Problem 53 - Prealgebra

Question

For each pair of numbers, choose the number that is closest to 10. $$10.5 \text { and } 10.05$$

Step-by-Step Solution

Answer

10.05 is closest to 10.

Step 1: Calculate the distance of 10.5 from 10

To determine how close 10.5 is to 10, we subtract 10 from 10.5: \[ |10.5 - 10| = 0.5 \] So, 10.5 is 0.5 units away from 10.

Step 2: Calculate the distance of 10.05 from 10

To find out how close 10.05 is to 10, we subtract 10 from 10.05: \[ |10.05 - 10| = 0.05 \] Therefore, 10.05 is 0.05 units away from 10.

Step 3: Compare the distances calculated

Now, we compare the distances: 0.5 for 10.5 and 0.05 for 10.05. Since 0.05 is less than 0.5, 10.05 is closer to 10 than 10.5 is.

Key Concepts

Distance Calculations

When we talk about distance calculations in mathematics, we're essentially looking at how far one value is from another. Imagine standing on a number line. The space between you and another point reflects the distance between those two numbers.

In our exercise, we're interested in finding out which of the two numbers, 10.5 or 10.05, is closer to the number 10. To do this, we find out how far each number is from 10 by calculating the distance between them. This involves the simple process of subtraction:

• Distance from 10.5 to 10: \[ |10.5 - 10| = 0.5 \]
• Distance from 10.05 to 10: \[ |10.05 - 10| = 0.05 \]

These calculations give us the exact distances: 0.5 units and 0.05 units, respectively. The smaller the number, the shorter the distance. Like stepping stones on a path, the closer the stone (or number), the less distance you have to travel.

Absolute Value

The concept of absolute value is key when considering distance calculations on a number line. It is like looking at numbers without considering direction. Whether the number is positive or negative, the absolute value is simply how far that number is from zero.

When we write the absolute value of a number, we use vertical bars like this: \[ |x| \]. So, for example, \[ |-5| = 5 \] because the distance from -5 to 0 is 5 units. Similarly, \[ |3| = 3 \], because it's already 3 units away from zero without any negative sign consideration.

In finding which number is closer to 10, we utilize absolute value to disregard any negative sign that might arise from subtracting numbers. This allows us to focus purely on the distance without worrying about which direction on the number line we are moving.

Comparison of Numbers

Comparison of numbers is at the heart of so many mathematical problems. It involves determining which number is larger, smaller, or closer to another. The beauty of comparison is that it provides clarity in understanding relationships between numbers.

In our specific exercise, we compared the distances we calculated

• Distance for 10.5 was 0.5 units
• Distance for 10.05 was 0.05 units

Since 0.05 is smaller than 0.5, we concluded that 10.05 is closer to 10 than 10.5. This simple comparison allows us to easily determine which number is nearer to another point of reference. By comparing their distances to 10, we could make a confident decision on which was closer just by seeing which distance was less.