# Thread: Age Word Problem

1. ## Age Word Problem

In 7 years, I will be half my mother's age. Thirteen years ago, she was 6 times as old as I was then.
How old is Anna now?

2. Originally Posted by magentarita
In 7 years, I will be half my mother's age. Thirteen years ago, she was 6 times as old as I was then.
How old is Anna now?

Is there something missing? Who is Anna?

3. Hello, magentarita!

I assume your mother's name is Anna . . .

In 7 years, I will be half Anna's age.
Thirteen years ago, she was 6 times as old as I was then.
How old is Anna now?

Let $\displaystyle M$ = my age (now).
Let $\displaystyle A$ = Anna's age (now).

Seven years from now, both of us are 7 years older.

. . $\displaystyle \begin{array}{ccc}\text{My age} &=& M + 7 \\ \text{Her age} &=& A + 7 \end{array}$

. . $\displaystyle \underbrace{\text{My age}}_{M+7}\: \underbrace{\text{will be}}_{=}\: \underbrace{\text{half her age}}_{\frac{1}{2}(A+7)}$

We have: .$\displaystyle M + 7 \:=\:\tfrac{1}{2}(A+7) \quad\Rightarrow\quad A - 2M \:=\:7\;\;{\color{blue}[1]}$

Thirteen years ago, both of us were 13 years younger.

. . $\displaystyle \begin{array}{ccc}\text{My age} &=& M-13 \\ \text{Her age} &=& A - 13 \end{array}$

. . $\displaystyle \underbrace{\text{Her age}}_{A-13}\: \underbrace{\text{was}}_{=}\: \underbrace{\text{6 times my age}}_{6(M-13)}$

We have: .$\displaystyle A - 13 \:=\:6(M-13) \quad\Rightarrow\quad A - 6M \:=\:-65\;\;{\color{blue}[2]}$

Subtract [2] from [1]: .$\displaystyle 4M \:=\:72 \quad\Rightarrow\quad M \:=\:18$

Substitute into [1]: .$\displaystyle A -2(18) \:=\:7 \quad\Rightarrow\quad\boxed{ A \:=\:43}$

Therefore, Anna is 43 years old.

4. ## You know...............

Originally Posted by Soroban
Hello, magentarita!

I assume your mother's name is Anna . . .

Let $\displaystyle M$ = my age (now).
Let $\displaystyle A$ = Anna's age (now).

Seven years from now, both of us are 7 years older.

. . $\displaystyle \begin{array}{ccc}\text{My age} &=& M + 7 \\ \text{Her age} &=& A + 7 \end{array}$

. . $\displaystyle \underbrace{\text{My age}}_{M+7}\: \underbrace{\text{will be}}_{=}\: \underbrace{\text{half her age}}_{\frac{1}{2}(A+7)}$

We have: .$\displaystyle M + 7 \:=\:\tfrac{1}{2}(A+7) \quad\Rightarrow\quad A - 2M \:=\:7\;\;{\color{blue}[1]}$

Thirteen years ago, both of us were 13 years younger.

. . $\displaystyle \begin{array}{ccc}\text{My age} &=& M-13 \\ \text{Her age} &=& A - 13 \end{array}$

. . $\displaystyle \underbrace{\text{Her age}}_{A-13}\: \underbrace{\text{was}}_{=}\: \underbrace{\text{6 times my age}}_{6(M-13)}$

We have: .$\displaystyle A - 13 \:=\:6(M-13) \quad\Rightarrow\quad A - 6M \:=\:-65\;\;{\color{blue}[2]}$

Subtract [2] from [1]: .$\displaystyle 4M \:=\:72 \quad\Rightarrow\quad M \:=\:18$

Substitute into [1]: .$\displaystyle A -2(18) \:=\:7 \quad\Rightarrow\quad\boxed{ A \:=\:43}$

Therefore, Anna is 43 years old.
You know, I'm not so bad with age word problems but this one threw me off just a bit.