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Some loose change (pennies, nickels, dimes, and quarters) found behind the couch. Brianna noticed more than half the coins were pennies. Geofrey noticed that there was at least one coin of each type. Michele noticed that you could not make change for the same amount using fewer coins. In the space below, write the total amount of change, in cents, that was found.

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The total amount of change, in cents, that was found is determined as 96 cents.

Total amount of change

The total amount of change is calculated as follows;

let the total = T

there was at least one coin of each type

T = p + n + d + q

more than half of the coins were pennies

[tex]\frac{p+n + d + q }{2} \leq p[/tex]

Value of the coins

  • pennies = $0.01
  • nickels = $0.05
  • dimes = $0.1
  • quarters = $0.25

more than half = pennies

next, we have; nickels + dimes + quarters

number of dimes

y(dimes) = $0.25

y(0.1) = 0.25

y = 0.25/0.1

y = 2.5 ≅ 3

dimes = 3 coins

number of nickels

since dimes = 3 coins, number of nickels is now calculated as;

x(0.05) = 3(0.1)

x = (0.3)/(0.05)

x = 6 coins

nickels = 6 coins

Total coins

Total coins = nickels + dimes + quarters + pennies

  • pennies must greater than 6 coins, since we can't have fewer coins.

Total coins = nickels + dimes + quarters + pennies

let pennies = 7

Total coins = 6 + 3 + 1 + 7

Total coins = 17

half of 17 = 8.5

7 is less than half (8.5), number of pennies cannot be 7, since more than half of the coins are pennies.

let pennies = 11

Total coins = 6 + 3 + 1 + 11

Total coins = 21

half = 10.5

  • 11 is more than half (10.5)

more than half = number of pennies = 11

Total amount in cents

11($0.01) + 6($0.05) + 3($0.1) + 1($0.25) = $0.96 = 96 cents

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