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Math Help - Counting

  1. #1
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    Counting

    Is there an easier way to do questions like these? (I stuggle with them sooo much and all the overlap/non-overlap).

    In a class of 27 students, 9 students take Japanese, 12 students take government, 9 neither take Japanese nor government. How many students take both Japanese and government?

    (And the answer is 3).
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  2. #2
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    Hello mjoshua
    Quote Originally Posted by mjoshua View Post
    Is there an easier way to do questions like these? (I stuggle with them sooo much and all the overlap/non-overlap).

    In a class of 27 students, 9 students take Japanese, 12 students take government, 9 neither take Japanese nor government. How many students take both Japanese and government?

    (And the answer is 3).
    Easier than what? Guesswork?

    The standard way is to draw a Venn diagram like the one that I've attached.


    You'll see that I have drawn two overlapping loops representing the students who take Japanese (J) and government (G). Since we don't know how many take both, write x in the overlap. Then work outwards:
    There are 9 altogether in J; we've placed x of them in the overlap, so there must be 9 - x left in the other part of the loop.

    In the same way, there are 12 - x in the right-hand part of loop G.

    There are 9 who aren't in either loop. These go outside the loops, then.
    Then add up the numbers in each of the four regions, putting the total equal to 27:
    x + (9-x) + (12-x) + 9 = 27

    \Rightarrow 30 - x = 27

    \Rightarrow x = 3
    OK now?

    Grandad
    Attached Thumbnails Attached Thumbnails Counting-untitled.jpg  
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  3. #3
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    This is a classic Venn diagram question. In a class of 27 students, 9 of them take neither Japanese nor Government, which leaves 18 students that take at least one of Japanese and Government. If we add the number of students that take Japanese and the number of students that take Government together, we get 9 + 12 = 21. So the number of students that take both Japanese and Government is 21 - 18 = 3, because the 3 students that take both Japanese and Government were counted twice when adding the number of students that take Japanese and the number of students that take Government together, whereas the figure of 18 counts each student once.
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