Problem 12 - A History of Mathematics: An Introduction

Question

Another problem from Mah?v?ra: Of a collection of mango fruits, the king took \(1 / 6\); the queen took \(1 / 5\) of the remainder, and three chief princes took \(1 / 4,1 / 3,1 / 2\) of what remained at each step; and the youngest child took the remaining three mangoes. O you, who are clever in working miscellaneous problems on fractions, give out the measure of that collection of mangoes.

Step-by-Step Solution

Answer

Answer: There were 6 mangoes in the original collection.

Step 1: Represent the problem in terms of fractions and variables

Let x represent the total number of mangoes in the collection. The king took 1/6, so the remaining mangoes are 5/6 of the total. Then, the queen took 1/5 of the remainder, so the number of mangoes left after the king and queen had their shares is (5/6)(4/5) times the total mangoes. Next, the princes took their shares of the remaining mangoes, which can be represented as (1 - 1/4)(1 - 1/3)(1 - 1/2) times the number of mangoes left after the king and queen took their shares. Finally, the youngest child takes the final 3 mangoes.

Step 2: Set up the equation

We can now write an equation representing the youngest child taking the final 3 mangoes: \((5/6)(4/5)((3/4)(2/3)(1/2))x = 3\)

Step 3: Solve the equation

First, simplify the equation: \((5/6)(4/5)((3/4)(2/3)(1/2))x = (1/2) x = 3\) Next, you need to solve for x: \(x = 3 \cdot 2\) \(x = 6\)

Step 4: Interpret the result

Since x = 6, the total number of mangoes in the original collection was 6.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of solving problems involving fractions, especially when dealing with scenarios that include a sequence of divisions or proportions like in our mango problem. In algebra, we use numbers, variables, and mathematical operations to create expressions that represent real-world situations.

In the mango problem, we start by using a variable to represent an unknown quantity. Here, we let \(x\) denote the total number of mangoes. This variable helps us keep track of how each participant's actions reduce the total number of mangoes. Initially, the king's share reduced the mango count by a sixth, which leaves us with \(\frac{5}{6}\) of \(x\). By using algebraic expressions, we represent each step of distribution in terms of fractions, continually updating the remainder as each subsequent individual takes their share.

The beauty of algebra is in its ability to encapsulate complex real-life processes into manageable equations, which we can then manipulate to find our solution. The process shows how algebraic expressions are essential for clear and effective problem representation.

Problem-Solving Strategies

When tackling problems like the mango distribution, it's crucial to adopt a systematic approach. A good problem-solving strategy simplifies complex situations and helps in finding the correct solution steps systematically.

1. **Understand the problem:** Start by reading the problem carefully and identifying what is being asked. Here, we need to find the total number of mangoes, so we introduce a variable \(x\) to represent that unknown quantity.

2. **Break down the steps:** Split the problem into smaller, manageable parts. Determine how each participant's share affects the total number of mangoes. Calculate the fractions step-by-step for each person's share, starting with the king to the youngest child. Each step should seamlessly flow into the next.

3. **Translate to equations:** Use the insights from the breakdown to set up an equation that culminates in the problem's end state. In our case, this means forming an equation that reflects the youngest child having precisely three mangoes left.

4. **Simultaneous checking and simplification:** After setting up the equations, simplify them to find \(x\). Then check each step to ensure no mistakes were made along the way. This is crucial for verifying the accuracy of the final solution.

These strategies lead to logical, organized problem-solving, ensuring all steps are accounted for reliably and comprehensibly.

Mathematical Reasoning

Mathematical reasoning involves drawing logical conclusions from sets of premises or hypotheses—an integral part of solving algebraic problems with fractions.

In our case, reasoning helps us interpret the effect of each person's actions on the total number of mangoes sensibly. From the initial action of the king, taking \(\frac{1}{6}\) of the mangoes, we reason that the remainder is \(\frac{5}{6}x\). This continues, with each fraction calculated representing another fraction of the already reduced quantities. Through reasoning, we ensure each calculation respects the previous results, maintaining logical consistency.

At the end of the distribution process, we are left with the mathematical task of interpreting and solving for \(x\) using the provided equation—a key part of applying mathematical reasoning. Solving the equation \(3(\frac{1}{2}x) = 3\) requires understanding that multiplying the equations lets us isolate \(x\), effectively solving the problem.

Reasoning helps handle intricate multi-step problems with ease, creating a coherent path from assumptions to solutions, which is essential for proficient mathematical comprehension.