Given:

The resultant ratio is \(\frac{10}{21}\)

The numerator of the fraction increased by 500%

The denominator of the fraction increased by 600%

Calculation:

Let the fraction be \(\frac{x}{y}\)

The increased numerator = \(x \times \left( {1 + \frac{{500}}{{100}}} \right)\;\)

The increased denominator =  \(y \times \left( {1 + \frac{{600}}{{100}}} \right)\;\)

then, the resultant fraction = \(\frac{10}{21}\)

⇒ \(\frac{{\left[ {x \times \left( {1 + \frac{{500}}{{100}}} \right)} \right]}}{{\left[ {y \times \left( {1 + \frac{{600}}{{100}}} \right)} \right]}}\) = \(\frac{10}{21}\)

⇒  \(\frac{6x}{7y}\) = \(\frac{10}{21}\)

⇒ \(\frac{x}{y}\) = \(\frac{5}{9}\)

The original fraction is \(\frac{5}{9}\)

Alternate Method:

The  original fraction = \(\frac{{\left[ {10 \times \left( {1 + \frac{{600}}{{100}}} \right)} \right]}}{{\left[ {21 \times \left( {1 + \frac{{500}}{{100}}} \right)} \right]}}\) = \(\frac{{\left[ {10 \times \left( { \frac{{700}}{{100}}} \right)} \right]}}{{\left[ {21 \times \left( { \frac{{600}}{{100}}} \right)} \right]}}\)  = \(\frac{5}{9}\)

The original fraction is \(\frac{5}{9}\)