Reduce relations to a single-unknown equation

3.OA.D.83.OA.A.3 · adapt · grade 3

Archetype: Find Two Unknowns from Sum and Difference · step in a 8-type progression

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?

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Understand

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

Plan

#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.

Execute

#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.

#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.

#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.

#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-third of Noah's age, found by dividing.

Answer: 4 years old

Review

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, and a 4-year-old being much younger than the 12-year-old twins is sensible.

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

Standards · min 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

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