Answer:
He correctly determines that she has <u>300</u> nickels and <u>330</u> dimes.
Step-by-step explanation:
I'll first explain how Mark got that system of equations. Then I'll solve the system of equations to find the numbers of coins.
Moira has a collection of nickels and dimes. We don't know the number of nickels and the number of dimes she has.
First, we define two variables to represent the unknowns in this problems, the numbers of coins.
Let n = number of nickels.
Let d = number of dimes.
The sum of the numbers of coins is n + d. We are told she has 630 coins, so the first equation is
n + d = 630
Since we have two unknowns, we need two equations. Now we write an equation based on the values of the coins. A nickel is worth $0.05. A dime is worth $0.1. n nickels are worth 0.05n, and d dimes are worth 0.1d. The total value of the coins is 0.05n + 0.1d. We are told the value of the coins is $48. Now we can write the second equation.
0.05n + 0.1d = 48
Our system of equations is:
n + d = 630
0.05n + 0.1d = 48
These are the same equations Mark got.
Now we solve the system of equations. We will use the substitution method. First, we solve one equation for one variable. Then we substitute that into the other equation.
Let's solve the first equation for n:
n + d = 630
Subtract d from both sides:
n = 630 - d
Now that we know that n is the same as 630 - d, we replace n of the second equation with 630 - d.
0.05n + 0.1d = 48
0.05(630 - d) + 0.1d = 48
Distribute the 0.05:
31.5 - 0.05d + 0.1d = 48
Combine the terms in d:
0.05d + 31.5 = 48
Subtract 31.5 from both sides.
0.05d = 16.5
Divide both sides by 0.05.
d = 330
Now that we know d is 330, we substitute d with 330 in the first original equation and solve for n.
n + d = 630
n + 330 = 630
Subtract 330 from both sides.
n = 300
Since we let n = the number of nickels, and d = the number of dimes, now we can fill in the blanks.
n = number of nickels = 300
d = number of dimes = 330
Answer: He correctly determines that she has <u>300</u> nickels and <u>330</u> dimes.
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The question is already answered, but we can check with the given information to confirm that our answer is correct.
We check the number of coins:
300 nickels + 330 dimes = 630 coins (the number of coins checks out.)
Now, we check the value of the coins:
300 * $0.05 + 330 * $0.1 = $15 + $33 = $48 (the value of the coins checks out.)
Since both the number of coins and the value of coins check correctly, our answer, 300 nickels and 330 dimes, is correct.
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