Age & time puzzles

Every age puzzle rests on one fact: people age at the same rate, so the difference between any two people's ages never changes, and if $t$ years pass, everyone's age grows by exactly $t$. Give each person's present age a variable, such as $x$ for Tommy and $y$ for papa. Translate each clue into an equation: "six times as old" means $y=6x$; "ten years ago" means subtract $10$ from every variable; "in five years" means add $5$ to every variable. Solve the resulting system, then check that the ages make sense in the original story. Do not treat ages as fixed numbers that only one person changes.

Alice is now half as old as Betty. In 12 years, Alice will be two-thirds as old as Betty. How old is Betty now?

Step 1. Let Alice's present age be $a$ and Betty's present age be $b$.
a = rac{1}{2}b
Step 2. "In 12 years, Alice will be two-thirds as old as Betty" means
a + 12 = rac{2}{3}(b + 12)
Step 3. Substitute a = rac{1}{2}b into the second equation:
rac{1}{2}b + 12 = rac{2}{3}(b + 12)
Step 4. Multiply by 6 to clear fractions: $3b+72=4b+48$, so $b=24$.
Step 5. Thus Betty is 24 and Alice is 12. Check: in 12 years they are 36 and 24, and 24 = rac{2}{3}(36).