# How can I work out how many extra are required to reach a certain percentage?

I'm devising a system to work out if a student will get a particular grade based on their first year result.

If they get 90% of marks in their second year and 80% of marks overall, they get an A* (a British A+).

As an example, let's say they've gotten 80/100 marks in year one and are about to embark on year two, in which 120 marks are available.

How can I calculate how many marks (out of 120) they'd need to get in year two to get an A*, preferably expressed as a single equation?

Thanks.

P.S. Sorry if I've made a mistake with the tag. I'm new here and there wasn't a 'percent' tag, but a few other percent qs use the stats tag. Thanks.

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If I understand your question correctly, every student will have one grade $x$ out of 100 for year 1 (which you know) and will get one grade $y$ out of 120 at the end of year 2. They get an A* if y is above 108 (90%) and the average grade is also above 80%.

The average grade must be larger than 80% so : $$\frac{\frac{x}{100}+\frac{y}{120}}{2}\geq 0.80$$

which can be rearranged to $$y \geq 192 -1.2x.$$

So to answer your question a student will get an A* if their grade is larger than 108 and also greater than $192-1.2x$ where $x$ is their grade out of 100 from year 1.

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Thanks. I think I understand the first equation you give. To be clear, the total marks available in years can change, but I guess I can just replace these with more letters? However, I'm clear how I would adjust the second equation accordingly. I guess that '192' number changes based on the total marks available, and that 192 isn't a constant? – samiles Mar 8 '12 at 17:42
If $x$ and $y$ are both grades in percentages, then the average grade will be above 80% if $(x+y)/2 \geq 80$. So a student would get an A* if y was both greater than 90, and greater than 160-x. – user16124 Mar 8 '12 at 17:51
Thank you. I am still not sure how to put the three numbers in and get a number out but I will keep researching. Thank you for your help. :) – samiles Mar 8 '12 at 19:14

You only need $90\%$ of $120$ since the conditions are

1) To get $90\%$ second year.

2) To get $80\%$ overall.

Once the person obtains the $90\%$ required for the second year, then automatically the second condition will be satisified because he already had $80\%$ that first year, and given that he did better in his second his percentage will only increase.

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Sorry, I don't think I was clear enough. The first year grade can change, I was just using those numbers as an example. – samiles Mar 8 '12 at 17:37