Solving Coin Combination Puzzles: A Mathematical Approach

Mathematics is a powerful tool, and it can be used to solve intriguing puzzles, such as the one involving nickels, dimes, and quarters. In this article, we will explore how to solve the coin problem: 'I have only nickels, dimes, and quarters, and have at least one of each type of coin. The total number of coins I have is fifteen and the total value of all the coins is $1.00. How many of each coin do I have?'

Problem Statement and Variables

The problem requires us to find the number of each type of coin that satisfies the conditions:

- The total number of coins is fifteen: n d q 15

- The total value of the coins is $1.00: 5n 10d 25q 100 (in cents)

- There is at least one of each type of coin: n 1, d 1, q 1

Solution Approach

To solve this problem, we need to translate the words into mathematical equations and solve them step by step.

Step 1: Simplify the Value Equation

Dividing the second equation by 5, we get:

n 2d 5q 20

Step 2: Set Up the System of Equations

Now, we have the following system of equations:

n d q 15 (Equation 1) n 2d 5q 20 (Equation 2)

Step 3: Eliminate n from the Equations

Subtract Equation 1 from Equation 2:

n 2d 5q - (n d q) 20 - 15

This simplifies to:

d 4q 5

Step 4: Express d in Terms of q

From the equation d 4q 5, we can express d in terms of q:

d 5 - 4q

Step 5: Substitute d into Equation 1

Substituting d 5 - 4q into Equation 1:

n (5 - 4q) q 15

This simplifies to:

n 5 - 3q 15

Rearranging, we get:

n 10 - 3q (Equation 4)

Step 6: Determine Valid Values for q

The values of n, d, and q must be at least 1. Using Equation 4:

10 - 3q 1

This implies:

q 3

From d 5 - 4q 1:

5 - 4q 1

This simplifies to:

4q 4

Which implies:

q 1

Since q must be a positive integer, the only possible value is:

q 1

Step 7: Calculate n and d

Substituting q 1 into d 5 - 4q:

d 5 - 4(1) 1

Substituting q 1 into n 10 - 3q:

n 10 - 3(1) 7

However, this contradicts the requirement that there is at least one dime and one quarter. Therefore, we must re-evaluate the solution.

Using q 1, we get:

n 10 - 3(1) 7, d 5 - 4(1) 1, q 1

Summing them, we get:

n d q 7 1 1 9

This is incorrect, so we must try another approach. Instead, let's try q 1, and re-evaluate:

d 5 - 4(1) 1, n 15 - (1 1) 13

This satisfies all the conditions:

13 1 1 15 (number of coins)

5(13) 10(1) 25(1) 65 10 25 100 (total value in cents)

Final Solution

The number of each type of coin is:

Nickels: 13 Dimes: 1 Quarters: 1

Summary

You have 13 nickels, 1 dime, and 1 quarter, which totals 15 coins and a value of $1.00.

Using mathematical methods, we can uncover the solution to these types of puzzles, showcasing the practical applications of algebra and logical reasoning in everyday life.