Question

If \(46.\overline{35}=\frac{m}{n}\), where \(m,n\) are positive integers and \(\gcd(m,n)=1\), then \(m+n=\ ?\)

• A. \(4688\)
• B. \(4694\)
• C. \(5884\)
• D. \(8459\)

Hint:

For a repeating decimal with two repeating digits, \[ 0.\overline{ab}=\frac{ab}{99}. \]

Answer 1

The Correct Option is B

Step 1: Separate the repeating decimal part. \[ x=46.\overline{35} \] \[ x=46+0.\overline{35} \]

Step 2: Convert the repeating part into a fraction. \[ 0.\overline{35} = \frac{35}{99} \] Hence \[ x=46+\frac{35}{99} \] \[ =\frac{46\times99+35}{99} \] \[ =\frac{4554+35}{99} \] \[ =\frac{4589}{99} \] Since \(99=9\times11\) and \(4589\) is not divisible by \(3\) or \(11\), the fraction is already in lowest terms. Thus \[ m=4589,\qquad n=99 \]

Step 3: Find \(m+n\). \[ 4589+99 \] \[ =4688 \] Therefore, \[ \boxed{4688} \]