  This paper provides bounds for the number of terms, denoted by $f$, of a
harmonic sum with the condition that it starts from any arbitrary unit fraction
$\frac{1}{m}$, $m > 1$, until another unit fraction $\frac{1}{m+f-1}$ such that
the sum is the highest sum less than a particular positive integer $q$. Also,
we consider the number of terms of Egyptian fractions whose terms are
consecutive multiples of $r$, $r \geq 1$, under the same above condition. We
end the paper with a formula for the case: $q=1$ and $r=1$.
