a bag contains only nickels and dimes. there are exactly 18 coins in the bag. the amount of money in the bag is $1.25.
let x represent the number of nickels in the bag. let y represent the number of dimes in the bag.
1. write a system of equations that correctly models this situation
2. does your answer represent integer values for the nickels and dimes? why must this be the case (or why does it not matter) ?
1. To write a system of equations for this situation, we can use the following information:
- The number of nickels, represented by x.
- The number of dimes, represented by y.
- There are exactly 18 coins in the bag, so x + y = 18.
- The amount of money in the bag is $1.25. Since each nickel is worth $0.05 and each dime is worth $0.10, we can express the total value as 0.05x + 0.10y = 1.25.
So, the system of equations that correctly models this situation is:
x + y = 18
0.05x + 0.10y = 1.25
2. In this problem, it is important that the values for x and y are integers. This is because the number of coins cannot be a fraction or a decimal, as they are whole units of count. You cannot have a fraction of a nickel or a decimal number of dimes. Therefore, x and y must be integer values.
Solving the system of equations x + y = 18 and 0.05x + 0.10y = 1.25 will give us specific integer values for x and y that satisfy the conditions of the problem.