Questions: How many ways can you make change for 40 using only nickels, dimes, and quarters? (Type a whole number.)

How many ways can you make change for 40 using only nickels, dimes, and quarters? (Type a whole number.)
Transcript text: How many ways can you make change for $40 using only nickels, dimes, and quarters? (Type a whole number.)
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Solution

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Solution Steps

To solve this problem, we need to find the number of combinations of nickels, dimes, and quarters that sum up to 40 dollars (or 4000 cents). We can use a nested loop approach where we iterate over possible counts of each coin type and check if their total value equals 4000 cents.

Step 1: Define the Problem

We need to determine the number of ways to make change for $40 using only nickels, dimes, and quarters. This translates to finding combinations of these coins that sum to 4000 cents.

Step 2: Set Up the Equation

Let \( n \) be the number of nickels, \( d \) be the number of dimes, and \( q \) be the number of quarters. The equation representing the total value in cents is: \[ 5n + 10d + 25q = 4000 \]

Step 3: Iterate Over Possible Values

We iterate over possible values of \( n \), \( d \), and \( q \) within their respective ranges:

  • \( n \) ranges from 0 to \(\frac{4000}{5}\)
  • \( d \) ranges from 0 to \(\frac{4000}{10}\)
  • \( q \) ranges from 0 to \(\frac{4000}{25}\)

For each combination, we check if the equation holds true.

Step 4: Count Valid Combinations

We count the number of valid combinations where the equation \( 5n + 10d + 25q = 4000 \) is satisfied.

Final Answer

The total number of ways to make change for $40 using nickels, dimes, and quarters is \(\boxed{32321}\).

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