Problem 51 - Prealgebra

Question

For each pair of numbers, choose the number that is closest to 10. $$9.9 \text { and } 9.99$$

Step-by-Step Solution

Answer

9.99 is closer to 10.

Step 1: Calculate the difference of 9.9 from 10

To determine how close 9.9 is to 10, calculate the absolute difference: \[ |9.9 - 10| = 0.1 \]

Step 2: Calculate the difference of 9.99 from 10

Next, calculate the absolute difference between 9.99 and 10:\[ |9.99 - 10| = 0.01 \]

Step 3: Compare the differences

Compare the differences calculated in Steps 1 and 2. The smaller the difference, the closer the number is to 10. In this case, 0.01 is less than 0.1.

Step 4: Select the closest number

Since the difference between 9.99 and 10 is smaller than the difference between 9.9 and 10, 9.99 is closer to 10.

Key Concepts

Understanding Comparison of Numbers

When we talk about comparing numbers, we are essentially evaluating which of them is larger, smaller, or nearer to a certain reference point—in this case, the number 10. One common way to compare numbers is by calculating their absolute differences from a particular value. This simply means subtracting each number from the reference value and considering the absolute value of the result, as negative distances don't make sense.
To compare two numbers with respect to a reference value:

• Calculate the absolute difference of each number from the reference value.
• Compare these differences to determine which is smaller.
• The number with the smaller difference is closer to the reference point.

Using these steps, you can easily determine the proximity of two numbers to a chosen reference.

Foundations of Prealgebra Concepts

Prealgebra serves as a foundation for all future mathematics learning by introducing important concepts like integers, fractions, decimals, and negative numbers. Understanding absolute difference is part of this basis.
Absolute difference helps us measure how far one number is from another. It's similar to calculating distance in geometry, where direction doesn’t matter because we’re only interested in how far apart two points are:

• Use the formula for absolute difference: \[ |a - b| \]
• This calculation tells us how close one number is to another, or in this exercise, to the number 10.

Being comfortable with absolute differences is crucial for more advanced mathematical concepts. It's a simple yet fundamental aspect of math that will be regularly used in various applications.

Approach to Math Problem Solving

Approaching math problems strategically can greatly enhance your ability to find solutions efficiently. In this context, solving which number is closest to 10 involves following a systematic method:

• Start by identifying what is being asked: How close is each number to 10?
• List the steps necessary to find the answer—here, calculating absolute differences.
• Execute the calculations step-by-step, checking your work at each phase.
• Compare results logically to make the final determination.

This kind of organized problem-solving approach is extremely useful, as it ensures that you can tackle even larger and more complex math challenges with confidence. With practice, you'll be able to identify patterns and use these strategies almost instinctively.