Percentage of Relatively Prime Integers

By Euler's Totient Function , the percentage of integers not larger than n that are relatively prime to n is the percentage representation of $\frac{\phi \left ( n \right )}{n}$ .

It follows that $a$ must be the smallest integer that has the maximum number of prime factors within the range $\Rightarrow \; a=2\times 3\times 5=30$
Percentage of positive integers not larger than $30$ that are relatively prime to $30$ is $\left ( \frac{1}{2} \right )\left ( \frac{2}{3} \right )\left ( \frac{4}{5} \right )\approx 27\%$

$b$ must be the largest multiple of $30$ within the range $\Rightarrow \; b=2^2\: 3^2\: 5=180$ . Percentage of positive integers not larger than $180$ that are relatively prime to $180$ is also $\left ( \frac{1}{2} \right )\left ( \frac{2}{3} \right )\left ( \frac{4}{5} \right )\approx 27\%$

Finally, $c$ must be the largest prime number within the range $\Rightarrow \; c=199$ . Percentage of positive integers not larger than $199$ that are relatively prime to $199$ is $\frac{198}{199}\approx 99.50\%$

Hence $a+b+c=30+180+199=409$