Question

convert the following repeating decimal to a fraction in simplest form..\overline{37}

Explanation:

Step1: Let \( x = 0.\overline{37} \)

Let \( x \) represent the repeating decimal. So \( x = 0.373737\ldots \)

Step2: Multiply \( x \) by 100

Since the repeating part has 2 digits, multiply both sides by 100. So \( 100x = 37.373737\ldots \)

Step3: Subtract the original equation from the new one

Subtract \( x = 0.373737\ldots \) from \( 100x = 37.373737\ldots \). We get \( 100x - x = 37.3737\ldots - 0.3737\ldots \)
Simplify the left side: \( 99x \)
Simplify the right side: \( 37 \)
So \( 99x = 37 \)

Step4: Solve for \( x \)

Divide both sides by 99: \( x = \frac{37}{99} \)
Check if the fraction can be simplified. The greatest common divisor of 37 and 99 is 1 (since 37 is a prime number and does not divide 99), so the fraction is in simplest form.

Answer:

\(\frac{37}{99}\)