A Boy's Journey at a Fair: An Interesting Mathematical Problem

Imagine a cheerful day at the fair, where a young boy embarks on an intriguing journey involving entrance fees and strategic spending. This fascinating problem not only challenges the mind but also provides a delightful insight into solving equations step by step. Let's delve into the details and unravel the mystery of how much money the boy started with.

The Challenge: A Mathematical Puzzle

A young boy visits three stalls at a fair. At each stall, he pays Re. 1 as entrance fee, spends half of the remaining money, and then exits the stall paying Re. 1 as exit fee. At the end of his journey, he has Rs. 3 left. Can you figure out how much he started with?

To solve this puzzle, let's designate the initial amount of money the boy started with as x. We will follow the sequence of his visits step by step to determine the value of x.

Step-by-Step Solution

At the first stall:

The boy pays Re. 1 as entrance fee, leaving him with x - 1. Then he spends half of the remaining money, which is (x - 1) / 2. After spending, he has x - 1 - (x - 1) / 2 (x - 1) / 2. He then pays Re. 1 as exit fee, leaving him with (x - 1) / 2 - 1 (x - 3) / 2.

At the second stall:

He pays Re. 1 as entrance fee, leaving him with (x - 3) / 2 - 1 (x - 5) / 2. He then spends half of the remaining money, which is (x - 5) / 4. After spending, he has (x - 5) / 2 - (x - 5) / 4 (x - 5) / 4. He pays Re. 1 as exit fee, leaving him with (x - 5) / 4 - 1 (x - 9) / 4.

At the third stall:

He pays Re. 1 as entrance fee, leaving him with (x - 9) / 4 - 1 (x - 13) / 4. He spends half of the remaining money, which is (x - 13) / 8. After spending, he has (x - 13) / 4 - (x - 13) / 8 (x - 13) / 8. He pays Re. 1 as exit fee, leaving him with (x - 13) / 8 - 1 (x - 21) / 8.

At the end:

He is left with Rs. 3, which means:

(x - 21) / 8 3

Solving for x:

x - 21 24

x 45

Therefore, the boy started with Rs. 45.

Another Approach: Simplified Solution

Let's assume the boy started with Rs. x:

After he exits the first stall, he has x - 1 - (x - 1) / 2 (x - 1) / 2. After the exit, he has (x - 1) / 2 - 1 (x - 3) / 2. After the second stall, he has (x - 3) / 2 - 1 - (x - 3) / 4 (x - 5) / 4. After the third stall, he has (x - 5) / 4 - 1 - (x - 5) / 8 (x - 9) / 8. Finally, he has (x - 9) / 8 - 1 (x - 21) / 8. He ends with Rs. 3, so (x - 21) / 8 3. Solving for x, we get x 45.

Following this method, he started with Rs. 45.

Verification: A Real-Life Example

To verify, let's use the value of x 45 and trace his journey:

He enters the first stall with Rs. 45, spends half, and exits with Rs. 21. He then enters the second stall with Rs. 20, spends half, and exits with Rs. 9. He finally enters the third stall with Rs. 8, spends half, and exits with Rs. 3, which matches the given condition.

Thus, the boy indeed started with Rs. 45, providing a fascinating lesson in mathematical problem-solving.