Solving A Marble Problem A Step-by-Step Math Guide

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Introduction

In this article, we will explore a classic math problem involving marbles and bags. This problem requires careful reading, logical deduction, and a bit of algebraic thinking to solve. These types of problems are an excellent way to sharpen your problem-solving skills and your ability to translate word problems into mathematical expressions. So, let's dive into this interesting problem and break it down step by step.

Problem Statement: Jaimito's Marbles

The problem states: Jaimito has two bags of marbles. One bag contains 3 more marbles than the other. If 6 marbles are transferred from the bag with more marbles to the other bag, the product of the number of marbles in both bags is less than 70. What is the maximum number of marbles the bag with fewer marbles could have initially?

Breaking Down the Problem

To effectively solve this problem, we need to break it down into smaller, manageable parts. This involves understanding the given information, identifying the unknowns, and formulating a plan to connect the information to find the solution. Effective problem-solving often begins with a meticulous analysis of the problem statement. We will begin by defining variables to represent the unknowns and then express the given conditions mathematically. This methodical approach will help us unravel the complexities and arrive at the correct answer. The key is to translate the words into mathematical expressions, making the problem easier to visualize and solve.

Identifying the Unknowns and Givens

First, let's identify the unknowns. We need to find the maximum number of marbles the bag with fewer marbles could have initially. Let's represent this unknown with a variable. Let:

  • x = the number of marbles in the bag with fewer marbles initially

Now, let's translate the given information into mathematical expressions.

  • The other bag has 3 more marbles than the first bag, so it has x + 3 marbles.

  • 6 marbles are transferred from the bag with more marbles to the other bag.

    • The bag with more marbles will then have (x + 3) - 6 = x - 3 marbles.
    • The bag with fewer marbles will then have x + 6 marbles.

  • The product of the number of marbles in both bags after the transfer is less than 70. This gives us the inequality (x - 3)(x + 6) < 70.

Setting Up the Inequality

We've now translated the word problem into a mathematical inequality. This is a crucial step in solving the problem. The inequality (x - 3)(x + 6) < 70 represents the core relationship described in the problem. Solving this inequality will lead us to the possible values of x, which represents the initial number of marbles in the bag with fewer marbles. The ability to convert verbal descriptions into algebraic expressions is a fundamental skill in mathematics and is vital for problem-solving in various contexts. This step allows us to use algebraic techniques to find the solution efficiently and accurately.

Now, let's solve the inequality:

(x - 3)(x + 6) < 70

Expand the left side:

x2 + 6x - 3x - 18 < 70

Simplify:

x2 + 3x - 18 < 70

Move all terms to one side:

x2 + 3x - 88 < 0

Solving the Quadratic Inequality

We now have a quadratic inequality. To solve it, we first need to find the roots of the corresponding quadratic equation. This involves setting the quadratic expression equal to zero and solving for x. Factoring or using the quadratic formula are common methods to find these roots. Once we have the roots, we can determine the intervals where the inequality holds true. This process allows us to identify the range of possible values for x that satisfy the original condition of the problem. Understanding how to solve quadratic inequalities is a valuable skill in algebra and is applicable in various mathematical and real-world scenarios.

Finding the Roots

To find the roots, we solve the quadratic equation:

x2 + 3x - 88 = 0

We can factor this quadratic equation:

(x - 8)(x + 11) = 0

So, the roots are x = 8 and x = -11.

Determining the Interval

The roots divide the number line into three intervals: x < -11, -11 < x < 8, and x > 8. We need to test a value from each interval to determine where the inequality x2 + 3x - 88 < 0 holds true.

  • For x < -11, let's test x = -12: (-12)2 + 3(-12) - 88 = 144 - 36 - 88 = 20 > 0 (False)
  • For -11 < x < 8, let's test x = 0: (0)2 + 3(0) - 88 = -88 < 0 (True)
  • For x > 8, let's test x = 9: (9)2 + 3(9) - 88 = 81 + 27 - 88 = 20 > 0 (False)

Thus, the inequality holds true for -11 < x < 8.

Considering the Constraints

Since x represents the number of marbles, it must be a non-negative integer. This constraint is crucial because it limits the possible solutions to realistic values. We cannot have a negative number of marbles, and fractions or decimals are not applicable in this context. Therefore, we need to consider only the integer values within the interval we found that are greater than or equal to zero. This step ensures that our solution is not only mathematically correct but also makes sense in the real-world context of the problem. Applying such constraints is essential in many problem-solving situations to ensure the final answer is practical and meaningful.

Given this constraint, the possible values for x are 0, 1, 2, 3, 4, 5, 6, and 7.

Finding the Maximum Value

We are looking for the maximum number of marbles the bag with fewer marbles could have initially. Among the possible values of x (0, 1, 2, 3, 4, 5, 6, and 7), the largest integer is 7. Therefore, the maximum number of marbles the bag with fewer marbles could have initially is 7.

Verification

To ensure our solution is correct, it's always a good practice to verify it by plugging the value back into the original problem statement. This step helps to catch any potential errors in our calculations or reasoning. By substituting the value we found into the initial conditions, we can confirm that it satisfies all the requirements of the problem. This verification process reinforces our confidence in the solution and demonstrates a thorough understanding of the problem-solving process. It is a crucial step in mathematical problem-solving to ensure accuracy and completeness.

Let's verify our answer:

  • Initially, the bag with fewer marbles has 7 marbles.

  • The other bag has 7 + 3 = 10 marbles.

  • After transferring 6 marbles, the bags have:

    • 7 + 6 = 13 marbles
    • 10 - 6 = 4 marbles

  • The product of the number of marbles is 13 * 4 = 52, which is less than 70.

Our solution satisfies all the conditions of the problem.

Conclusion

Therefore, the maximum number of marbles the bag with fewer marbles could have initially is 7. This problem demonstrates the importance of careful reading, breaking down a problem into smaller parts, translating words into mathematical expressions, and verifying the solution. Problem-solving skills like these are valuable not only in mathematics but also in many other areas of life. By practicing these techniques, you can become a more effective and confident problem solver.

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