UNO High School Problem of the Week Competition
Problem 2
Ben took the average of all of the pennies, nickels, dimes, and quarters in his pocket, and found that it was 20 cents. If he added one more quarter, the average would go up to 21 cents. How many dimes does Ben have in his pocket?
Solution
The average of Ben's coins is 20 cents, and to find the average we need to take the amount of money in cents divided by the amount of coins he has. Therefore,
(X amount in cents) / (N coins) = 20 cents
We also know that if Ben adds one more quarter to his pocket, the average will increase to 21 cents. In other words,
(X + 25 cents) / (N + 1 coins) = 21
We can solve the first equation for X, and then plug it into the second equation.
X = 20N
(X + 25) / (N + 1) = 21
X + 25 = 21 * (N + 1)
X + 25 = 21N + 21
(20N) + 25 = 21N + 21
N = 4
Therefore, Ben originally had 4 coins in his pocket. We can solve the equation for X, to find out how much he originally had.
X = 20N
X = 20 * 4 = 80 cents
Ben had 4 coins that added up to 80 cents. Using only pennies, nickels, dimes, and quarters, there is only one possible way to get 80 cents with 4 coins, which
is 3 quarters and 1 nickel, since (3 * 25) + (1 * 5) = 80.
Ben had no dimes in his pocket.