The sum of three numbers is 120 . The third number is 4 times the first. The first number is 6 more than the second. What are the numbers?

Question

The sum of three numbers is
120
. The third number is
4
times the first. The first number is
6
more than the second. What are the numbers?

Answer 1

Let the numbers be x, y, and z.

 Translate:

  "sum of three numbers is 120"  means   x+y+z=120

  "third number is 4 times the first"      means  z = 4x

  "first number is 6 more than the second"    means   x = 6+y 

 

With problems that express the relationship between unknown numbers, look for the one that others depend on.  Here it is y (the second), put everything in terms of y.

 

x         +   y    +   z   =  120                   [first equation]

x         +   y    +   (4x)   =  120               [substitute for z]

(6+y)   +   y   +   (4)(6+y)  = 120          [substitute for x]

 

6 + y + y + 24 + 4y = 120

    6y + 30 = 120

    6y = 90

      y=15

x = 6+y = 21

z = 4x = (4)(21)=84

 

Checking (very important):

  Is  21 + 15 + 84 = 120   ?

         120 = 120   ? yes

  Is  84 = 4(21)   ?

       84 = 84   ? yes

  Is  21 = 6+15   ?

       21 = 21   ? yes

Answer 2

From this information you can create three equations:

 

x + y + z = 120

 

z = 4x

 

x = y + 6

 

 

Now, substitute information from the second two equations into the first in order to reduce the variables to one.  For instance:

 

(y + 6) + y + z =120

 

(y + 6) + y + 4x = 120

 

x + (x-6) + 4x = 120

 

Once you solve for x you can then go back and plug that value into the 2nd from last equation above, to get an equation which you can solve for y.  Then take the values of x and y and plug them into the original equation to solve for z.

 

Good luck!