Question

what is 0.43 with the 3 repeating converted to a fraction in simplest form

Answer 1

To convert the repeating decimal \( 0.43\overline{3} \) into a fraction, we can follow these steps:

1. Let \( x = 0.43\overline{3} \).

2. To express the repeating part, multiply \( x \) by 10 to shift the decimal point one place to the right:
\[
10x = 4.3\overline{3}
\]

3. Next, multiply \( x \) by 100 to shift the decimal point two places to the right:
\[
100x = 43.3\overline{3}
\]

4. Now, we have the two equations:
\[
10x = 4.3\overline{3} \quad \text{(1)}
\]
\[
100x = 43.3\overline{3} \quad \text{(2)}
\]

5. Subtract equation (1) from equation (2):
\[
100x - 10x = 43.3\overline{3} - 4.3\overline{3}
\]
\[
90x = 39
\]

6. Solve for \( x \):
\[
x = \frac{39}{90}
\]

7. Finally, simplify the fraction. The greatest common divisor (GCD) of 39 and 90 is 3:
\[
\frac{39 \div 3}{90 \div 3} = \frac{13}{30}
\]

Thus, the decimal \( 0.43\overline{3} \) converted to a fraction in simplest form is \( \frac{13}{30} \).