Finding Present Ages

Maria is 4 years older than her brother Juan. The sum of their ages is 22. How old is each person?

Define the variable: Let $x$ = Juan's age = $x$ represents Juan's age

Express Maria's age: Maria is 4 years older: $x + 4$ = Maria's age = $x + 4$

Set up the equation: Sum of ages = 22: $x + (x + 4) = 22$ = $2x + 4 = 22$

Solve for x: $2x = 18$, so $x = 9$ = Juan is 9 years old

Find Maria's age: $9 + 4 = 13$ = Maria is 13 years old

Check the answer: $9 + 13 = 22$ ✓ and $13 - 9 = 4$ ✓ = Both conditions satisfied!

Answer: Juan is 9 years old and Maria is 13 years old.

Ages in the Past

Tom is currently 35 years old. Five years ago, Tom was twice as old as his daughter. How old is Tom's daughter now?

Define the variable: Let $x$ = daughter's current age = $x$ represents daughter's age now

Express ages 5 years ago: Tom was $35 - 5 = 30$, daughter was $x - 5$ = Past ages: Tom = 30, daughter = $x - 5$

Set up the equation: Tom was twice daughter's age: $30 = 2(x - 5)$ = $30 = 2x - 10$

Solve for x: $30 + 10 = 2x$, so $40 = 2x$, $x = 20$ = Daughter is 20 years old

Check the answer: 5 years ago: Tom was 30, daughter was 15. $30 = 2 \times 15$ ✓ = Correct!

Answer: Tom's daughter is currently 20 years old.

Ages in the Future

A father is currently 3 times as old as his son. In 12 years, the father will be twice as old as his son. Find their current ages.

Define the variable: Let $x$ = son's current age = $x$ represents son's age now

Express father's current age: Father is 3 times son's age: $3x$ = Father's current age = $3x$

Express ages in 12 years: Son: $x + 12$, Father: $3x + 12$ = Future ages expressed

Set up the equation: Father will be twice son's age: $3x + 12 = 2(x + 12)$ = $3x + 12 = 2x + 24$

Solve for x: $3x - 2x = 24 - 12$, so $x = 12$ = Son is 12 years old

Find father's age: $3 \times 12 = 36$ = Father is 36 years old

Check both conditions: Now: $36 = 3 \times 12$ ✓ In 12 years: $48 = 2 \times 24$ ✓ = Both conditions satisfied!

Answer: The son is 12 years old and the father is 36 years old.

Mistake: Forgetting to adjust ALL ages when moving through time

Why: When the problem says '5 years ago' or 'in 10 years', you must add or subtract from EVERY person's age, not just one.

Correct: If Juan is $x$ now and Maria is $x + 4$ now, then 5 years ago Juan was $x - 5$ and Maria was $(x + 4) - 5 = x - 1$.

Mistake: Confusing 'older than' with 'times as old as'

Why: '5 years older' means +5, while 'twice as old' means $\times 2$. These are very different operations!

Correct: Read carefully: 'Maria is 5 years older' → $x + 5$. 'Maria is twice as old' → $2x$.

Mistake: Not checking if the answer makes sense

Why: Ages must be positive numbers, and the relationships must hold at all time points mentioned in the problem.

Correct: Always verify: Are all ages positive? Do the original conditions work? If someone was '5 years ago', is their current age at least 5?

Family Relationships

Understanding age relationships helps in genealogy research and family planning.

If a mother is 28 years older than her daughter, and the mother is currently 45, the daughter is $45 - 28 = 17$ years old.

Retirement Planning

Calculating when someone will reach a certain age is essential for financial planning.

If you are 42 now and can retire at 65, you have $65 - 42 = 23$ years until retirement.

Q: How do I know which person's age to use as my variable?

A: Usually, choose the younger person's age as $x$. This makes other expressions positive (e.g., 'older by 5' becomes $x + 5$ instead of $x - 5$).

Q: What if I get a negative age as my answer?

A: A negative age means you made an error in setting up the problem, or the problem's conditions are impossible. Re-read the problem and check your equation.

Q: How do I handle problems with three or more people?

A: Define one variable for the youngest person, then express all other ages in terms of that variable. You may need multiple equations to solve.