Explain The Math Riddle About The Three Friends, The Restaurant Bill Of P25, Their Combined P30, And The Waiter Giving Back P1 Each.

Have you ever encountered a situation where the numbers just don't seem to add up? Where a simple calculation leads to a perplexing discrepancy? These types of scenarios, often presented as riddles or puzzles, can be a fun and engaging way to exercise our mathematical thinking. This article delves into a classic brain-teaser, exploring the logic and the common misstep that leads to its confusing conclusion.

The Puzzling Restaurant Bill

The math riddle begins with a seemingly straightforward situation three friends decide to dine together at a restaurant. Each friend has P10, resulting in a total of P30 available for their meal. They select a menu that suits their budget and pool their money. After enjoying their meal, the friends receive their bill, which amounts to P25. This leaves them with P5 in change. The waiter, in a gesture of goodwill, returns P1 to each of the three friends, and keeps the remaining P2 as a tip. Now, here's where the puzzle begins:

Each friend initially contributed P10 and received P1 back, effectively spending P9 each. If we multiply P9 (the amount each friend spent) by 3 (the number of friends), we get P27. Then, if we add the P2 that the waiter kept as a tip, we arrive at a total of P29. Where did the missing P1 go? This is the crux of the problem that often leads to confusion and debate. Many people find themselves scratching their heads, trying to reconcile the seemingly missing peso. The key to unraveling this puzzle lies in understanding the correct way to frame the calculation. It's not about adding the tip to the amount the friends effectively spent; it's about distributing the costs and the change correctly.

Deconstructing the Misleading Calculation

The mistake in the puzzle stems from an incorrect attempt to reconcile the expenses. It's tempting to add the amount the friends effectively spent (P27) to the tip the waiter received (P2). This leads to the erroneous conclusion that there is a missing peso. The flaw in this approach is that it mixes two different perspectives: the friends' expenses and the total cost of the meal. A more accurate way to analyze the situation is to focus on where the money went. The friends started with P30. The restaurant received P25 for the meal, and the friends received P5 in change. This accounts for all the money. The P5 change was then divided, with P3 going back to the friends (P1 each) and P2 going to the waiter as a tip. The confusion arises when we try to combine the P27 (the apparent total spent by the friends) with the P2 tip. The P27 already includes the P25 paid to the restaurant; it's the actual cost of the meal plus the waiter's tip. Adding the P2 tip again is a form of double-counting.

The Correct Way to Frame the Problem

To understand where the missing peso isn't, it's essential to reframe the problem and track the flow of money correctly. Start with the initial amount the friends had: P30. The total expense can be broken down into two parts: the cost of the meal (P25) and the total amount of money returned (P5). The change received by the friends (P5) is further divided into the amount they kept (P3, P1 each) and the tip for the waiter (P2). There's no missing money when the calculation is done correctly. The error in the riddle lies in the way the information is presented, which misdirects our mathematical instincts. We are drawn to add the tip to the amount the friends spent, but this is not the correct operation. Instead, we should be accounting for the total expenditure and the distribution of the change. The money paid for the meal plus the money received as change should equal the initial amount the friends had. This simple equation resolves the perceived discrepancy.

Unraveling the Math Riddle A Step-by-Step Explanation

To truly grasp the solution to this math puzzle, let's break it down step by step. This meticulous approach will illuminate the flaw in the initial, misleading calculation and demonstrate how to accurately account for the money.

Initial Investment

The three friends each contribute P10, resulting in a total of P30. This is our starting point, the initial amount of money in play. It's crucial to keep this total in mind as we trace the flow of funds throughout the scenario.

The Bill and the Change

The restaurant bill comes to P25. This means that P25 of the initial P30 goes to the restaurant. The remaining P5 is the change, the money that is left over after paying the bill. At this stage, we can clearly see that P25 (bill) + P5 (change) = P30 (initial investment). There is no missing money yet; the total amount is accounted for.

Dividing the Change

The waiter returns the P5 change. However, instead of dividing it equally, he gives P1 to each friend (totaling P3) and keeps P2 as a tip. This is where the misdirection begins. We now have the P5 change divided into two parts: P3 returned to the friends and P2 kept as a tip. It's important to note that P3 (friends' share) + P2 (tip) = P5 (total change). Again, no money is missing.

The Misleading Calculation

The puzzle's trick lies in how we frame the next calculation. Each friend received P1 back, meaning they effectively spent P9 (P10 initial contribution - P1 returned). If we multiply this P9 by the three friends, we get P27. Now, here's the critical mistake: the puzzle asks us to add the P2 tip to this P27. This is where the error occurs. The P27 already includes the P25 paid to the restaurant. It represents the total amount the friends effectively spent, which incorporates both the meal cost and the tip. Adding the tip again is double-counting it.

The Correct Accounting

To correctly account for the money, we need to focus on the two main components of the initial P30: the cost of the meal and the total change. The meal cost P25, and the total change returned was P5. These two amounts add up to P30, the initial investment. We can further break down the change into the P3 received by the friends and the P2 tip. However, we should not add the tip to the amount the friends effectively spent because that amount already includes the tip. The key is to understand that the P27 (effective spending) encompasses both the P25 meal cost and the P2 tip. There is no missing peso; it's simply a matter of how we frame the arithmetic.

The Psychology of Math Puzzles Why We Get Confused

Math puzzles like the missing peso riddle aren't just about numerical calculations; they also tap into the way our minds process information. Understanding the psychological aspects of these puzzles can shed light on why we sometimes fall for their traps and how to avoid them. Several cognitive biases and framing effects contribute to the confusion we experience when trying to solve these problems.

Cognitive Biases at Play

Cognitive biases are systematic patterns of deviation from norm or rationality in judgment. They are mental shortcuts that our brains use to simplify information processing, but they can sometimes lead to errors in thinking. One relevant bias in the context of math puzzles is the anchoring bias. This is the tendency to rely too heavily on the first piece of information offered (the