Answer: Let x be the number of nickels, and y be the number of pennies. Then we can write two equations based on the given information:
The total number of coins is 20: x + y = 20
The total value of the coins is $0.68: 0.05x + 0.01y = 0.68
We can solve this system of equations using any method we like, such as substitution or elimination. Here, we will use the elimination method.
To eliminate y, we can multiply the first equation by 100 to get 100x + 100y = 2000, and then subtract the second equation from it:
100x + 100y - 5x - y = 2000 - 0.68
Simplifying, we get:
95x + 99y = 1932
Now we have one equation with only x and y. We can use this equation to solve for one variable in terms of the other. For example, we can solve for y:
y = (1932 - 95x) / 99
To check if the solution is valid, we need to make sure that x and y are both non-negative integers, since we are dealing with coins. We can start by checking if x is an integer between 0 and 20 (inclusive). We can use a loop in a programming language, or we can simply try each integer one by one. If x is an integer between 0 and 20, we can plug it into the equation for y and check if y is a non-negative integer. If we find a pair of x and y that satisfy all the conditions, then Hannah's brother counted the money correctly. If we cannot find such a pair, then he made an error.
Alternatively, we can use the fact that y must be divisible by 1 cent (since it represents the number of pennies), and check if the expression (1932 - 95x) is divisible by 99 for any integer x between 0 and 20 (inclusive). If it is divisible, then the quotient is the number of pennies, and we can check if it is a non-negative integer. If it is not divisible for any x, then there is no solution.
Explanation: