Solving Math Word Problems: A Fun Approach to Coin Combinations

Math word problems can often feel like a daunting challenge, but they offer valuable real-world applications of mathematical concepts. One such problem involves a piggy bank containing a mix of nickels and dimes, totaling 50 coins and $4.55. Let's break down the solution step by step to understand how algebra can help us find the answer.

Algebraic Method to Solve the Problem

First, let's introduce some variables to represent the number of coins. Let n be the number of nickels and d the number of dimes.

Step 1: Setting Up the Equations

Since there are 50 coins in total, we have:

n d 50

The total value of the coins is $4.55, so:

0.05n 0.10d 4.55

Step 2: Simplifying the Equations

It's helpful to eliminate one variable. From the first equation, we can express n in terms of d:

n 50 - d

Substituting this into the second equation:

0.05(50 - d) 0.10d 4.55

2.50 - 0.05d 0.10d 4.55

2.50 0.05d 4.55

0.05d 2.05

d 41

Step 3: Finding the Number of Nickels

Now that we know d 41, we can find n using our initial equation:

n 50 - 41 9

Therefore, there are 41 dimes and 9 nickels.

Alternative Methods and Verification

Let's explore an alternative method to ensure our solution is correct.

Using Substitution and Linear Equations

Let's use n and d to represent the number of nickels and dimes, respectively. We know:

n d 50

0.05n 0.10d 4.55

From the first equation, we get:

n 50 - d

Substitute into the second equation:

0.05(50 - d) 0.10d 4.55

2.50 - 0.05d 0.10d 4.55

2.50 0.05d 4.55

0.05d 2.05

d 41

Therefore, n 50 - 41 9.

Conclusion

Algebra provides a systematic approach to solving complex math problems. By breaking down the problem and using equations, we can determine the exact number of nickels and dimes without much difficulty. This example not only reinforces algebraic concepts but also demonstrates the practical application of these concepts in everyday life.

To further your understanding, try solving similar problems with different values, such as a total of 30 coins with a combined value of $3.20. Practice is key to mastering these skills.