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Chapter 3: Problem 60

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. $$\frac{15}{16}$$

Short Answer

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

Step by step solution

01

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 \(\frac{1}{8}\).
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Find the following sums. (Add.) \(\frac{3}{4}+8 \frac{1}{4}+5\)

The following problems all involve the concept of borrowing. Subtract in case. \(12 \frac{3}{10}-5 \frac{7}{10}\)

Use the rule for order of operations to combine the following. \(6 \cdot 5^{2}+2 \cdot 3^{3}\)

The following problems all involve the concept of borrowing. Subtract in case. \(5-3 \frac{1}{3}\)

Divide. $$207 \div 26$$
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