Problem 17 - A History of Mathematics: An Introduction

Question

In a vessel full of wine there are three taps such that if one opens the largest it will empty the vessel in 3 hours, if one opens the middle tap it will empty it in 4 hours, and if one uses the smallest tap it will empty it in 6 hours. How long would it take to empty the vessel if all three taps were open? (This problem and the next are also from Chuquet's work.)

Step-by-Step Solution

Answer

Answer: 4 hours.

Step 1: Determine the capacity of the vessel

Let the capacity of the wine vessel be C liters. Now, we have the rates of each tap: - Largest tap: C/3 L/h (since it takes 3 hours to empty the vessel) - Middle tap: C/4 L/h - Smallest tap: C/6 L/h

Step 2: Find the combined rate of the three taps

When all three taps are open at the same time, their rates of emptying the vessel are added together. Therefore, the combined rate R can be found using the following equation: R = (C/3) + (C/4) + (C/6)

Step 3: Simplify the combined rate equation

To simplify the equation, first find the least common multiple (LCM) of 3, 4, and 6, which is 12. Then, express the combined rate in terms of C and the LCM, as follows: R = (4C/12) + (3C/12) + (2C/12) R = (4C + 3C + 2C)/12 R = 9C/12

Step 4: Simplify further

We can further simplify the combined rate by dividing the numerator and denominator by their common factor, 3: R = (9C/12) * (1/3) R = 3C/12

Step 5: Find the time it takes to empty the vessel

Now, we have the combined rate, R = 3C/12, to empty the vessel when all three taps are open. To find the time it takes, we can use the following formula: Time = Capacity / Combined Rate Time = C / (3C/12)

Step 6: Simplify and solve for time

As a final step, we need to simplify the formula for time and solve for the value. The capacity C will cancel out: Time = (C / (3C/12)) * (12/12) Time = 12 / 3 = 4 hours So, it would take 4 hours to empty the vessel when all three taps are open.

Key Concepts

Mathematics

Mathematics is the language that allows us to describe quantities and relationships. In this rate problem, we're dealing with the rates at which taps can empty a vessel. Each tap operates at a specific rate, effectively defined by how much of the vessel it can empty per hour. The key mathematical concept here is in the addition of these individual rates to determine a combined rate when all taps are working together.
To accomplish this, one must understand how to manipulate fractions, as each rate is represented in terms of capacity over time. In practical scenarios, capacity is a constant, so our focus is on how rates sum up to create a new, collective rate. The idea is straightforward: if each component works together toward the same goal, their efforts combine and intensify the result. This concept is foundational in working on rate problems, and it relies heavily on understanding mathematical operations like addition and division.

Problem Solving

In problem solving, particularly with rate problems, it is crucially important to approach the solution methodically. The main problem-solving strategy here involves breaking down the problem into smaller, manageable pieces and solving each in a step-by-step fashion.
Start by identifying what you know. We know rates for each tap and how they function independently. The next step is to find how they function together. Rate problems like this exemplify how combining individual components can lead to a holistic solution.

• Define your variables clearly, such as the capacity of the vessel and individual rates.
• Utilize basic principles, such as finding a common denominator, to simplify the rates into a usable form.
• Use logical progression by turning these simplified components into an actionable solution, i.e., finding the combined rate and subsequently the total time.

This strategy enhances clarity in problem-solving by dissecting complex tasks into more achievable steps.

Algebra

Algebra plays a pivotal role here as it provides the tools necessary to manipulate equations and solve rate problems effectively. From defining variables to using equations, algebra provides a systematic method to derive solutions.
In the exercise, we're using algebra to express the problem in mathematical terms. This involves:

• Writing equations to represent each tap's rate of contributing to emptying the vessel.
• Solving for a common denominator to simplify rate addition.
• Rearranging terms to isolate the variable of interest, in this case, the time needed to empty the vessel.

Through this exercise, the power of algebra becomes evident in how it can break down complex, word-based problems into solvable mathematical forms. By understanding how to establish, manipulate, and solve algebraic expressions, students can tackle a variety of rate problems with increased confidence.