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Math Help - Age Word Problem

  1. #1
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    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?

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  2. #2
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    Quote Originally Posted by magentarita View Post
    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?
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  3. #3
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    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 M = my age (now).
    Let A = Anna's age (now).


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

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

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

    We have: . 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.

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

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

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

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

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


    Therefore, Anna is 43 years old.

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

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

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


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


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

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

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

    We have: . 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.

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

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

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

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

    Substitute into [1]: . 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.
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