the sum of two numbers is 54. if the larger is divided by the smaller. the quotient is 2 and the remainder is 3. find the number.

Question

the sum of two numbers is 54. if the larger is divided by the smaller. the quotient is 2 and the remainder is 3. find the number.

Answer 1

I would do this by setting up a system of equations.

 

Let x be the larger and y be the smaller number.

 

x+y = 54 is the first relationship.

 

Now it would be nice if the division problem worked out nicely, but we have a remainder. I'm not sure how your teacher would have done this, but I'll compare this to a regular division problem.

 

example...24/7 = 3 and 3/7

so x/y is 2 with a remainder of 3 is x/y = 2+3/y........multiply everything by y and you get

x = 2y + 3.

 

So your 2 equations are

 

x+y=54

x=2y+3

 

This is set up perfectly for the substitution method, so I can replace x with 2y+3 in the first equation to get

 

(2y+3) + y =54

3y             = 51

y               = 17

 

Substituting back in to either equation, you get x= 37.

 

so to check, 37+17    IS    54 and 37/17 = 2 with a remainder of 3.

 

Hope this helped.

Answer 2

Let x = larger number

     y = smaller number

 

Translate:

  "sum of two numbers is 54"        means    x + y = 54      [eq1]

  "larger is divided by the smaller. the quotient is 2 and the remainder is 3"  means

          x/y = 2 remainder 3

          x  = 2y + 3    [eq2]

 

Let's use elimination:

   x   +   y   =   54             [eq1 again]

   x   -   2y  =     3             [from eq2]

  --------------------      [elimination; subtract equations]

           3y    =   51

            y    =   17

  x   +  y    =   54        [eq1 again]

 --------------------      [elimination; subtract equations]

 -x             =  -37

  x             =   37

 

The numbers are 17 and 37.

 

Check:

   Is      37 + 17 = 54   ?        [eq1]

                     54 = 54   ?yes

  Is     37  = 2(17) + 3   ?   [eq2]

           37 = 34 + 3   ?

            37 = 37   ?yes