Problem 60 - Prealgebra

Question

The number line below extends from 0 to 2, with the segment from 0 to 1 and the segment from 1 to 2 each divided into 8 equal parts. Locate each of the following numbers on this number line. $$\frac{15}{16}$$

Step-by-Step Solution

Answer

\(\frac{15}{16}\) is located at the 7th tick mark between 0 and 1.

Step 1: Understanding the interval division

The given number line ranges from 0 to 2, with each half (0 to 1 and 1 to 2) divided into 8 equal parts. This means that each part represents an increment of \(\frac{1}{8}\).

Step 2: Find position within 0 to 1

To locate \(\frac{15}{16}\) on the number line, first consider its value in relation to 1. Since \(\frac{15}{16} < 1\) (because 15 is less than 16), it lies within the 0 to 1 section of the number line.

Step 3: Calculate the specific position

Next, convert \(\frac{15}{16}\) into a form using eighths. Because \(1 = \frac{16}{16}\), subtract \(\frac{15}{16}\) from \(1\) to find its relation to \(\frac{1}{8}\) segments: \(\frac{15}{16} = 8 \times \frac{1}{8} + 7 \times \frac{1}{16} = \frac{7}{8}\). Therefore, \(\frac{15}{16}\) corresponds to the 7th point on the line from 0.

Key Concepts

Interval Division

Interval division on a number line involves breaking the entire line into equal parts. This ensures each segment represents a consistent unit of measurement. For example, if a number line stretches from 0 to 2, we could divide it at regular intervals that are appropriately spaced, based on the total endpoints. In the exercise, the section from 0 to 1 and from 1 to 2 is divided into 8 parts.

This division means each mark on the line represents an increment of \(\frac{1}{8}\). By viewing each part as equivalent, it becomes easier to pinpoint fractions like \(\frac{15}{16}\), as each fraction will have a specific segment on the line.

• Each part of the segment represents equal increments.
• Division into 8 parts means each part is \(\frac{1}{8}\) of the total distance.
• Understanding these increments helps in locating fractions accurately.

Number Line

The number line is a visual tool that helps us understand the position of numbers in a linear space. It's a straightforward way to compare values visually. On this line, each point corresponds to a value, making the number line a helpful guide in understanding fractions and their relative sizes.

In this case, the number line is marked from 0 to 2, and it helps to visualize where \(\frac{15}{16}\) falls relative to whole numbers. When numbers are plotted on the line, you can quickly determine which fractions are larger or smaller and by how much.

• The number line shows the relation of numbers to each other.
• It is used to locate fractions between whole numbers.
• Helps in visualizing the size and order of fractions.

Fraction Representation

Fraction representation on a number line is about finding the exact fractional position between whole numbers. Given the division into eighths from our exercise, \(\frac{15}{16}\) must be placed accurately. This fraction is just \(\frac{1}{16}\) less than 1, implying it is located near the end of the first section (0 to 1) of the number line.

To represent \(\frac{15}{16}\), note each increment moves by eighths, so \(\frac{15}{16}\) aligns closely with \(\frac{7}{8}\) on the line. By converting the fractions, one can identify \(\frac{15}{16}\) sandwiched right before 1 but after \(\frac{7}{8}\).

• Fractions divide the space between whole numbers.
• Accurate placement requires understanding of fractional value.
• Fractions can be directly correlated to their visual placement on the line.