First, what coins are available:
p = Penny
n = Nickel
d = Dime
q = Quarter
h = Half Dollar
 
Some combination of these coins must equal $1.00, therefore
 
.01p + .05n + .10d + .25q + .5h = $1.00
p + n + d + q + h = 19
 
Questions we should consider:
  1. Can we achieve a solution with only one type of coin? No.
  2. 100 pennies = $1
  3. 20 nickels = $1
  4. 10 dimes = $1
  5. 4 quarters = $1
  6. 2 half dollars = $1
  7.  
What happens if we use pennies?
  1. Pennies must be used in multiples of 5, otherwise we cannot reach a whole number
  2. Since we cannot use 20 pennies, we can only use 5, 10, or pennies
 
If we use 5 pennies
  1. .05n + .10d + .25q + .5h = $0.95
  2. n + d + q + h = 14
Now consider nickels
  1. 14 nickels only equals $0.70, so we need to reduce the number of nickels
  2. If we use 12 nickels
  3. .10d + .25q + .5h = $0.35
  4. d + q + h = 2
Now consider the other coins
  1. It seems clear from this point that we will need 1 dime and 1 quarter
  2. .01(5) + .05(12) + .10(1) + .25(1)= $1.00
 
Are there other combinations that can be made using 5 pennies?
 
Let’s consider nickels:
  1. 18 nickels = $0.90
  2. We can only use 1 coin and it must equal $010, so in this case it must be 1 dime
  3. .05(18) + .10(1) = $1.00
  4.  
Now let’s look at dimes:
  1. 9 dimes = $0.90
  2. We must use 10 more coins and they can’t sum to more than $0.10
  3. 1 nickel and 5 pennies sums to $0.10, but only requires 6 coins
  4. Therefore, the only option is to use 10 pennies
  5. .01(10) + .10(9) = $1.00
 
Now let’s look at quarters:
  1. 3 quarters = $0.75
  2. We must use 16 more coins and they can’t sum to more than $0.25
  3. 2 dimes and 1 nickel sums to $0.25, but only requires 3 coins
  4. 15 pennies and 1 dime has the correct sum and correct number of coins
  5. .01(15) + .10(1) + .25(3)= $1.00
  6.  
  7. 1 quarter = $0.25
  8. We must use 18 more coins and they can’t sum to more than $0.85
  9. 2 dimes and 1 nickel sums to $0.25, but only requires 3 coins
  10. 15 pennies and 1 dime has the correct sum and correct number of coins
  11. .01(10) + .05(1) + .10(7) + .25(1)= $1.00
 
Now let’s look at half dollars:
  1. 1 half dollar = $0.50
  2. We must use 18 more coins and they can’t sum to more than $0.50
  3. 15 pennies and 1 dime sums to $0.25, but we can’t reach $1 with only 2 coins
  4. 10 pennies, 1 dime, and 1 nickel sums to $0.25, but does not use enough coins
  5. .01(15) + .10(1) + .25(3)= $1.00
  6.  
Does adding the equations help?
 
 
Coins
Molly McKee
How many ways can 19 coins equal exactly one dollar?
The Problem
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  2. Molly McKee
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