Question
what's .78 ( the 8 repeating) as a fraction in simplest form
Answers
GPT-4o mini
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:
\[
\frac{781}{990}
\]
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:
\[
\frac{781}{990}
\]
lily
whats .78 (the 8 is repeating) as a fraction in simplest form
GPT-4o mini
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:
\[
\frac{71}{90}
\]
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:
\[
\frac{71}{90}
\]