Case (1): It is impossible to have 19 coins of the same type that equal one dollar.
19 pennies = 19 cents
19 nickels = 95 cents
19 dimes = $1.90 (too big so quarters and half dollars are also eliminated)
Case (2): There are TWO ways to have one dollar using only two types of coins.
10 pennies and 9 dimes = $1
18 nickels and 1 dime = $1
Case (3): There are THREE ways to have one dollar using only three types of coins.
15 pennies, 1 dime, and 3 quarters = $1
10 pennies, 8 nickels, and 1 half dollar = $1
5 pennies, 9 nickels, and 5 dimes = $1
Case (4): There are FOUR ways to have one dollar using only four types of coins.
15 pennies, 2 nickels, 1 quarter, and 1 half dollar = $1
10 pennies, 3 nickels, 5 dimes, and 1 quarter = $1
10 pennies, 6 nickels, 1 dime, and 2 quarters = $1
5 pennies, 12 nickels, 1 dime, and 1 quarter = $1
Case (5): It is impossible to have all five types of coins.
Suppose we did have all five types of coins. We could have at most one half dollar. So we have accounted for 50 cents. That leaves 50 cents for the other four types of coins. If we are to have all four types then we can have at most one quarter (25 cents). We must now split the remaining 25 cents among the dimes, nickels, and pennies. If we have just one dime, one nickel, and one penny (16 cents), then the remaining 9 cents must be in pennies (i.e. 9 pennies) or made up of another nickel and 4 more pennies. We now have
10 pennies, 1 nickel, 1 dime, 1 quarter, and 1 half dollar = $1
or 5 pennies, 2 nickels, 1 dime, 1 quarter, and 1 half dollar = $1
However, we only have 10 + 1 + 1 + 1 + 1 = 14 coins or 5 + 2 + 1 + 1 + 1 = 10 coins
( not the 19 coins we wanted).
Therefore, it is impossible to use all five types of coins to get 19 coins to equal one dollar.
How many dimes do you need to make 70 cents? seven (7)
How many nickels do you need to make 25 cents? five (5)
How many pennies does it take to make 1 dollar? one-hundred (100)
How many nickels does it take to make 50 cents? ten (10)
Answer: 100 pennies, 20 nickels, 10 dimes, or 4 quarters; each = 1 dollar. Can you make a dollar using only 7 coins? Answer: 2 quarters + 5 dimes = 1 dollar.
Goodluck!!