Problem 23 - Elementary and Intermediate Algebra

Question

Graph each rational number on the number line. $$ \frac{10}{3} $$

Step-by-Step Solution

Answer

Locate \( \frac{10}{3} \) at one-third distance between 3 and 4 on the number line.

Step 1 - Convert to mixed number or decimal

First, convert the given fraction \( \frac{10}{3} \) to a mixed number or a decimal to easily locate it on the number line. To convert \( \frac{10}{3} \), divide 10 by 3 which gives 3 as the quotient and a remainder of 1. So, \( \frac{10}{3} = 3 \frac{1}{3} \). Alternatively, as a decimal, \( 10 \div 3 = 3.33 \ldots \) (repeating).

Step 2 - Identify position on number line

Sketch a number line and mark the integers around your mixed number or decimal. Since \( 3 \frac{1}{3} \) or 3.33 lies between 3 and 4, place tick marks for numbers 3 and 4 on the number line.

Step 3 - Locate the fraction

Now, divide the segment between 3 and 4 into equal parts. Since 1/3 lies between 3 and 4, you could divide it into thirds. Locate \( 3 \frac{1}{3} \), which is one-third of the way from 3 towards 4.

Step 4 - Mark the point

Mark the point at one-third distance from 3 towards 4. Label this point as \( \frac{10}{3} \).

Key Concepts

number line

A number line is a visual representation of numbers on a straight line. It helps in comparing the size of numbers, understanding their sequence, and performing operations. The line has equally spaced markings, which can represent whole numbers, fractions, or decimals.
To graph a rational number like \(\frac{10}{3}\), start by drawing a horizontal line and mark the integer values closest to your number. For instance, since \(\frac{10}{3}\) converts to approximately 3.33, identify the range between 3 and 4. This makes it simpler to place your number precisely on the line.
Remember to:

• Identify your starting point (in this case, 3)
• Mark equal divisions within that range (e.g., thirds or tenths)
• Plot your number exactly according to its fraction or decimal value

mixed number

A mixed number combines a whole number and a fraction. It represents a value between whole numbers. Converting improper fractions to mixed numbers can simplify the graphing process.
Consider the fraction \(\frac{10}{3}\). When you divide 10 by 3, you get 3 complete groups and a remainder of 1. So, \(\frac{10}{3}\) becomes the mixed number \(3 \frac{1}{3}\). This format is helpful because:

• It shows the value's whole part (3, in this case)
• It clearly indicates how much more there is beyond that whole part (\(\frac{1}{3}\))

To use a mixed number on a number line, locate the whole number first, then move fractionally onward. For \(3 \frac{1}{3}\), find 3 and move one-third of the way towards 4.

decimal

Decimals are another way to express fractions, using a base-10 system. They are helpful for graphing rational numbers, as they illustrate exact positions on a number line. For example, the fraction \(\frac{10}{3}\) converts to the repeating decimal 3.33.
Decimals make it easier to mark exact points, especially on a finely divided number line. If you have 3.33, you'd:

• Locate 3
• Identify the next whole number (4)
• Mark sections between 3 and 4 into tenths
• Find approximately one-third of the way past 3 (i.e., 3.33)

Understand that repeating decimals (like 3.33...) indicate a value that never fully resolves, but can still be accurately placed on a number line. Remember that consistency in decimal places helps in precise plotting.