Question

write \\(0.\overline{76}\\) as a fraction.

Explanation:

Define the repeating decimal equation

Let \(x\) represent the repeating decimal:

$$x = 0.\overline{76} = 0.767676\dots$$

Multiply to shift the decimal point

Since there are two repeating digits, multiply \(x\) by \(100\):

$$100x = 76.767676\dots$$

Subtract the equations to eliminate the repeating part

Subtract the first equation from the second equation:

$$100x - x = 76.767676\dots - 0.767676\dots$$

$$99x = 76$$

Solve for x as a fraction

Divide both sides by \(99\):

$$x = \frac{76}{99}$$

Check for simplification

Using the Simplifying Fractions knowledge point, we check if \(76\) and \(99\) share any common factors.
The prime factorization of \(76\) is \(2^2 \times 19\).
The prime factorization of \(99\) is \(3^2 \times 11\).
Since they share no common factors, the fraction is in simplest form:

$$x = \frac{76}{99}$$

Answer:

Write \(0.\overline{76}\) as a fraction. <blank>\(\frac{76}{99}\)</blank>