In three years time Ann will one third of her mother's age. In another 18 years from now, Ann will be half her mother's age. How old is her mother now??
Hello, Spixol123!
In three years time Ann will one-third of her mother's age.
18 years from now, Ann will be half her mother's age.
How old is her mother now?
Let $\displaystyle A$ = Ann's age now.
Let $\displaystyle M$ = Mother's age now.
In three years, they will both be three years older.
. . Ann will be $\displaystyle A + 3$ years old.
. . Mother will be $\displaystyle M + 3$ years old.
At that time, Ann will be one-half her Mother's age.
. . $\displaystyle A + 3 \:=\:\tfrac{1}{2}(M + 3) \quad\Rightarrow\quad M - 3A \:=\:6$ .[1]
In eighteen years, they will both be eighteen years older.
. . Ann will be $\displaystyle A + 18$ years old.
. . Mother will be $\displaystyle M + 18$ years old.
At that time, Ann will be one-third her Mother's age.
. . $\displaystyle A + 18 \:=\:\tfrac{1}{3}(M + 18) \quad\Rightarrow\quad M - 2A \:=\:18 $ .[2]
We have: .$\displaystyle \begin{Bmatrix}M - 3A &=& 6 & {\color{blue}[1]} \\ M - 2A &=& 18 & {\color{blue}[2]} \end{Bmatrix}$
Subtract [2] - [1]: .$\displaystyle A \:=\:12$
Substitute into [2]: .$\displaystyle M - 2(12) \:=\:18 \quad\Rightarrow\quad M \:=\:42$
Her mother is 42 years old now.