Solving Coin Combinations: A Step-by-Step Guide

Imagine you have a coin collection consisting of nickels, dimes, and quarters, and the total value is $3.30. There are 3 times as many nickels as quarters, and the number of dimes is 3 less than the average of nickels and quarters. How many coins of each kind are there? This article will guide you through the process using simple steps and equations to find the answer.

Understanding the Problem

We have a collection of coins with a total value of $3.30. The coins consist of nickels (5 cents), dimes (10 cents), and quarters (25 cents). We know two critical pieces of information:

There are 3 times as many nickels as quarters. The number of dimes is 3 less than the average of the number of nickels and quarters.

Setting Up Equations

Let's denote the number of quarters by x. Then we can express the number of nickels and dimes as follows:

Number of quarters x Number of nickels 3x Number of dimes (x 3x) / 2 - 3 2x - 3

The total value equation is:

0.05x   0.10(2x - 3)   0.25(3x)  3.30

Solving the Equations

Now, let's solve the equation step by step:

Start with the total value equation:

0.05x   0.10(2x - 3)   0.25(3x)  3.30

Simplify inside the parentheses:

0.05x   0.2 - 0.30   0.75x  3.30

Combine like terms:

1.0 - 0.30  3.30

Add 0.30 to both sides:

1.0  3.60

Solve for x:

x  3.60 / 1.00  3.60

Rounding x to the nearest whole number (since the number of coins must be an integer), we get:

x  3

Therefore, the number of each type of coin is as follows:

Number of quarters 3 Number of nickels 3 * 3 9 Number of dimes (3 9) / 2 - 3 12 / 2 - 3 6 - 3 3

Verification

Let's verify by calculating the total value:

$0.25 * 3 (quarters) $0.10 * 3 (dimes) $0.05 * 9 (nickels) $0.75 $0.30 $0.45 $1.50 $1.80 $3.30

The calculations are correct, and we have 3 quarters, 9 nickels, and 3 dimes.

Conclusion

By breaking down the problem into simpler forms, we can easily find the number of each type of coin in the collection. The key steps are to set up the correct equations and solve them systematically. This method can be applied to similar problems involving coin collections or any other value-based problems.