Question

Now that you have converted a terminating decimal number into a fraction, try converting a repeating decimal number into a fraction. Repeating decimal numbers are more difficult to convert into fractions. The first step is to assign the given decimal number to be equal to a variable, . For the decimal number , that means . If , what does 10x equal? What would happen if you subtracted the equation for x from the equation for 10x?

Option 1: 10x equals 3.33... and subtracting the equation for x from the equation for 10x would result in 9x = 2.99...
Option 2: 10x equals 3.333... and subtracting the equation for x from the equation for 10x would result in 9x = 2.999...
Option 3: 10x equals 0.33... and subtracting the equation for x from the equation for 10x would result in 9x = 0.299...
Option 4: 10x equals 0.3333... and subtracting the equation for x from the equation for 10x would result in 9x = 0.2999...

Answer 1

When you assign the repeating decimal 0.333... to a variable x and then calculate 10x, the result will be 3.333.... Subtracting x from 10x gives you 9x = 3, which when solved for x indicates that x equals 1/3. Therefore, the repeating decimal 0.333... is equivalent to the fraction 1/3.

Step-by-step explanation:

When converting a repeating decimal to a fraction, let's assign the decimal number 0.333... to a variable, say x. Therefore, x equals 0.333.... If we multiply x by 10, we get 10x equals 3.333.... Now, if we subtract the original equation from this new one, specifically subtract x from 10x, we will see a pattern:

Let's set up the two equations:

• x = 0.333...
• 10x = 3.333...

Subtracting the first equation from the second gives us:

10x - x = 3.333... - 0.333...

This results in:

9x = 3.000...

Therefore, we can clearly see that 9x equals 3. Solving for x, we divide both sides of the equation by 9:

x = 3/9

And simplifying this fraction gives us:

x = 1/3

The repeating decimal 0.333... is equivalent to the fraction 1/3.