Abstract: The set of positive integers can be divided into different sub-sets, such as the sets of even numbers, of odd numbers, of primes and of composites. Unlike the patterns for even numbers and odd numbers, no repetitive pattern has been identified for readily locating all the primes within the number line and hence by definition to readily locate all the composites. Despite this, mathematicians have historically been able to look at boundaries for the estimated population of primes within the set of positive integers, most notably by applying the Prime Number Theorem. Initially this paper starts out by looking to identify an accurate formula for computing the number of composites and hence the number of primes within a group of eight arithmetic progressions (APs), knowing that the eight APs taken together contain all the prime numbers except 2, 3 and 5, and that the composites therein are all computed from the primes therein. However, under the methodology applied for computing the occurrence of composites in the eight APs, duplicates became an issue and consequently their occurrence also needs to be quantified. However though some results were obtained to work with, they were not sufficient to extrapolate properly for significantly larger integer populations. Instead, therefore, focus of the work changed abruptly, with the aim now being to directly identify three formulae for computing the estimated number of primes within a population of positive integers. Two of those formulae make use of the reduced population of positive integers in the eight APs whilst the third is derived from information obtained from a log10-log10 graph. All three formulae calculate an estimate of the number of primes for the population of positive integers, one of which produces a very good estimate for an upper-bound.