Solving Coin Problems: A Mathematical Approach to Counting Clay's Coins

Clay has a mixed collection of 10 centavo and 25 centavo coins. Given that he has twice as many 10 centavo coins as 25 centavo coins, and these coins amount to a total of 6.75, how many coins does he have in total? Let's dive into the problem step-by-step to find the solution. We will also explore an alternative method and provide a concise precis for quick reference.

Step-by-Step Solution Using Algebra

In this method, we use algebra to solve the problem. We start by representing the number of 25 centavo coins as x. Since Clay has twice as many 10 centavo coins, the number of 10 centavo coins he has is 2x.

The value of the 25 centavo coins can be calculated as 0.25x. Similarly, the value of the 10 centavo coins is 0.10 * 2x 0.2. The total value of all the coins is given as 6.75. Therefore, we can write the following equation:

[0.25x 0.2 6.75]

Combining the terms on the left side:

[0.45x 6.75]

To solve for x, divide both sides by 0.45:

[x frac{6.75}{0.45} 15]

So, Clay has 15 twenty-five centavo coins. Since he has twice as many ten centavo coins:

[2x 2 times 15 30]

The total number of coins is the sum of the number of ten centavo and twenty-five centavo coins:

[15 30 45]

Alternative Solution Using a Simplified Approach

This method provides a quick and easy way to solve the problem. Let x represent the number of fifty-centavo (10 centavo) coins and y represent the number of twenty-five centavo coins. The total value of the coins is given as 6.75. The equation based on the value can be written as:

[0.1 0.25y 6.75]

Since x is twice y, substitute x 2y into the equation:

[0.10(2y) 0.25y 6.75]

Combine the terms:

[0.20y 0.25y 6.75]

Simplify the equation:

[0.45y 6.75]

Solve for y:

[y frac{6.75}{0.45} 15]

Since x 2y:

[x 2 times 15 30]

The total number of coins is:

[15 30 45]

Precis

Clay has twice as many 10 centavo coins as 25 centavo coins amounting to 6.75. Here, we let the number of 25 centavo coins be x. We form the equation 0.25x 0.2 6.75. Solving for x gives x 15 (25 centavo coins), and 2x 30 (10 centavo coins). Thus, the total number of coins is 45.