ten ​coins, consisting of dimes and​ nickels, into a vending machine to purchase a Snickers bar for 70 cents. How many coins of each type did she​ use?

Question

inserted ten coins, consisting of dimes and nickels, into a vending machine to purchase a Snickers bar for 70 cents. How many coins of each type did she use?

Answer 1

Let:

   D = number of dimes

   N = number of nickels

 

Translate:

"ten coins, consisting of dimes and nickels"   means    D + N = 10      [eq1]

 

"purchase a Snickers bar for 70 cents"    means     10D + 5N = 70     [eq2]

 

Important note:

     10D is the value of the dimes in cents

      5B is the value of the nickels in cents

 The total is 70 cents.

 

You may substitute.  First isolate one variable:

      D = 10 - N           [from eq1]

Put that into the other equation [eq2]:

     10(10-N)   +  5N   = 70

 

    100 - 10N + 5N = 70        ]distribute]

    100 - 5N = 70                  [collect terms]

      -5N = -30               [subtract 100 from both sides]

        N = 6             [divide by (-5)]

     There are 6 nickels.

 

Use either equation to determine N (eq1 is easy]:

      D + N = 10

      D = 10 - N

      D = 4

   There are 4 dimes.

 

Check (very important):

  Is    10(4) + 5(6) = 70   ?

          40  +  30  = 70   ?

             70 = 70   ?yes

 

Answer 2

Let d = # dimes   and   n = # nickels

10d + 5n = 70

d + n = 10 coins

n = 10 - d

Substituting

10d + 5 ( 10 - d) = 70

10d + 50 - 5d = 70

5d = 20

d=4

n = 10 - d = 10 - 4 = 6

6 nickels and 4 dimes

Checking

6 (5) + 4(10) = 70

ok !