Maximize A * B: Solving Number Inequalities

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Maximize a * b: Solving Number Inequalities

Let's dive into a fun math problem where we need to find the largest possible value of a * b given the inequalities 119 < a < 132 < b < 140. This problem is all about understanding inequalities and how to strategically pick numbers to get the biggest product. So, grab your thinking caps, and let’s get started!

Understanding the Problem

First, let's break down what the inequalities 119 < a < 132 < b < 140 tell us. We know that a must be a natural number greater than 119 but less than 132. Similarly, b must be a natural number greater than 132 but less than 140. Our goal is to find the largest possible values for a and b that fit these conditions and then multiply them together.

To maximize the product a * b, we need to choose a and b as large as possible within their respective ranges. Since a must be less than 132, the largest natural number we can choose for a is 131. Similarly, since b must be less than 140, the largest natural number we can choose for b is 139. So, we'll set a = 131 and b = 139.

Now, let’s calculate the product of these values:

a * b = 131 * 139

To compute this, we can break it down:

131 * 139 = 131 * (100 + 30 + 9) = 131 * 100 + 131 * 30 + 131 * 9 = 13100 + 3930 + 1179 = 13100 + 3930 + 1179 = 18209

So, the largest possible value for a * b is 18209.

Detailed Explanation

When faced with a problem like this, it's crucial to understand why we choose the largest possible values for a and b. Think of it this way: if we chose smaller values, their product would naturally be smaller. For instance, if we picked a = 120 and b = 133, their product would be significantly less than when we picked a = 131 and b = 139.

The inequality 119 < a < 132 constrains a to be between 119 and 132. The largest integer that satisfies this is 131. Similarly, the inequality 132 < b < 140 constrains b to be between 132 and 140, making 139 the largest possible integer. Multiplying these two values gives us the maximum possible product under the given conditions.

Why Not Other Values?

To illustrate further, consider a few alternative choices for a and b:

  • If a = 120 and b = 133, then a * b = 120 * 133 = 15960
  • If a = 125 and b = 135, then a * b = 125 * 135 = 16875
  • If a = 130 and b = 138, then a * b = 130 * 138 = 17940

As you can see, all these products are less than 18209, which we obtained by choosing the largest possible values for a and b.

Step-by-Step Solution

Let's walk through the solution step by step to make it crystal clear.

  1. Identify the Range for a:

    The inequality 119 < a < 132 tells us that a must be greater than 119 and less than 132. The largest natural number that satisfies this condition is a = 131. Remember, natural numbers are positive integers. Think of the numbers on a number line; a has to be bigger than 119, but smaller than 132. The biggest whole number that fits is 131. Don't overthink it!

  2. Identify the Range for b:

    The inequality 132 < b < 140 tells us that b must be greater than 132 and less than 140. The largest natural number that satisfies this condition is b = 139. Just like with a, b has to fit within a certain range. It's like finding the perfect fit for shoes, but with numbers! The biggest whole number that works is 139.

  3. Calculate the Product a * b:

    Now that we have the largest possible values for a and b, we multiply them together: a * b = 131 * 139. This is where the magic happens! We're taking the two biggest numbers we could find and multiplying them to get the biggest possible result.

  4. Perform the Multiplication:

    131 * 139 = 18209. We can break this down: 131 * (100 + 30 + 9) = (131 * 100) + (131 * 30) + (131 * 9) = 13100 + 3930 + 1179 = 18209.

    So, after all that calculating, we find that 131 * 139 = 18209. This is the largest possible value for a * b given the constraints.

  5. State the Final Answer:

    The largest possible value of a * b is 18209.

Key Concepts

  • Inequalities: Understanding inequalities is crucial for determining the possible values of variables. Inequalities define a range of numbers rather than a single value. For example, x > 5 means x can be any number greater than 5. Mastering inequalities is key to solving many math problems.
  • Natural Numbers: Natural numbers are positive integers (1, 2, 3, ...). They do not include zero or negative numbers. Knowing this helps you narrow down the possible values for a and b. Natural numbers are like the building blocks of math. They're the positive whole numbers we use for counting.
  • Maximization: Maximization involves finding the largest possible value that satisfies given conditions. In this case, we maximized the product a * b by choosing the largest possible values for a and b within their defined ranges. Think of maximization as finding the peak of a mountain. You want to get as high as possible within the given constraints.
  • Strategic Thinking: Choosing the correct values for a and b requires strategic thinking. Understanding that the largest product is achieved by multiplying the largest possible values within the given constraints is essential. Strategic thinking is like planning your moves in a game of chess. You need to think ahead and choose the best possible course of action.

Common Mistakes to Avoid

  • Incorrectly Interpreting Inequalities: Make sure you correctly understand the inequalities. For example, a < 132 means a must be strictly less than 132, so 132 itself is not a valid value for a. Misinterpreting inequalities is a common trap. Always double-check what the symbols mean.

  • Choosing Non-Integer Values: Remember that a and b must be natural numbers (integers). Do not choose decimal or fractional values. Sticking to whole numbers is crucial. Natural numbers are integers, so no decimals allowed!

  • Not Maximizing a and b: A common mistake is not choosing the largest possible values for a and b within their ranges. Always aim for the highest possible values to maximize the product. Don't settle for less. Always aim for the biggest possible numbers to get the largest product.

  • Arithmetic Errors: Be careful when performing the multiplication. Double-check your calculations to avoid errors. Arithmetic errors can be sneaky. Always double-check your work to make sure you haven't made any mistakes.

Practice Problems

To reinforce your understanding, try these practice problems:

  1. Given the inequality 20 < x < 30 < y < 40, where x and y are natural numbers, what is the largest possible value of x * y?
  2. Given the inequality 5 < p < 15 < q < 25, where p and q are natural numbers, what is the largest possible value of p * q?
  3. Given the inequality 100 < m < 110 < n < 120, where m and n are natural numbers, what is the largest possible value of m * n?

Conclusion

In summary, to find the largest possible value of a * b given the inequalities 119 < a < 132 < b < 140, we identify the largest natural numbers within the ranges for a and b, which are 131 and 139, respectively. Multiplying these values gives us the largest possible product, 18209. Understanding inequalities, natural numbers, and strategic maximization are key to solving this type of problem. By avoiding common mistakes and practicing similar problems, you can master these concepts and improve your problem-solving skills. Keep practicing, and you'll become a math whiz in no time! Remember guys, it's all about practice and understanding the basics. Keep grinding!