A collection of nickels (5¢) and quarters (25¢) is worth $5.00. How many nickels and quarters are there in the collection if there are ten more nickels than quarters?

Steps to Solve

1. Define variables for the coins

Let $q$ represent the number of quarters.
Let $n$ represent the number of nickels.
According to the problem, we have:
$$ n = q + 10 $$
This equation states that there are ten more nickels than quarters.

2. Set up the value equation

We know that the total value of the nickels and quarters is $5.00.
The value of the nickels is $0.05n$ and the value of the quarters is $0.25q$.
We can create the equation:
$$ 0.05n + 0.25q = 5.00 $$

3. Substitute for nickels

Now, we can substitute the expression for $n$ from the first equation into the value equation:
$$ 0.05(q + 10) + 0.25q = 5.00 $$

4. Simplify the equation

Distributing $0.05$ gives:
$$ 0.05q + 0.50 + 0.25q = 5.00 $$
Combine like terms:
$$ 0.30q + 0.50 = 5.00 $$

5. Isolate the variable

Subtract $0.50$ from both sides of the equation:
$$ 0.30q = 4.50 $$

6. Solve for quarters

Now, divide by $0.30$ to find $q$:
$$ q = \frac{4.50}{0.30} = 15 $$

7. Find the number of nickels

Substitute $q = 15$ back into the equation for $n$:
$$ n = q + 10 = 15 + 10 = 25 $$