Integer Insertion In Ordered Number List: A Math Problem

by Alex Johnson 57 views

Have you ever been faced with a seemingly simple math problem that makes you pause and think? This is one of those! Let's break down a question about inserting an integer into an ordered list of numbers. We'll walk through the problem, understand the concepts, and find the solution together.

Understanding the Problem

At its core, this question tests our understanding of number ordering, integers, and how to compare different forms of numbers (like fractions and decimals). The given list is:

-15, -3 4/5, -1.5, __, 5, 2/3

Our mission is to figure out which integer from the options can fit into the blank space while maintaining the ascending order of the list. This requires us to understand where each number sits on the number line.

Key Concepts

Before we dive into solving, let's refresh some essential concepts:

  • Integers: These are whole numbers (no fractions or decimals) and can be positive, negative, or zero (e.g., -3, -2, -1, 0, 1, 2, 3).
  • Number Ordering: Numbers increase in value as you move from left to right on the number line. Negative numbers are smaller than positive numbers, and numbers closer to zero are larger than those further away.
  • Fractions and Decimals: To compare these with integers, it's often helpful to convert fractions to decimals or visualize them on a number line. For example, 2/3 is approximately 0.67.

Converting to a Common Format

To make comparisons easier, let's convert all the numbers in the list to decimals:

  • -15 (already an integer)
  • -3 4/5 = -3.8
  • -1.5 (already a decimal)
  • 5 (already an integer)
  • 2/3 ≈ 0.67

Now our list looks like this:

-15, -3.8, -1.5, __, 5, 0.67

This makes it much clearer to see the order of the numbers and where our integer needs to fit.

Analyzing the Options

Now let's look at the options provided and see which one makes sense:

  • A. -2
  • B. -1 1/3
  • C. 0
  • D. 1.2

We need to determine which of these numbers, when inserted into the blank space, will keep the list in ascending order.

Option A: -2

If we insert -2 into the list, it becomes:

-15, -3.8, -1.5, -2, 5, 0.67

Notice that -2 is smaller than -1.5. This disrupts the ascending order, as -2 should come before -1.5. So, option A is not correct.

Option B: -1 1/3

First, let's convert -1 1/3 to a decimal: -1 1/3 = -1.33 (approximately). Now the list looks like this with -1.33 inserted:

-15, -3.8, -1.5, -1.33, 5, 0.67

Here, -1.33 is greater than -1.5, so it should come after -1.5. However, 0.67 which is approximately is less than 5, therefore this is not an integer. Option B is incorrect as well.

Option C: 0

Inserting 0 into the list gives us:

-15, -3.8, -1.5, 0, 5, 0.67

Now, let's check the order: -15 < -3.8 < -1.5 < 0. Here, 0 is indeed greater than -1.5. However, 0 is less than 0.67 is false, as 0.67 is approximately 2/3, which is greater than 0. Therefore, this disrupts the ascending order too. This makes the entire list invalid. Option C is therefore, not correct.

Option D: 1.2

If we put 1.2 in the blank, the list is:

-15, -3.8, -1.5, 1.2, 5, 0.67

Again, let’s verify the order. Is -15 < -3.8 < -1.5 < 1.2? Yes, so far so good. But wait! 1.2 should be less than 5, and 0.67 should follow 1.2. Thus, this order is incorrect. Option D, also, will not work.

Finding the Correct Integer

It seems we made a slight mistake in our analysis. Let's re-evaluate option C. When we insert 0, the list becomes:

-15, -3.8, -1.5, 0, 5, 0.67

We need to correct the order of the list. The correct ascending order should be:

-15, -3.8, -1.5, 0, 0.67, 5

With 0 inserted, it fits perfectly between -1.5 and the positive numbers. So, option C is indeed the correct answer!

Conclusion

Therefore, the integer that can be inserted in the blank space to maintain the ascending order of the list is 0. This problem highlights the importance of understanding number ordering, converting between fractions and decimals, and carefully comparing values. Math problems like these can seem tricky at first, but by breaking them down step-by-step and revisiting key concepts, we can find the solution!

For more in-depth information and practice problems on number systems and ordering, you can visit resources like Khan Academy's Number System Section.