Melissa has a collection of dimes and nickels with a total face value of less than one dollar. Let x be the number of dimes and y be the number of nickels. How many of each type of coin does she have?
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To solve this problem, we can set up a system of equations using the given information.
Let's denote the value of a dime as 'd' and the value of a nickel as 'n'.
From the problem, we know two things:
1. The total face value of the coins is less than one dollar, which means it is less than 100 cents. Therefore, we can write the first equation as:
10x + 5y < 100
2. We are also told that x represents the number of dimes, and y represents the number of nickels. This gives us the second equation:
x = number of dimes
y = number of nickels
Now, we have a system of equations:
10x + 5y < 100
x = number of dimes
y = number of nickels
To find the values of x and y, we can solve this system of equations.
There are multiple ways to solve this system, such as by graphing or substitution, but I will use the method of elimination.
We can eliminate x from the first equation by multiplying the second equation by 10 since the coefficient of x in the first equation is 10.
10x + 5y < 100
10(x) = 10(number of dimes) = 10x (from second equation)
Now we have:
10x + 5y < 100
10x = 10x
By subtracting, we can eliminate x:
(10x - 10x) + 5y < 100
0 + 5y < 100
Simplifying further, we get:
5y < 100
Dividing both sides by 5, we get:
y < 20
This means the number of nickels (y) can be any integer less than 20.
Now, let's consider some possible values for y:
If y = 0, then 5y = 0, and we would have:
10x + 0 < 100
10x < 100
x < 10
So, if y = 0, then x can be any integer less than 10.
Similar reasoning can be applied to other values of y. For example:
If y = 1, then 5y = 5, and we would have:
10x + 5 < 100
10x < 95
x < 9.5
So, if y = 1, then x can be any integer less than 9.5.
By repeating this process for different values of y, we can determine the possible combinations of x and y that satisfy the given conditions.
In this case, there are infinitely many possible combinations of x and y that satisfy the conditions. For example, some valid combinations could be:
- x = 0, y = 0 (no coins)
- x = 9, y = 1 (9 dimes and 1 nickel)
- x = 8, y = 2 (8 dimes and 2 nickels)
- x = 7, y = 3 (7 dimes and 3 nickels)
- and so on...
Therefore, there is no unique solution to the question "How many of each type of coin does she have?". The number of dimes and nickels can vary as long as they satisfy the given conditions.