How would I write an equation for this, also an explanation please?

Question

"Paula's purse contains twice as many dimes as quarters and three fewer nickels than dimes. The total of these coins $3.15" 

Answer 1

The key to solving word (story) problems is to correctly translate the words and phrases into precise, concise math notation (with variables and operators).

 

First, try to assign variables to the values the problem asks you to find.  Since this problem only wants you to write an equation, let's use:

      N = number of nickels

      D = number of dimes

      Q = number of quarters

 

Translate:  

      "twice as many dimes as quarters"      means     D = 2Q 

                        [the number of dimes is twice the number of quarters]

      "three fewer nickels than dimes"    means   N = D - 3         (read this out loud)

      "The total of these coins $3.15"     means     5N + 10D + 25Q = 315

 

Just a minute --  "5 times the number of nickels (N) equals the number of cents for those nickels"    and

        $3.15 has been changed to 315 cents.   O.K.? go on.

 

Now, that we have the equation:      5N + 10D + 25Q = 315       we can solve it by changing everything to Q

                                         5(D-3)   + 10D  + 25Q  = 315             (replace N with D-3)

                                      5((2Q)-3))  +  10(2Q)   + 25Q    = 315     (replace D with 2Q)

                                      10Q - 15   + 20Q   + 25Q   =  315               (expand terms)

                                         50Q + 15 = 315                                   (collect terms)

                                             50Q = 300                                 (subtract 15 from both sides)

                                               Q=6                                  )divide both sides by 50)

 

Now, put Q=50 into the first equation to find D

          D = 2Q

          D = 12

 

Then, put that value into the equation N=D-3 to find N.

          N = 9

 

Now, check (very important):

   Is   5N + 10D + 25Q = 315  ?

         5(9) + 10(12) + 25(6)  =  315    ?

         45  + 120  + 150  = 315   ?

              315 = 315  ?    yes !

 

Answer 2

Assign x to any of the coins.

 

Let the number of dimes = x

Let the number quarters = (1 / 2)x = 0.5x

Let the number nickels = x - 3

 

Each of these coins have their own value, and the total is $3.15.  A dime is worth 10¢.  A quarter is worth 25¢.  A nickel is worth 5¢.

 

 

0.10x + 0.25(0.5x) + 0.05(x - 3) = 3.15