[Math] What’s the Total Number of Candidates who Applied for the Exam

percentagesproblem solvingword problem

In an examination, 5% of the applicants were found ineligible and 85% of the eligible candidates belonged to the general category. If 4275 eligible candidates belonged to other categories, then how many candidates applied for the examination?

The choices are as follow:

30,000

35,000

37,000

None of these

Kindly help me solve this problem. I'm a first year college student taking up an algebra class and I'm really having trouble with translating word problems into equation. I tried to solve this but my answer doesn't match the answer keys in this problem.

4275 + 85/100x + 5/100x = x [total # of candidates]

x = 4275.9 <–my answer is none of these but the answer key says otherwise, that the answer is 30,000. It shows the solution:

x = number of aplicants

95% of x = number of eligible applicants

eligible candidates of each categories = 15% of (95% of x) –> (57/400)x

therefore, (57/400)x = 4275
x = 30,000

I can't understand why the answer key looks for the 15% of the eligible candidates in both categories when it should only look for 10% (which represents the candidates in "other categories") of the 95% of x (since the 85% belongs to the general category).

Best Answer

Your looking at the question innaccurately. Take it slowly and break it down.

In an examination, 5% of the applicants were found ineligible and 85% of the eligible candidates belonged to the general category.

First take the first part

5% of the applicants were found ineligible

What does this tell you? It tells you that 5% were ineligible and 95% were eligible. Let x be the number of total applicants, and e be the number of eligible candidates. We would get the following formula

x = (5/95) e + e

Now we look at the next line

85% of the eligible candidates belonged to the general category.

This is saying 85% of e (not x) belong to the general category. So letting n be the number of people not in the general category we would get

e = n + (85/15)n
e = n * 20/3

We are then given n

 If 4275 eligible candidates belonged to other categories

So we have

n = 4275

Thus we have

e = 4275 * 20/3
e = 28500

And plugging this into our previous equation we get

x = 28500 + (5/95) * 28500
x = 30000

Hope that helps a little more.

Edit Where does 85/15 come from?

Let e = eligible, n = non-generally eligible, and g = generally eligible. So we have

e = n + g

What we are trying to do is convert g to n since we don't know what g is. We do know that:

n = 0.15e
g = 0.85e

Thus we get that

n/0.15 = e
g = (0.85) * (n/0.15)
g = (0.85/0.15) n
g = (85/15) n

And that's where we get the 85/15. Basically think of it as "how much bigger is g than n?". If n is 15% of e and g is 85% of e then g is (85/15) times bigger than n so we get (85/15)n = g.

2nd edit

Why can't we convert (5/95) to (5/100)x?

Because of the following:

x = (5/95) e + e
x = (5/95) e + (95/95) e
x = (100/95) e
e = (95/100) x

You can put it all into one nice equation, but you can't just flop things around without going line by line to see what everything does.

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