an envelope contains 20 bills in denominations of $1,$5,$10, the total of the envelope is $54. how many of each bill do i have
To determine the number of each bill in the envelope, we can set up a system of equations. Let's denote the number of $1 bills as x, the number of $5 bills as y, and the number of $10 bills as z.
We know that there are a total of 20 bills, so we can write the first equation as:
x + y + z = 20
We also know that the total value of the bills in the envelope is $54. We can write the second equation as:
1x + 5y + 10z = 54
Now, we have a system of two equations with three variables. To solve this system, we can use a method called substitution or elimination.
Let's use the substitution method to solve this system:
1. Solve the first equation for x:
x = 20 - y - z
2. Substitute the value of x in the second equation:
1(20 - y - z) + 5y + 10z = 54
3. Simplify and solve for z:
20 - y - z + 5y + 10z = 54
20 + 4y + 9z = 54
4y + 9z = 34 (equation 3)
Now, we have two equations in two variables. We can solve this system by substitution or elimination method again.
Since we're looking for whole numbers of bills, we can try different values for y and z until we find a solution that satisfies both equations.
Let's start by trying the values of y from 0 to 5 (one by one) and calculate the corresponding value of z using equation 3. If the value of z is a whole number, we can substitute y and z into equation 1 to find the value of x.
By substituting different values, we find that when y = 2 and z = 2, both equations are satisfied and give us a whole number solution.
So, the solution is:
x = 20 - y - z = 20 - 2 - 2 = 16
y = 2
z = 2
Therefore, you have 16 $1 bills, 2 $5 bills, and 2 $10 bills in the envelope.