Skeet shooting

Algebra Level 5

Three persons Bob, bill, and Bop go skeet shooting. Bob hits every \(4^{th}\) bird (these are what people shoot at when skeet shooting), bill hits every \(5^{th}\) bird, and Bop hits every \(6^{th}\) bird. However, on every second pull, the skeet machine shoots out \(2\) birds instead of one. The ratio of birds hit to total birds fired after \(31\) pulls can be represented as \(\frac{a}{b}\), where both \(a\) and \(b\) are prime numbers. Find \(a+b\).

Notes & Assumptions:

If two people shoot at the same time and one bird is fired, then it counts as one bird being destroyed. If two people shoot when two birds are fired, then both are destroyed If one person shoots when two birds are fired, then only one is destroyed.

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