Expressing Repeating Decimals as Fractions

We are asked to express the repeating decimal \(8.\overline{48}\) as a fraction. A repeating decimal is a decimal in which one or more digits repeat infinitely.

To convert a repeating decimal into a fraction, we can follow these steps:

1. Let the given decimal be equal to a variable, say \(x\).

   So, let \(x = 8.\overline{48}\)

   This means \(x = 8.484848...\) (Equation 1)

2. Identify the number of repeating digits. In \(8.\overline{48}\), the repeating digits are '48', which is a block of 2 digits.

3. Multiply the equation by \(10^n\), where \(n\) is the number of repeating digits. Since there are 2 repeating digits, we multiply by \(10^2 = 100\).

   \(100x = 100 \times 8.484848...\)

   \(100x = 848.484848...\) (Equation 2)

4. Subtract Equation 1 from Equation 2. This step eliminates the repeating part of the decimal.

   \(100x - x = 848.484848... - 8.484848...\)

   \(99x = 840\)

5. Solve for \(x\) by dividing both sides by the coefficient of \(x\).

   \(x = \frac{840}{99}\)

6. Simplify the fraction if possible. In this case, both 840 and 99 are divisible by 3 (since the sum of digits of 840 is 8+4+0=12, divisible by 3; the sum of digits of 99 is 9+9=18, divisible by 3).

   \(x = \frac{840 \div 3}{99 \div 3} = \frac{280}{33}\)

The simplified fraction is \(\frac{280}{33}\). However, the options provided are not in simplified form. Let's check the options against the unsimplified fraction \(\frac{840}{99}\).

Comparing \(\frac{840}{99}\) with the given options:

• Option 1: \(\frac{169.6}{20}\). This is not a simple fraction. Converting it to a simple fraction: \(\frac{169.6 \times 10}{20 \times 10} = \frac{1696}{200} = \frac{212}{25}\). This is not \(\frac{840}{99}\).
• Option 2: \(\frac{840}{99}\). This matches our result before simplification.
• Option 3: \(\frac{848}{100}\). This represents \(8.48\), not \(8.\overline{48}\).
• Option 4: \(\frac{848}{99}\). This would represent a different repeating decimal.

Therefore, the expression of \(8.\overline{48}\) as a fraction among the given options is \(\frac{840}{99}\).