a vending machine takes only nickels and dimes. there five times as many dims as nickels in the machine. the face value of the coins is $4.40. how many of each are there?
Let x = # of nickels
Let 5x = # of dimes
10(5x) + 5x = 4.40
Solve for x and then 5x.
Let's solve this problem step-by-step.
Step 1: Let's assume the number of nickels in the machine is 'x'. Since the problem states that there are five times as many dimes as nickels, the number of dimes will be 5x.
Step 2: The value of a nickel is 5 cents, and the value of a dime is 10 cents. So, the total value of nickels can be calculated as 5x (nickels) multiplied by 5 cents, and the total value of dimes can be calculated as 5x (dimes) multiplied by 10 cents.
Step 3: The total value of the coins is $4.40, which can be represented as 440 cents.
Step 4: Now, let's create an equation to represent the total value of the coins: 5x (nickels) * 5 cents + 5x (dimes) * 10 cents = 440 cents.
Step 5: Simplifying the equation, we get: 25x + 50x = 440.
Step 6: Combining like terms, we have: 75x = 440.
Step 7: Now, we divide both sides of the equation by 75 to solve for x: x = 440 / 75.
Step 8: Evaluating the above expression, x ≈ 5.87.
Step 9: Since you can't have a fraction of a coin, we round x to the nearest whole number, which is 6.
Therefore, there are 6 nickels and 5 times that number, which is 30 dimes in the vending machine.
To determine the number of nickels and dimes in the vending machine, we can set up a system of equations based on the information provided.
Let's assume that the number of nickels is represented by 'n', and the number of dimes is represented by 'd'. Now, let's translate the given information into equations.
1) "There are five times as many dimes as nickels in the machine":
d = 5n
2) "The face value of the coins is $4.40":
The value of a nickel is $0.05, and the value of a dime is $0.10. So, the value of 'n' nickels is 0.05n, and the value of 'd' dimes is 0.10d. According to the given information, the sum of these values is $4.40:
0.05n + 0.10d = 4.40
Now, we have a system of two equations that we can solve simultaneously to find the values of 'n' and 'd'.
Substituting the value of d from equation (1) into equation (2):
0.05n + 0.10(5n) = 4.40
0.05n + 0.50n = 4.40
0.55n = 4.40
n = 4.40 / 0.55
n ≈ 8
Now, substitute the value of 'n' back into equation (1) to find the value of 'd':
d = 5n
d = 5(8)
d = 40
Therefore, there are approximately 8 nickels and 40 dimes in the vending machine.