Jeremy has 19 coins of 2 different values. the total value of the coins is $1.00. What are the coins? How many of each coin does he have?
how about 18 nickels and 1 dime
or
9 dimes and 10 penny
or
..
suppose we restrict ourselves to p pennies, n nickels, d dimes, q quarter
then we need integer solutions to
pennies + nickels = 100
p + 5(19-p) = 100
p + 95 -5p = 100
-4p = 5 --->no integer solution
pennies + dimes = 100
p + 10(19-p) = 100
p + 190 - 10p = 100
-9p = -90
p = 10 ---> 10 pennies, and 19-10 or 9 dimes, see above
pennies + quarters = 100 ?
25q + 1(19-q) = 100
-24q = 81 , no way
nickels and dimes = 100 ?
5n + 10(19-n) = 100
5n + 190 - 10n = 100
-5n = -90
n = 18 ---> 18 nickels, and 1 dime , see above
nickels + quarters ?
5n + 25(19-n) = 100
-20n = 375 , no integer for n
dimes and quarters ??
10d + 25(19-d) = 100
-15d = -375
d = 25 , but we only had 19 coins, so no way
looks like we only have the two solutions stated above
To find out the types and quantities of coins Jeremy has, we can set up a system of equations. Let's assume he has x coins of one value and y coins of the other value.
From the given information, we can establish two equations:
1. The total number of coins: x + y = 19
2. The total value of the coins: 1x + 0.01y = 1.00 (since the value of one of the coins is $1 and the value of the other coin is $0.01)
Let's solve this system of equations to find the values of x and y.
We can rewrite equation 1 as:
x = 19 - y
Substituting this expression for x into equation 2:
1(19 - y) + 0.01y = 1.00
Simplifying the equation:
19 - y + 0.01y = 1.00
19 - 0.99y = 1.00
-0.99y = 1.00 - 19
-0.99y = -18.00
Dividing through by -0.99:
y = -18.00 / -0.99
y ≈ 18.18
Since we can't have a fractional number of coins, we'll round y to the nearest whole number, which is 18.
Substituting the value of y back into equation 1:
x = 19 - y
x = 19 - 18
x = 1
Therefore, Jeremy has 1 coin with a value of $1 and 18 coins with a value of $0.01.