A cash register contains only quarters and twenty-dollar bills. It contains three times as many quarters as twenty-dollar bills, and the total amount of money in the cash register is $8249.00. How many twenty-dollar bills are in the cash register?
The Correct Answer and Explanation is:
Step 1: Define the variables
Let
x = number of twenty-dollar bills
Since there are three times as many quarters as twenty-dollar bills, the number of quarters is 3x
Step 2: Determine the value of the money
Each twenty-dollar bill is worth $20, so the value of the bills is:
20 * x = 20x dollars
Each quarter is worth $0.25, so the value of the quarters is:
0.25 * 3x = 0.75x dollars
The total amount of money is given as $8249.00
Step 3: Set up the equation
We add the value of the twenty-dollar bills and the value of the quarters:
20x + 0.75x = 8249
Combine like terms:
20.75x = 8249
Step 4: Solve for x
Divide both sides of the equation by 20.75:
x = 8249 / 20.75 = 397.5301…
This result is not a whole number. Since the number of bills and coins must be whole numbers, this suggests that either there is a mistake in the model or a unit mismatch.
Let’s convert all amounts to cents to eliminate decimals.
Step 5: Convert to cents
Twenty-dollar bill = 2000 cents
Quarter = 25 cents
Total = 824900 cents
Using the same variables:
Bills: x
Quarters: 3x
Value from bills: 2000x
Value from quarters: 25 * 3x = 75x
Now set up the equation:
2000x + 75x = 824900
Combine terms:
2075x = 824900
Solve:
x = 824900 / 2075 = 397
Final Answer:
There are 397 twenty-dollar bills in the cash register.
Explanation
To solve the problem, we begin by assigning a variable to represent the number of twenty-dollar bills, since that’s one of the unknowns. Let x represent the number of twenty-dollar bills. We’re told that the number of quarters is three times this number, so we represent the number of quarters as 3x. The problem also tells us the total value in the register is $8249.00.
Each twenty-dollar bill is worth $20, and each quarter is worth 25 cents. At first, we try solving using decimal dollars: the value of the bills is 20x dollars, and the value of the quarters is 0.25 multiplied by 3x, which is 0.75x dollars. Adding them gives 20.75x. Setting that equal to 8249 and solving for x gives a non-whole number, which is not valid because we cannot have a fraction of a bill or coin.
To avoid decimal values and get more accurate results, we convert all dollar amounts to cents. Twenty-dollar bills become 2000 cents each, and quarters become 25 cents each. The total value becomes 824900 cents. The value of the twenty-dollar bills is now 2000x, and the value of the quarters is 75x (since 3x quarters at 25 cents each). Adding those gives 2075x, and setting that equal to 824900 results in a simple equation. Dividing both sides by 2075 gives x = 397.
So, the cash register contains 397 twenty-dollar bills.
