Age Relationship Problem: Solve for Hector's Age Using 3x and -10 Relations

To solve this problem, let's define the variables and set up the equation:

• Let Ivana's age be $x$.
• Then, Luciana's age is $3x$, since her age is three times Ivana's age.
• Hector, being 10 years younger than Luciana, would be $3x - 10$.

We know the sum of their ages is 32, so we set up the equation:

$x + 3x + (3x - 10) = 32$

Simplify the equation:

$x + 3x + 3x - 10 = 32$

$7x - 10 = 32$

Next, solve for $x$:

Add 10 to both sides:

$7x = 42$

Divide both sides by 7:

$x = 6$

Now that we know Ivana is 6 years old, we can find Hector's age:

Hector's age = $3x - 10 = 3(6) - 10 = 18 - 10 = 8$

Therefore, Hector is 8 years old.