what's .78 ( the 8 repeating) as a fraction in simplest form

Answer 1

Let \( x = 0.78\overline{8} \), where \( 0.78\overline{8} \) means \( 0.788888...\).

To convert this repeating decimal into a fraction, we can manipulate the equation as follows:

1. Multiply \( x \) by 10 to shift the decimal point: \[ 10x = 7.88888... \]

2. Now, multiply \( x \) by 1000 to shift the decimal point three places to the right: \[ 1000x = 788.88888... \]

3. We now have two equations: \[ 10x = 7.88888... \] \[ 1000x = 788.88888... \]

4. To eliminate the repeating part, subtract the first equation from the second: \[ 1000x - 10x = 788.88888... - 7.88888... \] \[ 990x = 781 \]

5. Now, solve for \( x \): \[ x = \frac{781}{990} \]

Next, we can simplify \( \frac{781}{990} \) if possible.

1. Check for common factors. Both 781 and 990 do not seem to have any common factors. In fact, 781 is a prime number.

Since there are no common factors between the numerator and the denominator, the fraction is already in its simplest form.

Thus, the decimal \( 0.78\overline{8} \) as a fraction in simplest form is: \[ \frac{781}{990} \]

Answer 2

whats .78 (the 8 is repeating) as a fraction in simplest form

Answer 3

Let \( x = 0.78\overline{8} \), which represents the repeating decimal \( 0.788888...\).

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

1. Set up the equation: \[ x = 0.788888... \]

2. Multiply both sides by 10: \[ 10x = 7.88888... \]

3. Multiply both sides again by 10 (to eliminate the repeating part after two decimal places): \[ 100x = 78.88888... \]

4. Now you have the system: \[ 10x = 7.88888... \] \[ 100x = 78.88888... \]

5. Subtract the first equation from the second: \[ 100x - 10x = 78.88888... - 7.88888... \] \[ 90x = 71 \]

6. Solve for \( x \): \[ x = \frac{71}{90} \]

Now, check if \( \frac{71}{90} \) is in simplest form. The number 71 is a prime number and does not share any factors with 90 (which factors into \( 2 \times 3^2 \times 5 \)).

Therefore, the repeating decimal \( 0.78\overline{8} \) as a fraction in simplest form is:

\[ \frac{71}{90} \]