UPSC CDS I 2019 Math Paper

Concept:Adding two repeating decimals by aligning their decimal expansions or converting them to fractions.Explanation:Write $0.\overline{53}$ as $0.535353\ldots$.Write $0.5\overline{3}$ as $0.533333\ldots$.Add them column-wise: $0.535353\ldots + 0.533333\ldots = 1.068686\ldots$.The sum has a single non-repeating digit (the tenths place is $0$) and then the repeating block $68$.Thus, the result is $1.0\overline{68}$.Alternatively, convert to fractions: $0.\overline{53} = \frac{53}{99}$.$0.5\overline{3} = \frac{5}{10} + \frac{3}{90} = \frac{1}{2} + \frac{1}{30} = \frac{15}{30} + \frac{1}{30} = \frac{16}{30} = \frac{8}{15}$.Add: $\frac{53}{99} + \frac{8}{15} = \frac{265}{495} + \frac{264}{495} = \frac{529}{495} = 1 + \frac{34}{495}$.Since $\frac{34}{495} = 0.06868\ldots = 0.0\overline{68}$, the total is $1.0\overline{68}$.Answer:$1.0\overline{68}$ (Option A).