The Grand Old Duke of York
What percentage of his 10,000 men did the Grand Old Duke of York have left when he arrived back at the bottom of the hill?
Age
14 to 16
Challenge level
Problem
The Grand Old Duke of York had $10 000$ men.
He lost $10\%$ of them on the way to the top of the hill.
He then lost $15\%$ of the rest on the way back down.
What percentage of his $10,000$ men remained when he got to the bottom of the hill?
If you liked this probelm, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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.