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I started working for a company with salary of X per annum. Now that I started in the last part of month.

Now when I got paid after 1 month + 5 working days, they calculated my 5 days extra pay based on following formula,

(AS / 12) x (Number of days I worked / Number of working days in month)

But if I use a online calculator and use following formula,

 Daily Rate x Number of days I worked 

The difference is that I am getting more if use online calculator calculations, about £60 MORE for 5 days, not a big amount but I only want to know if there calculations are right or not.

Their calculation is correct - as a salaried employee you are considered doing what the government classes as "Salaried hours":

Paid an annual salary
A worker is doing ‘salaried hours’ work if they’re paid:

a set basic number of hours each year under their contract
an annual salary in equal weekly or monthly amounts
Salaried hours workers’ contracts might not state the basic number of hours as an annual figure, but it must be possible to work this out. Workers and employers can then use this figure to make sure the rate of pay is at least the minimum wage.

And as such you don't have a fixed "day rate". You have an annual salary that is split into equal amounts over a set pay period frequency (monthly or weekly) - it sounds as if you're on monthly pay periods.

Given that each pay period doesn't have the same number of working days in it there is no set amount that a working day is paid at - the finest granularity is in terms of pay periods or part thereof.

Since you you've worked one partial pay period and one full pay period they need to pay you the proportional amount of what you worked in that partial period. In this case it's 5 days divided by the total number of working days in the full period.

So for example, if you annual salary was £12000, your salary per pay period would be 12000/12 = £1000. So if you started in September 2018 there were 20 working days (assuming a typical 5-day working week) of which you worked 5 so you should be paid 5/20 * 1000 = £250

If on the other hand you had started in January 2018 there were 23 working days, so you'd get paid 5/23 * 1000 = £217.39

If you were paid weekly then all pay periods would be the same length and therefore any pro-rata day rate would always work out to be the same (even though strictly speaking the same calculations would be run for any part-weeks you worked - it's just that the answer would always be the same)

Their formula split the annual salary in 12 equal parts, one per month. But not all months have the same amount of working days so the "daily rate" vary between months.

The advantage of their formula is that it will give the same monthly salary every month. If your non-prorated monthly salary will be the same every month, then their formula for the prorated 5 days is correct.