The equation becomes 2(Liam + 8) + 2(Liam) = 64, which simplifies to 4 times Liam plus 16 equals 64.

2(\text{Liam}+8) + 2\,\text{Liam} = 64 \;\Rightarrow\; 4\,\text{Liam} + 16 = 64

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Liam's (and Noah's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 16 from 64 to get 48, then divide by 4. Liam is 12, and since Noah is his twin, Noah is also 12.

4\,\text{Liam} = 64 - 16 = 48 \;\Rightarrow\; \text{Liam} = 12,\quad \text{Noah} = 12

Undoing 'add 16' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Noah is 12, and Noah's age is 3 times his younger brother's age. So the younger brother's age is 12 divided by 3, which is 4.

\text{younger brother} = 12 \div 3 = 4

'3 times as old' means the younger child is one-3th of Noah's age, found by dividing.

Answer: 4 years old

4 · Reviewdoes it hold up?

Ella = 20, Mia = 20, Liam = 12, Noah = 12 add to 64, and Ella is indeed 8 more than Liam. The younger brother is 4, and 3 times 4 is 12 = Noah's age. Everything checks.

Another way: Guess and check (tool 6): try Liam = 12, then Ella = 20, total = 2(20) + 2(12) = 64, which matches; then 12 / 3 = 4.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 12 by 3 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 8-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Liam plus 16 equals 64 for Liam's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 8 hard

Lily and Rose are twin sisters, and Owen and Cole are twin brothers. The four of them are $72$ years old in all, and Rose is $8$ years older than Owen. If Cole's age is $2$ times the age of Cole's younger brother, how old is Cole's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Lily and Rose) are the same age, and two brothers (Owen and Cole) are the same age. The four ages add to 72. Rose is 8 years older than Owen. Also, Cole is 2 times as old as his younger brother. We must find the younger brother's age.

Givens

Lily and Rose are twins, so they have the same age
Owen and Cole are twins, so they have the same age
The four ages together total 72
Rose is 8 years older than Owen
Cole's age is 2 times his younger brother's age

Unknowns

The age of Cole's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Rose is 8 more than Owen'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Cole is 2 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Lily and Rose are the same age; Owen and Cole are the same age. So the total is twice Rose's age plus twice Owen's age, and that total is 72.

2 \times (\text{Rose}) + 2 \times (\text{Owen}) = 72

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 8-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Rose is 8 older than Owen, so replace Rose with Owen + 8. The equation becomes 2(Owen + 8) + 2(Owen) = 72, which simplifies to 4 times Owen plus 16 equals 72.

2(\text{Owen}+8) + 2\,\text{Owen} = 72 \;\Rightarrow\; 4\,\text{Owen} + 16 = 72

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Owen's (and Cole's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 16 from 72 to get 56, then divide by 4. Owen is 14, and since Cole is his twin, Cole is also 14.

4\,\text{Owen} = 72 - 16 = 56 \;\Rightarrow\; \text{Owen} = 14,\quad \text{Cole} = 14

Undoing 'add 16' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Cole is 14, and Cole's age is 2 times his younger brother's age. So the younger brother's age is 14 divided by 2, which is 7.

\text{younger brother} = 14 \div 2 = 7

'2 times as old' means the younger child is one-2th of Cole's age, found by dividing.

Answer: 7 years old

4 · Reviewdoes it hold up?

Rose = 22, Lily = 22, Owen = 14, Cole = 14 add to 72, and Rose is indeed 8 more than Owen. The younger brother is 7, and 2 times 7 is 14 = Cole's age. Everything checks.

Another way: Guess and check (tool 6): try Owen = 14, then Rose = 22, total = 2(22) + 2(14) = 72, which matches; then 14 / 2 = 7.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 14 by 2 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 8-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Owen plus 16 equals 72 for Owen's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 9 hard

Maya and Tess are twin sisters, and Adam and Eric are twin brothers. The four of them are $80$ years old in all, and Tess is $8$ years older than Adam. If Eric's age is $4$ times the age of Eric's younger brother, how old is Eric's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Maya and Tess) are the same age, and two brothers (Adam and Eric) are the same age. The four ages add to 80. Tess is 8 years older than Adam. Also, Eric is 4 times as old as his younger brother. We must find the younger brother's age.

Givens

Maya and Tess are twins, so they have the same age
Adam and Eric are twins, so they have the same age
The four ages together total 80
Tess is 8 years older than Adam
Eric's age is 4 times his younger brother's age

Unknowns

The age of Eric's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Tess is 8 more than Adam'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Eric is 4 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Maya and Tess are the same age; Adam and Eric are the same age. So the total is twice Tess's age plus twice Adam's age, and that total is 80.

2 \times (\text{Tess}) + 2 \times (\text{Adam}) = 80

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 8-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Tess is 8 older than Adam, so replace Tess with Adam + 8. The equation becomes 2(Adam + 8) + 2(Adam) = 80, which simplifies to 4 times Adam plus 16 equals 80.

2(\text{Adam}+8) + 2\,\text{Adam} = 80 \;\Rightarrow\; 4\,\text{Adam} + 16 = 80

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Adam's (and Eric's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 16 from 80 to get 64, then divide by 4. Adam is 16, and since Eric is his twin, Eric is also 16.

4\,\text{Adam} = 80 - 16 = 64 \;\Rightarrow\; \text{Adam} = 16,\quad \text{Eric} = 16

Undoing 'add 16' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Eric is 16, and Eric's age is 4 times his younger brother's age. So the younger brother's age is 16 divided by 4, which is 4.

\text{younger brother} = 16 \div 4 = 4

'4 times as old' means the younger child is one-4th of Eric's age, found by dividing.

Answer: 4 years old

4 · Reviewdoes it hold up?

Tess = 24, Maya = 24, Adam = 16, Eric = 16 add to 80, and Tess is indeed 8 more than Adam. The younger brother is 4, and 4 times 4 is 16 = Eric's age. Everything checks.

Another way: Guess and check (tool 6): try Adam = 16, then Tess = 24, total = 2(24) + 2(16) = 80, which matches; then 16 / 4 = 4.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 16 by 4 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 8-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Adam plus 16 equals 80 for Adam's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Variant 10 hard

Ruby and Jade are twin sisters, and Wade and Kent are twin brothers. The four of them are $88$ years old in all, and Jade is $8$ years older than Wade. If Kent's age is $3$ times the age of Kent's younger brother, how old is Kent's younger brother?

Your answer

Show solution

1 · Understandwhat's really being asked

Two sisters (Ruby and Jade) are the same age, and two brothers (Wade and Kent) are the same age. The four ages add to 88. Jade is 8 years older than Wade. Also, Kent is 3 times as old as his younger brother. We must find the younger brother's age.

Givens

Ruby and Jade are twins, so they have the same age
Wade and Kent are twins, so they have the same age
The four ages together total 88
Jade is 8 years older than Wade
Kent's age is 3 times his younger brother's age

Unknowns

The age of Kent's younger brother

Constraints

All ages are whole numbers
Twins share one age value each, so there are really only two unknown ages among the four

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems

Because each pair of twins shares one age, the four people collapse into just two unknown ages tied together by 'Jade is 8 more than Wade'. Writing the total as a single-unknown equation lets us solve for one twin's age, then a second small subproblem (Kent is 3 times the younger brother) gives the final answer.

3 · Execute4 carry out the plan

1Collapse the twins into two ages

#13 Convert to Algebra 3.OA.A.3

Ruby and Jade are the same age; Wade and Kent are the same age. So the total is twice Jade's age plus twice Wade's age, and that total is 88.

2 \times (\text{Jade}) + 2 \times (\text{Wade}) = 88

Twins having equal ages means we only have two different numbers to track, not four.

2Use the 8-year relation to get one unknown

#13 Convert to Algebra 3.OA.D.8

Jade is 8 older than Wade, so replace Jade with Wade + 8. The equation becomes 2(Wade + 8) + 2(Wade) = 88, which simplifies to 4 times Wade plus 16 equals 88.

2(\text{Wade}+8) + 2\,\text{Wade} = 88 \;\Rightarrow\; 4\,\text{Wade} + 16 = 88

Substituting the relationship turns two unknowns into one, so a single equation remains.

3Solve for Wade's (and Kent's) age

#13 Convert to Algebra 3.OA.A.4

Subtract 16 from 88 to get 72, then divide by 4. Wade is 18, and since Kent is his twin, Kent is also 18.

4\,\text{Wade} = 88 - 16 = 72 \;\Rightarrow\; \text{Wade} = 18,\quad \text{Kent} = 18

Undoing 'add 16' then 'times 4' isolates the one unknown age.

4Find the younger brother's age

#7 Identify Subproblems 3.OA.A.3

Kent is 18, and Kent's age is 3 times his younger brother's age. So the younger brother's age is 18 divided by 3, which is 6.

\text{younger brother} = 18 \div 3 = 6

'3 times as old' means the younger child is one-3th of Kent's age, found by dividing.

Answer: 6 years old

4 · Reviewdoes it hold up?

Jade = 26, Ruby = 26, Wade = 18, Kent = 18 add to 88, and Jade is indeed 8 more than Wade. The younger brother is 6, and 3 times 6 is 18 = Kent's age. Everything checks.

Another way: Guess and check (tool 6): try Wade = 18, then Jade = 26, total = 2(26) + 2(18) = 88, which matches; then 18 / 3 = 6.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Setting up the doubled-twin total and dividing 18 by 3 for the younger brother
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the total and the 8-year relation into one equation
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 times Wade plus 16 equals 88 for Wade's age

💡Takeaway. When twins share an age, four people become just two numbers, so the whole puzzle fits into one neat equation you can solve!

Where next?

← Back to the problem: Reduce relations to a single-unknown equation