The Grand Old Duke of York

Answer: $76.5\%$


Using numbers of men
$10 000$ lose $10\% = 1000$ leaves $9000$
$9000$ lose $15\% = 900 + 450$
                      $= 1350$ leaves $9000- 1350 = 7650$

$\dfrac{7650}{10000}=\dfrac{76.5}{100}$


Using proportion
After he gets to the top, the Grand Old Duke of York has $90\%$ of his men remaining. When he gets back down, he has $85\%$ of the men he had at the top. Therefore he has $85\%$ of $90\%$ of his original army.

Percentages
$85\%$ of $90\% = 90\% - 9\%-4.5\%=76.5\%$

Decimals
Written as decimals, $90\%$ is $0.9$ and $85\%$ is $0.85$. Therefore, $85\%$ of $90\%$ is $0.9 \times 0.85 = 0.765$. This is equivalent to $76.5\%$.

Fractions
$\dfrac{17}{20}$ of $\dfrac{9}{10} = \dfrac{17\times9}{200} = \dfrac{153}{200}=\dfrac{76.5}{100}$

Therefore, the Duke has $76.5\%$ of his army remaining.