SOLUTION: A collection of 100 coins consist of 5 and 10 if the total value is 725 find the number of coins of each denomination

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Question 1023819: A collection of 100 coins consist of 5 and 10 if the total value is 725 find the number of coins of each denomination
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Let the number of nickels be x
Let the number of dimes be y


                      Value      Value
Type       Number       of         of
 of          of        EACH       ALL
coin        coins      coin      coins
-------------------------------------------
nickels      x        $0.05      $0.05x
dimes        y        $0.10      $0.10y
-------------------------------------------
TOTALS       100      -----      $7.25

 The first equation comes from the second column.

  %28matrix%283%2C1%2CNumber%2Cof%2Cnickels%29%29%22%22%2B%22%22%28matrix%283%2C1%2CNumber%2Cof%2Cdimes%29%29%22%22=%22%22%28matrix%284%2C1%2Ctotal%2Cnumber%2Cof%2Ccoins%29%29

                 x + y = 100

 The second equation comes from the last column.
  %28matrix%284%2C1%2CValue%2Cof%2CALL%2Cnickels%29%29%22%22%2B%22%22%28matrix%284%2C1%2CValue%2Cof%2CALL%2Cdimes%29%29%22%22=%22%22%28matrix%285%2C1%2CTotal%2Cvalue%2Cof%2CALL%2Ccoins%29%29

         0.05x + 0.10y = 7.25

Get rid of decimals by multiplying every term by 100:

              5x + 10y = 725

 So we have the system of equations:

           system%28x+%2B+y+=+100%2C5x+%2B+10y+=+725%29.

We solve by substitution.  Solve the first equation for y:

           x + y = 100
               y = 100 - x

Substitute (100 - x) for y in 5x + 10y = 725

    5x + 10(100 - x) = 725
     5x + 1000 - 10x = 725
          -5x + 1000 = 725
                 -5x = -275
                   x = 55 = the number of nickels.

Substitute in y = 100 - x
              y = 100 - (55
              y = 45 dimes.

Checking:  55 nickels is $2.75 and 45 dimes is $4.50
            That's 100 coins.
            And indeed $2.75 + $4.50 = $7.25
Edwin