Write 0.32 repeating as a fraction in simplest form

Write 0.32 repeating as a fraction in simplest form.

The Correct Answer and Explanation is :

The correct answer is: {\frac{97}{300}}

To convert the repeating decimal (0.32\overline{3}) (which can be written as (0.323333…)) into a fraction, we can follow a systematic approach.

Step 1: Define the repeating decimal

Let (x = 0.323333…).

Step 2: Isolate the repeating part

To eliminate the repeating part, we can multiply (x) by 10 to move the decimal point one place to the right. This gives us:

[
10x = 3.23333…
]

Step 3: Align the repeating decimals

Next, we want to align the repeating decimals. To do this, we can multiply (x) by 100, which moves the decimal point two places to the right:

[
100x = 32.3333…
]

Now we have two equations:

1. (10x = 3.23333…) (equation 1)
2. (100x = 32.3333…) (equation 2)

Step 4: Subtract the equations

Next, we subtract equation 1 from equation 2:

[
100x – 10x = 32.3333… – 3.23333…
]

This simplifies to:

[
90x = 32.3333… – 3.23333…
]

Calculating the right side:

[
32.3333… – 3.23333… = 29.1
]

Thus, we have:

[
90x = 29.1
]

Step 5: Solve for (x)

To find (x), divide both sides by 90:

[
x = \frac{29.1}{90}
]

Step 6: Eliminate the decimal in the numerator

To eliminate the decimal in the numerator, we can multiply the numerator and denominator by 10:

[
x = \frac{291}{900}
]

Step 7: Simplify the fraction

Now, we simplify (\frac{291}{900}). The greatest common divisor (GCD) of 291 and 900 is 3. So we divide both the numerator and denominator by 3:

[
x = \frac{291 \div 3}{900 \div 3} = \frac{97}{300}
]

Final Result

Thus, the fraction representation of the repeating decimal (0.32\overline{3}) is:

[
\boxed{\frac{97}{300}}
]

This is the simplest form of the fraction.