Bottle Sales Puzzle Solving A Fraction Problem

Introduction

In this article, we're going to dive into a fascinating mathematical problem involving a group of friends selling water bottles. This problem, at its core, is about fractions, comparisons, and problem-solving skills. We will meticulously break down the problem, dissecting each piece of information provided to arrive at a logical and accurate solution. This isn't just about finding the answer; it's about understanding the process, the reasoning, and the mathematical principles at play. Through this exploration, we aim to not only solve the specific problem at hand but also enhance your problem-solving abilities in general. Whether you're a student grappling with mathematical concepts or simply someone who enjoys a good mental workout, this article is designed to provide you with a clear, concise, and engaging journey through the solution. So, grab your thinking caps, and let's embark on this mathematical adventure together!

Problem Statement: The Bottle-Selling Scenario

Our mathematical journey begins with a scenario: Imagine a group of friends, each equipped with an equal number of water bottles, ready to make some sales. At the end of a bustling day, we have three friends – Vinicio, Ricardo, and Henry – with varying degrees of sales success. Vinicio manages to sell ${\frac{7}{10}}$ of his total stock of bottles. Ricardo, displaying a slightly more robust sales performance, sells ${\frac{3}{4}}$ of his bottles. Now, the puzzle unfolds: Henry, with his sales figures yet to be explicitly stated, introduces an element of comparison. We know that Henry sold more bottles than Vinicio but fewer bottles than Ricardo. This comparative information is crucial, acting as a bridge between the known sales fractions of Vinicio and Ricardo and the unknown sales fraction of Henry. The central question we aim to answer is: What fraction of his water bottles could Henry have possibly sold? To solve this, we'll need to carefully analyze the fractions, understand the comparative relationships, and employ logical deduction to narrow down the possibilities. This is where the heart of the mathematical problem lies – in the interplay between fractions, comparisons, and logical reasoning.

Understanding the Key Information

Before we plunge into the solution, it's crucial to distill the problem down to its core components. Let's recap the vital information provided: 1. Equal Starting Point: Each friend in the group began with the same number of water bottles. This is a foundational piece of information, allowing us to directly compare the fractions of bottles sold by each person. If they started with different amounts, comparing fractions would be significantly more complex. 2. Vinicio's Sales: Vinicio sold ${\frac{7}{10}}$ of his bottles. This gives us a concrete benchmark – a specific fraction to work with. 3. Ricardo's Sales: Ricardo sold ${\frac{3}{4}}$ of his bottles. This provides another fixed point and allows us to compare Ricardo's sales performance directly against Vinicio's. 4. Henry's Sales (Comparative): This is where the puzzle gets interesting. We know Henry's sales fall between Vinicio's and Ricardo's. He sold more than Vinicio but less than Ricardo. This comparative information is the key to unlocking the possible solutions for Henry's sales fraction. To effectively use this information, we need to not only understand the individual fractions but also how they relate to each other. We'll need to compare ${\frac{7}{10}}$ and ${\frac{3}{4}}$ to establish the range within which Henry's sales fraction must lie. This understanding is the first step in our journey towards solving the problem.

Comparing Fractions: Vinicio vs. Ricardo

The problem's core lies in the comparison of fractions, specifically understanding how ${\frac{7}{10}}$ (Vinicio's sales) relates to ${\frac{3}{4}}$ (Ricardo's sales). To make a meaningful comparison, we need to express these fractions with a common denominator. This allows us to directly compare the numerators and, consequently, the fractions themselves. The least common multiple of 10 and 4 is 20. Therefore, we'll convert both fractions to have a denominator of 20.

• Converting ${\frac{7}{10}}$: Multiply both the numerator and denominator by 2: ${\frac{7}{10} \times \frac{2}{2} = \frac{14}{20}}$
• Converting ${\frac{3}{4}}$: Multiply both the numerator and denominator by 5: ${\frac{3}{4} \times \frac{5}{5} = \frac{15}{20}}$

Now, we have ${\frac{14}{20}}$ (Vinicio) and ${\frac{15}{20}}$ (Ricardo). A clear comparison emerges: ${\frac{15}{20}}$ is greater than ${\frac{14}{20}}$. This confirms that Ricardo sold a larger fraction of his bottles than Vinicio. This is a crucial piece of the puzzle. It establishes the upper and lower bounds for Henry's sales. We know Henry's sales fraction must be greater than ${\frac{14}{20}}$ and less than ${\frac{15}{20}}$. However, since we're looking for a fraction that Henry could have sold, we need to explore the fractions that fall between these two values.

Finding Fractions Between ${\frac{7}{10}}$ and ${\frac{3}{4}}$

Now, the challenge is to identify fractions that lie strictly between ${\frac{7}{10}}$ (or ${\frac{14}{20}}$) and ${\frac{3}{4}}$ (or ${\frac{15}{20}}$). This might seem tricky at first glance, as there appear to be no obvious whole-number fractions between ${\frac{14}{20}}$ and ${\frac{15}{20}}$. However, the key here is that there are infinitely many fractions between any two given fractions. To find such fractions, we can use a simple yet powerful technique: increasing the denominator. When we increase the denominator of a fraction, we effectively create smaller