The income from the school car wash is one, five, and ten dollar bills came to a total of $355. If there are 120 bills all together, and twice as many one dollar bills as five and ten dollar bills combined, how many bills of each denomination are there?
How do I get three questions so I can turn them into a matrix?
So far, I have,
n+f+t=355
n+f+t=120
2n=f+t
I have no clue as to if my equations are correct. I also have no clue what I'm doing, as far as word problems go. I can work out matrixes fine, but I don't understand word problems well.
The total value is 355. You have just counted the bills, rather than adding their values. Clearly, n+f+t cannot be both 355 and 120!
n+5f+10t = 355
n+f+t = 120
n = 2(f+t)
The way you wrote the last equation, you indicated that adding the 5's and 10's you had twice as many bills as the ones.
You really should not have had "no clue". You just need to check the meanings of the words against the meanings of the expressions.
To solve this word problem and convert it into a matrix equation, you can follow these steps:
1. Define the variables: Let's assign the following variables to represent the number of one dollar bills (n), five dollar bills (f), and ten dollar bills (t).
2. Translate the first sentence into an equation: The sentence "The income from the school car wash is one, five, and ten dollar bills came to a total of $355" gives you the equation: 1n + 5f + 10t = 355.
3. Translate the second sentence into an equation: The sentence "If there are 120 bills altogether" tells you that the total number of bills (n + f + t) is equal to 120: n + f + t = 120.
4. Translate the third sentence into an equation: The sentence "Twice as many one dollar bills as five and ten dollar bills combined" means that the number of one dollar bills (n) is equal to two times the sum of the number of five dollar bills (f) and ten dollar bills (t): 2(f + t) = n.
5. Combine the equations: You now have three equations:
- 1n + 5f + 10t = 355
- n + f + t = 120
- 2(f + t) = n
6. Rewrite the equations in matrix form: To represent the three equations in matrix form, you will need to create a coefficient matrix (A), a variable matrix (X), and a constant matrix (B).
- Coefficient matrix (A): The matrix A will contain the coefficients of the variables: [[1, 5, 10], [1, 1, 1], [-2, 1, -1]].
- Variable matrix (X): The matrix X will contain the variables: [[n], [f], [t]].
- Constant matrix (B): The matrix B will contain the constants: [[355], [120], [0]].
Now, you have a matrix equation: AX = B.
Solving this matrix equation will give you the values for n, f, and t.