I am confused by on how to answer this

Question

Shirley is currently 2 years more than 4 times Rose's age. In 12 years Rose will be one half of Shirley's age. How old are they at the moment

Answer 1

There must be some values that we can use as "reference points" to make expressions for other values.  In this problem, "how old are they at the moment" are the values we are asked to find, so:

    Let x = Shirley's age at the moment

         y = Rose's age at the moment

 

Now, let's translate the problem:

 

  "Shirley is currently 2 years more than 4 times Rose's age"     means

       x      =              2             +            4   *    y

 

  "In 12 years Rose will be one half of Shirley's age"       means

        (12   +   y)         =     (1/2)    *    (x + 12)

 

It is important to realize that, in English, the problem says, "one half of Shirley's age," but the "In 12 years" applies to the entire rest of the sentence, so it means, "one half of Shirley's age in 12 years."

 

Now, the math is easy:

      x = 4y + 2                     [re-write]              [eq1]

     y + 12 = (1/2)(x+12)                               [eq2]

   [note:   tutors and teachers have done this so much, they often start with these equations

      forgetting that students must work to translate the words.]

 

   We could first simplify eq2, but let's just substitute x from eq12 into it:

     y + 12 = (1/2)( (4y+2) + 12)

     y + 12 = 2y + 7

         -y = -5         [subtract 2y from both sides; subtract 12 from both sides]

         y = 5          [multiply both sides by (-1)]

 

and, from eq1, x= 22.

 

Now, to check, put the values of x and y back into the problem sentences:

    "22 is 4 more than 4 times 5.   Adding 12, (5+12) is half of (22+12), or 17 is half of 34.