Solving a Coin Problem Using Algebraic Equations

Algebra can be a powerful tool for solving numerous real-world problems. One common scenario involves determining the number of coins of different denominations given their total value and count. In this article, we will walk through a step-by-step solution to a coin problem that combines mathematical reasoning and algebraic methods.

Understanding the Problem

Anna has a total of 1.50 (or 150 cents) in 5-cent and 10-cent coins, and she has a total of 24 coins. The task is to determine the number of 5-cent coins.

Setting Up the Equations

To solve this problem, we can set up a system of linear equations using x and y, where:

x represents the number of 5-cent coins y represents the number of 10-cent coins

Equation 1: Total Number of Coins

The first equation represents the total number of coins:

x   y  24

Equation 2: Total Value of Coins in Cents

The second equation represents the total value of the coins:

5x   10y  150

To simplify the second equation, we can divide everything by 5:

x   2y  30

Solving the System of Equations

Now, we have the following system of equations:

Equation 1 Equation 2 x y 24 x 2y 30

We can solve this system by subtracting the first equation from the second equation:

x   2y - (x   y)  30 - 24y  6

Now, we substitute y 6 back into the first equation to find x:

x   6  24x  24 - 6x  18

Conclusion

Thus, Anna has 18 5-cent coins. To verify, we can check the total value and count of coins:

18 5-cent coins give 90 cents. 6 10-cent coins give 60 cents. Total value: 90 60 150 cents. Total number of coins: 18 6 24.

Alternative Solutions

Another way to approach this problem is by intuitive reasoning. If Anna had only 5-cent coins, she would have 24 5-cent coins, totaling 1.20 (or 120 cents). The remaining 30 cents (150 - 120) would be from the 10-cent coins. Thus, 30 cents divided by 5 cents per coin is 6 10-cent coins. Consequently, the number of 5-cent coins would be 18 (24 - 6).

Algebraic methods and intuitive reasoning both lead to the same solution, making it easier to find the answer without extensive calculations.

Conclusion and Relevance

Algebraic equations and system solving are fundamental skills that have practical applications in various fields, including finance, economics, and IT. Understanding these concepts can help in solving a wide range of real-world problems efficiently and effectively.