Justin is exchanging dollars for rolls of quarters and rolls of dime...
Justin is exchanging dollars for rolls of quarters and rolls of dimes at a bank. Each roll of quarters has a value of $10 and each roll of dimes has a value of $5. He exchanges $90 and receives 11 rolls of coins. How many of each type of roll of coins did he receive?
Solving the Coin Roll Problem
Justin's coin exchange problem can be solved using a system of equations. We know the total value of the exchange and the total number of rolls. This information allows us to determine the number of each type of roll.
Understanding the Problem
The problem states that Justin exchanges $90 for a total of 11 rolls of coins. These rolls consist of quarters (worth $10 per roll) and dimes (worth $5 per roll). We need to figure out how many rolls of quarters and how many rolls of dimes Justin received.
Setting up the Equations
Let 'q' represent the number of quarter rolls and 'd' represent the number of dime rolls. We can set up the following two equations:
Equation 1 (based on the number of rolls): q + d = 11 Equation 2 (based on the total value): 10q + 5d = 90
Solving the Equations
We can solve this system of equations using substitution or elimination. Here's the solution using substitution:
1. Solve Equation 1 for q: q = 11 - d 2. Substitute this value of q into Equation 2: 10(11 - d) + 5d = 90 3. Simplify and solve for d: 110 - 10d + 5d = 90 => -5d = -20 => d = 4 4. Substitute the value of d (4) back into Equation 1: q + 4 = 11 => q = 7
Answer
Justin received 7 rolls of quarters and 4 rolls of dimes.
Checking the Answer
Let's verify the solution: 7 rolls of quarters * $10/roll = $70, and 4 rolls of dimes * $5/roll = $20. $70 + $20 = $90, which is the total amount Justin exchanged. Also, 7 rolls + 4 rolls = 11 rolls, the total number of rolls he received.
Real-World Application
This type of problem is relevant to everyday situations involving money management, budgeting, and even coin collecting. Understanding how to solve these problems can help you make informed decisions about currency exchange and manage your finances effectively.
FAQ
What if the total value or number of rolls were different?
You would simply adjust the equations accordingly and follow the same steps to solve for the unknowns.
Can this method be used with other denominations of coins?
Yes, this method can be adapted to solve problems involving other coin denominations or even other items with different values.
Call to Action: Try solving similar problems with different values to practice your math skills! You can even create your own scenarios involving different coin denominations and amounts.
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