A cashier has 25 bills consisting of twice as many ones as tens, two fewer fives than than ones, and the rest are twenties. If the total value is $140, find how many of each bill she has?
start writing the data as math.
Let the number of bills (1,5,10,20) be w,x,y,z. Then
w+x+y+z = 25
w = 2y
...
You will wind up with 4 equations to solve for w,x,y,z.
Let's solve this problem step by step.
Step 1: Let's assume the number of tens as x.
So, the number of ones will be twice the number of tens, which is 2x.
Step 2: The number of fives is two fewer than the number of ones, which means the number of fives is 2x - 2.
Step 3: The total number of bills is given as 25, so we can write the equation as:
x + 2x + (2x - 2) + T = 25
where T represents the number of twenties.
Step 4: The value of the tens is 10x, the value of the ones is 1 * (2x) = 2x, the value of the fives is 5 * (2x - 2) = 10x - 10, and the value of the twenties is 20 * T.
Step 5: The sum of the values is given as $140, so we can write the equation as:
10x + 2x + (10x - 10) + 20T = 140
Step 6: Simplifying the above equation, we get:
22x + 20T = 150
Step 7: Since we have two variables, we need another equation to solve the system of equations.
Step 8: From Step 1, we know the number of tens is x, and from Step 2, we know the number of fives is 2x - 2. So, the number of ones, fives, and tens sums up to x + (2x - 2) = 3x - 2.
Step 9: Adding up the number of tens, ones, fives, and twenties, we get:
(x) + (2x) + (2x - 2) + T = 25
5x + T = 25
Step 10: Now we have two equations:
22x + 20T = 150
5x + T = 25
Step 11: To solve this system of equations, we can use substitution or elimination method.
Step 12: Let's solve by substitution method.
From the second equation, we can express T in terms of x as T = 25 - 5x.
Step 13: Substituting this value of T in the first equation, we get:
22x + 20(25 - 5x) = 150
22x + 500 - 100x = 150
-78x = -350
x = -350 / -78
x ≈ 4.487
Step 14: Since the number of bills should be whole numbers, we can approximate x to 4.
Step 15: Using the value of x = 4, we can find the value of T:
T = 25 - 5x
T = 25 - 5 * 4
T = 25 - 20
T = 5
Step 16: Now we can find the number of ones, fives, tens, and twenties using the values of x and T:
Number of ones = 2x = 2 * 4 = 8
Number of fives = 2x - 2 = 2 * 4 - 2 = 6
Number of tens = x = 4
Number of twenties = T = 5
Therefore, the cashier has 8 ones, 6 fives, 4 tens, and 5 twenties.
To solve this problem, we can first set up a system of equations based on the given information. Let's represent the number of tens as "x".
Given:
- Twice as many ones as tens: There are 2x ones.
- Two fewer fives than ones: There are 2x - 2 fives.
- The rest are twenties: The number of twenties can be calculated as (25 - x - (2x + 2)).
Next, let's determine the value of each type of bill:
- The value of each ten is $10, so the total value of tens is 10x.
- The value of each one is $1, so the total value of ones is (2x * 1) = 2x.
- The value of each five is $5, so the total value of fives is ((2x - 2) * 5) = 10x - 10.
- The value of each twenty is $20, so the total value of twenties is ((25 - x - (2x + 2)) * 20) = (25 - 3x - 2) * 20 = (23 - 3x) * 20.
The sum of these values should be $140, so we can set up an equation:
10x + 2x + (10x - 10) + (23 - 3x) * 20 = 140
Now, solve the equation to find the value of "x", which represents the number of tens.