Question : Three numbers are in the ratio $\frac{4}{5}: \frac{5}{6}: \frac{9}{10}$. The difference between the smallest and the greatest numbers is 12. Find the number which is neither the smallest nor the greatest.
Option 1: 96
Option 2: 108
Option 3: 100
Option 4: 104
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Correct Answer: 100
Solution : Three numbers are in the ratio $\frac{4}{5}: \frac{5}{6}: \frac{9}{10}$. LCM of 5, 6, and 10 is 30. So, $\frac{4}{5}: \frac{5}{6}: \frac{9}{10}$ = $\frac{4}{5}\times 30: \frac{5}{6}\times 30: \frac{9}{10}\times 30=24:25:27$ The numbers can be simplified and written as $24x,25x,$ and $27x$. According to the question, $27x -24x = 12$ $⇒3x = 12$ $\therefore x = 4$ The number which is neither the smallest nor the greatest $=25x = 25 \times 4 = 100$ Hence, the correct answer is 100.
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