Understanding Change and Transactions: A Simple Math Puzzle Explained

Imagine a common scenario where a child, Timmy, wishes to buy a piece of candy. However, something interesting happens along the way that tests our understanding of change and transactions. This article breaks down the puzzle and provides a clear explanation with a step-by-step approach to solving it, using simple logic and mathematical principles.

Scenario: Timmy's Candy Purchase

Timmy wants to buy a piece of candy that costs $0.98. However, when he counts his change, he realizes he is only $0.02 short. As a result, he hands the cashier a dollar bill ($1.00), expecting to receive his remaining change. But what is the total amount of change Timmy has now?

Step-by-Step Analysis

Let's denote the following variables:

h_1: The amount of change Timmy had before the transaction. h_2: The amount of change Timmy has after the transaction. a: The cost of the candy, which is $0.98.

According to the problem, we know that Timmy is two cents short:

Equation 1:

h_1 a - 0.02 (Timmy is two cents short of the candy price)

When Timmy gives the cashier a dollar bill, the cashier gives him his change:

Equation 2:

h_2 h_1 1.00 - a (Timmy gets the difference between a dollar and the candy price)

Now, let's combine these equations and simplify:

Add Equations 1 and 2 together:

h_1 h_2 (a - 0.02) (1.00 - a)

Cancel out the terms:

h_1 - a h_2 - a -0.02 1.00

Simplify further:

h_2 0.98

After performing the calculations, we find that:

h_2 0.98 (The amount of change Timmy has after the transaction).

Alternative Explanation by Substitution

Let's solve the problem using an alternative, more intuitive method. Let x be the amount of change Timmy had before the transaction. The candy costs x - 0.02. Timmy starts with 1 x and buys something worth x - 0.02.

Total change after the transaction: 1 x - (x - 0.02) 0.98

This confirms our previous result: Timmy ends up with 98 cents in change.

Conclusion: Applying Logical Reasoning

The transaction can be easily understood by applying logical reasoning. The two cents Timmy was short were directly taken from the dollar bill. Therefore, the transaction is equivalent to Timmy having 98 cents in total change after paying for the candy.

Key Takeaways:

Understanding the relationship between change and the amount spent. Using algebraic equations to simplify and solve the problem. Taking a logical approach to identify the total change after a transaction.

By breaking down the problem this way, we can see how simple mathematical principles and logical thinking can help us solve real-world scenarios involving transactions and change.