A man is now 4 times as old as his son, but in 20 years he will only be twice as old. How old is each now?

Question

A man is now 4 times as old as his son, but in 20 years he will only be twice as old. How old is each now?

Answer 1

Assign variables:    M = man's age now;   S = son's age now

 

Translate phrases into concise, precise math expressions (with variables and expressions):

  "A man is now 4 times as old as his son"     means     M = 4*S    eq1

  "in 20 years he will only be twice as old"      means    (M+20) = 2*(S+20)     eq2

  "How old is each now?"       means      report M,S

 

Now, the math:

  To get rid of M, plug value for M from eq1 into eq2

         4*S + 20 = 2*(S+20)

         4*S + 20  = 2*S + 40          (distribute 2)

               2*S    =  20                   (collect S and constants)

                  S   = 10                    (divide both sides by 2)

  Plug 10 into eq1:       M = 4*10

                                  M = 40

Checking (always a good idea):

  Is    40 = 4*10  ?    yes.

  Is    40+20 = 2*(10+20)  ?

             60   = 60    ?    yes.

Answer 2

To solve these problems you will want to rename each of these people's ages as variables to start. In this case I will call the man's age x, and the son's age y. From the first part of the sentence we know that the man's age currently, x, is equal to four times the son's age, y, or x = 4y. The second half of the sentence tells us that in twenty years the man will only be twice as old as his son, or (x+20) = 2(y+20). Now, we have a system of equations that can be solved for using either the methods of substitution or elimination.

By substitution:

x = 4y   From this equation we know that we can substitute 4y in wherever there is an x in the second equation

(x+20) = 2(y+20) ---Subsitute 4y for x---> (4y+20) = 2(y+20) ---Simplifying---->  4y+20 = 2y+40  

4y-2y = 40-20  Get the y's by themselves

2y=20

y=10

By elimination:

x = 4y

(x+20) = 2(y+20) You want to get rid of one of the variables in the second equation by subtracting one from the first.

(x+20)-x = 2(y+20) - 4y This is easily done here by simply subtracting the first equation from the second

20 = 2y-4y+40     This leaves us with only one variable and a much simpler equation to solve

-20 = -2y

y = 10

Now we can plug y=10 into both of the previous equations and make sure that we get the same answer of x both times.

x = 4y  ---->  x = 4(10)   ------>  x = 40

(x+20) = 2(y+20) -----> (x+20) = 2(10+20)  ------> (x+20) = 2(30) -------> x=40

Both of these come out to x = 40, so we can conclusively say that the man is 40 years old and the son is 10 years old right now.