Question

solve.
ann, betty and claire are sisters. ann is the youngest, and betty is in high school. betty’s age is ann’s age squared. claire’s age is one less than three times ann’s age. the sum of their ages is 31. ann’s age and claire’s age combined is 1 less than betty’s age.
let ( x = ) ann’s age. find all 3 girls’ ages.
ages
ann: ______ betty: ____ claire: ______

Explanation:

Step1: Define variables and equations

Let \( x \) be Ann's age. Then Betty's age is \( x^2 \), and Claire's age is \( 3x - 1 \).
From the sum of their ages: \( x + x^2 + (3x - 1) = 31 \).
From Ann and Claire's combined age: \( x + (3x - 1) = x^2 - 1 \).

Step2: Simplify the second equation

Simplify \( x + 3x - 1 = x^2 - 1 \):
\( 4x - 1 = x^2 - 1 \)
Subtract \( 4x - 1 \) from both sides: \( x^2 - 4x = 0 \)
Factor: \( x(x - 4) = 0 \)
Solutions: \( x = 0 \) (invalid, age can't be 0) or \( x = 4 \).

Step3: Verify with the first equation

If \( x = 4 \), Betty's age is \( 4^2 = 16 \), Claire's age is \( 3(4) - 1 = 11 \).
Sum: \( 4 + 16 + 11 = 31 \), which matches.

Answer:

Ann: \( 4 \)
Betty: \( 16 \)
Claire: \( 11 \)