How to calculate the lowest possible integer value of 100% from 5 percentage values?

[nattsam]

Hi guys!

I have a real life maths problem that's beyond me! I need to calculate the number of students who took an exam based on the percentage of students who got each grade.

I have 5 percentage values to 1 decimal place each. I know that each percentage value represents an integer (a number of students).

I'd like to know the lowest possible integer value for 100%.

For example if I have these percentages:

13.2%
19.4%
43.3%
18.9%
5.3%

and I know each percentage represents an integer (a number of students).

How can I find the lowest integer value for 100% that could give those percentages?



My thinking is yes, probably!

Because if I have 4 values of 25% and all of them are integers, then I know that the lowest possible integer value of 100% is 4.

But I don't know how to extend this to less convenient numbers with decimal places!



A massive thank you and lots of good karma to anyone who can help me out with this!

 

[Dr.Peterson]

I need to calculate the number of students who took an exam based on the percentage of students who got each grade.

I have 5 percentage values to 1 decimal place each. I know that each percentage value represents an integer (a number of students).

I'd like to know the lowest possible integer value for 100%.

For example if I have these percentages:

13.2%
19.4%
43.3%
18.9%
5.3%

and I know each percentage represents an integer (a number of students).

How can I find the lowest integer value for 100% that could give those percentages?



My thinking is yes, probably!

Because if I have 4 values of 25% and all of them are integers, then I know that the lowest possible integer value of 100% is 4.

But I don't know how to extend this to less convenient numbers with decimal places!

If the percentages were not rounded, this could definitely be done. You could convert them to fractions and find the LCD.

But being rounded, it would be much harder, and less certain.

I'm playing with your data in a spreadsheet to see if I can solve it. With some manual work, my guess is 5005. But a proper answer would require taking the amount of rounding into account explicitly, which I haven't done.

 

[Dr.Peterson]

Looking back at my results, I realize that if the total were 5000, your percentages would be exact -- except that they add up to 100.1%, not 100%, so they can't be exact; and the numbers I get add up to the 5005 I stated before, not 5000!

 

[nattsam]

Well, I'm not sure what the OP is asking anyway

Thanks for your help guys! To clarify as best I can:

I have this table of data:
http://gradestatistics.cambridgeenglish.org/2016/fce.html


It tells me the percentage of students who got each grade in an exam, grouped by country. I would like to know the lowest possible total number of students in each country who took the exam.

Looking back at my results, I realize that if the total were 5000, your percentages would be exact -- except that they add up to 100.1%, not 100%, so they can't be exact; and the numbers I get add up to the 5005 I stated before, not 5000!

I don't know why the numbers add up to 100.1% - that must be an error in the original set of data. Looking at more of the data there are other examples where it adds up to 0.1 more than or less than 100 as well.


Can you guys please share the methodology you used in your calculations? I'd like to understand how to do this.