1. Find the sum of all fractions in lowest terms with value greater than 10 but smaller than 100 and with denominator equal to 3 .
求所有大於 10 且小於 100 , 且以 3 為分母的最簡分數之和。
1. 9900
1. Such fractions include $10 \frac{1}{3}, 10 \frac{2}{3}, 11 \frac{1}{3}, 11 \frac{2}{3}, \ldots, 99 \frac{1}{3}, 99 \frac{2}{3}$. They can be grouped into 90 pairs, each with sum 110 (i.e. $10 \frac{1}{3}+99 \frac{2}{3}=110,10 \frac{2}{3}+99 \frac{1}{3}=110,11 \frac{1}{3}+98 \frac{2}{3}=110$ and so on). Hence the answer is $110 \times 90=9900$.