Lesson HOW TO algebraize and to solve age problems?

HOW TO algebraize and to solve age problems?

Problem 1

Let x be the young man's current age.


In 32 years his age will be (x+32) years.

8 years back his age was (x-8) years.


The problem says that the age (x+32) years is 5 times as big as the age of (x-8) years.


So, you write this equation

    x + 32 = 5*(x-8).


Next, you simplify the equation step by step

    x + 32 = 5x - 40

    32 + 40 = 5x - x

    72      = 4x

     x      = 72/4 = 18.


ANSWER.  The young man is 18 years old.


You may check it on your own that all problem's conditions are satisfied; hence, the answer is correct.

Problem 2

Let x be the Susan age.

Then the Raymond's age is (x-12), according to the condition.


Four years ago, Susan was (x-4) years old.

Four years ago, Raymond was ((x-12)-4) = x - 16 years old.



    +----------------------------------------------------------+
    |   "Four years ago, she was 4 times as old as he was."    |
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It means that


    x - 4 = 4*(x-16).    <<<---===  it is your setup equation.

                                    You take it from the condition, using the introduced variables.


Now you have this single equation and simplify it step by step


    x - 4  = 4x - 64

    64 - 4 = 4x - x

    60     = 3x

     x = 20.


ANSWER.  Susan is 20 years old now.  Raymond is 12 years younger, i.e. 20=12 = 8 years old.

Problem 3

Let x be the daughter's age now, and let y be the Ed's age.


Eight years ago, the daughter was (x-8) years old;

                 Ed was (y-8) years old.


So, your first equation is

    y - 8 = 10*(x-8).     (1)



In six years, Ed will be (y+6) years old;

              his daughter will be (x+6) years old.


So, your second equation is

    y + 6 = 3*(x+6).        (2)


Now our task is to solve equations (1) and (2).
For it, express y from equation (1)

    y = 10*(x-8) + 8 = 10x - 80 + 8 = 10x - 72     (3)


and substitute it into equation (2).  You will get

    10x - 72 + 6 = 3x + 18

    10x - 3x     = 18 + 72 - 6

    7x           = 84

     x           = 84/7 = 12.


Thus the daughter is 12 years old now.

Then the father is, according to (3), 10x - 72 = 10*12 - 72 = 120 - 72 = 48 years old.

Problem 4

Let "x" be the present age of the elder son.
Then the father's present age is 2x, while the younger son's present age is (x-15).

In 10 years: The father's age will be      (2x)+10 years;
             the younger son's age will be (x-15) + 10 = x-5 years

From the other side, in ten years the age of the father will be 3(x-5) years.

Thus you get an equation

    2x + 10 = 3(x-5).


Simplify and solve for x:

    2x + 10 = 3x - 15  --->  10 + 15 = 3x - 2x  --->  x = 25.


Answer. The elder son's present age is 25 years.

        The younger son's present age is 25-15 = 10 years.

        The father's present age is 2*25 = 50 years.