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The question is "90% of the students in school passed in English, 85% passed in Mathematics and 150 students passed in both the subjects. If no student failed in both the subjects, find the total number of students." options given are 120/220/200/300

I solved it as, x=.9x+.85x-150, thus x=200.

but I am not sure why I did so. The line about failed students is troubling me. Is my 'x' not the total number of passed students? Because there could be students who failed in either of the subject. In which factor are they being included?

and the given %ages, what do they tell us. That 10% fail math and 15% fail English. So total fail 25%? Perhaps that is true, because 25% of 200 is 50 and that when added in 150 gives 200. But I am not sure.

I have 2 objectives. A) to solve this question without letting 'x' B) to see in and out of this question.

need your guidance.

Edit after seeing comments :

we are getting 75% from 2 ways.

1) 90+85=175

2) 10+15=25

so, I wonder would these two cases always give us the same result. Or the line about "nobody failed both" has something to do with it?

aarbee
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    without drawwing venn diagram you are complicating your work and inviting error, if you're a beginner don't drop them! – RE60K Sep 07 '14 at 14:09
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    The "nobody failed both" does simplify things. Since there is no overlap between fails English and fails math, $25%$ failed something, so $75%$ passed both. – André Nicolas Sep 07 '14 at 14:29
  • @Andre Nicolas- I am really liking your approach. It would be nice if you can dwell more on it in Answer. – aarbee Sep 07 '14 at 16:37

1 Answers1

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You can use the inclusion-exclusion principle. For the two sets $A$ and $B,$ $$ |A \cup B| = |A| + |B| - |A \cap B| $$ where $|S|$ denotes the cardinality of $S$ (number of elements in $S$).

In your case, let $A$ be the set of students who passed math, $B$ the set who passed English. You know that $A \cup B$ is the set of all students in the school (since all passed at least one of the two subjects). You also know the ratios $|A|/|A \cup B|$ and $|B|/|A \cup B|,$ and you know the exact value of $|A \cap B|.$ Set $x = |A \cup B|,$ and then all these facts enable you to write the equation you wrote.

In other words, your definition of $x$ was a good one and the equation is correct.

The Venn Diagram method is more complicated to apply to this problem, I think. I barely even recall it being used to solve any problems in my math education; looking around the Web I found a large number of examples of its use, but found none that involve percentages. For two sets $A$ and $B,$ the Venn Diagram method would have you find (or copy from the problem statement) the values of $|A \cap B|,$ $|A \setminus B|,$ $|B \setminus A|,$ and $\overline{|A \cup B|}.$ If you follow through on this carefully, you can eventually come up with the equation you wrote.

David K
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