Alexandra has 36 coins valued at $6.00. She has only nickels and quarters. Which equation can be used to solve for the number of nickels, $n$?
| | Number | Value | Total |
| :------- | :----- | :---- | :--------- |
| Nickels | $n$ | 0.05 | $0.05n$ |
| Quarters | $36-n$ | 0.25 | $0.25(36-n)$ |
| Total | 36 | | 6 |
A. $0.05 n+0.25(36-n)=6$
B. $0.05 n+0.25(36-n)=36$
C. $n+0.05=0.05 n$
D. $n+(36-n)=36$
Answer (1)
Define n as the number of nickels, so 36 − n is the number of quarters.
Express the total value of nickels as 0.05 n and quarters as 0.25 ( 36 − n ) .
Set up the equation representing the total value: 0.05 n + 0.25 ( 36 − n ) = 6 .
The equation to solve for the number of nickels is 0.05 n + 0.25 ( 36 − n ) = 6 .
Explanation
Problem Analysis Let's analyze the given information. We know that Alexandra has 36 coins, which are a mix of nickels and quarters. The total value of these coins is $6.00 . We need to find the equation that can be used to determine the number of nickels, represented by n .
Equation Setup We can set up an equation based on the total value of the coins. The value of nickels is 0.05 n and the value of quarters is 0.25 ( 36 − n ) . The sum of these values must equal the total value, which is $6.00 . Therefore, the equation is: 0.05 n + 0.25 ( 36 − n ) = 6
Comparison with Options Now, let's compare this equation with the given options:
0.05 n + 0.25 ( 36 − n ) = 6 - This matches our equation.
0.05 n + 0.25 ( 36 − n ) = 36 - This equation equates the value to the number of coins, which is incorrect.
n + 0.05 = 0.05 n - This equation doesn't make sense in the context of the problem.
n + ( 36 − n ) = 36 - This equation simply states that the number of nickels plus the number of quarters equals the total number of coins, but it doesn't involve the values of the coins.
Final Answer Therefore, the correct equation is: 0.05 n + 0.25 ( 36 − n ) = 6
Examples
This type of problem is useful in everyday situations when you need to calculate the number of different denominations of coins or bills you have, given their total value. For example, if you have a certain amount of money in your piggy bank consisting of only dimes and quarters, and you know the total number of coins, you can use a similar equation to find out how many of each type of coin you have. This can also be applied to inventory management, where you need to track the quantities of different items with varying values to ensure the total value matches your records.
