Write $x+y=a / 60+b / 70=(7 a+6 b) / 420$. Since $a$ is relatively prime to 60 and $b$ is relatively prime to 70 , it follows that none of the primes $2,3,7$ can divide $7 a+6 b$, so we won't be able to cancel any of these factors in the denominator. Thus, after reducing to lowest terms, the denominator will still be at least $2^{2} \cdot 3 \cdot 7=84$ (the product of the powers of 2,3 , and 7 dividing 420 ). On the other hand, 84 is achievable, by taking (e.g.) $1 / 60+3 / 70=25 / 420=5 / 84$. So 84 is the answer.