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 October 14th, 2012, 04:27 AM #1 Senior Member     Joined: Jan 2012 Posts: 739 Thanks: 7 Venn diagram In a market survey, 100 traders sell fruits. 40 sell apples, 46 oranges, 50 mangoes, 14 apples and oranges, 15 apples and mangoes and 10 sell the three fruits. Each of the 100 traders sells atleast one of the three fruits. (i) Represent the information in a venn diagram. (ii) Find the number that sells oranges and mangoes only. It difficult for me to find the number that sells oranges and mangoes only. I drew three circles to show how the traders sell the fruits: all the three fruits n(AuOuM) = 10 oranges n(O) = 46-(4+10+x) = 32 mangoes n(M). = 50-(5+10+x) = 35 apple n(A) = 40-(4+10+5) = 21 apples and oranges n(AnO) = 14-10 = 4 apples and mangoes (AnM) = 15-10 = 5 I don't know how to find the number of traders that sells oranges and mangoes only n (OnM). Please let n(OnM) be x. What step or direction of thinking should I take to find x ?
 October 14th, 2012, 07:34 AM #2 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Re: Venn diagram first of all where the ovals are overlapping 3 times you know its value is 10. 10 of the 15 people who sells apples and mangoes are selling the 3 fruits so there are 5 which are only selling apples and mangoes. 10 of the 14 people who sells apples and oranges are selling the 3 fruits so there are 4 which are only selling apples and mangoes. 10+4+5 of the 40 people who sells apples are selling something else then apples so there are 21 which are only selling apples. there are in total 40+50+46-100=36 fruits which are sold in pairs. 10 of them are sold by 3 so we have to substract 10 of that 36=26. 4+5+10 of them aren't selling oranges and mangoes so there are 26-19=7 who do. 50-10-7-5=28 are selling mangoes only and 46-7-10-4=25 are selling oranges only.
October 14th, 2012, 04:34 PM   #3
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
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Re: Venn diagram

Hello, Chikis!

Quote:
 In a market survey: 100 traders sell fruits. [color=beige]. . [/color]40 sell apples, [color=beige]. . [/color]46 sell oranges, [color=beige]. . [/color]50 sell mangoes, [color=beige]. . [/color]14 apples and oranges, [color=beige]. . [/color]15 apples and mangoes [color=beige]. . [/color]10 sell all three fruits. Each of the 100 traders sells at least one of the three fruits. (a) Represent the information in a Venn diagram. (b) Find the number that sells oranges and mangoes only.

It looks like you were off to a good start.
Use the scroll bar to see the entire Venn diagram.

Code:
            *---------*  *---------*
/   Apples  \/  Oranges  \
/            /\            \
*      21    *  *   32-x     *
|            | 4|            |
[40] |        *---+--+---*        | [46]
|       /    |10|    \       |
*      /     *  *     \      *
\    *   5   \/    x  *    /
\   |       /\       |   /
*--+--+---*  *------+--*
|                |
*      35-x      *
\              /
\   Mangos   /
*----------*
[50]
Ten sold all three fruits.
The central intersection has "10".

15 sold apples and mangos.
Hence, there are 5 in the "apples and mangos only" region.

14 sold apples and oranges.
Hence, there are 4 in the "apples and oranges only" region.

The total in the Apples circle is 40.
[color=beige]. . [/color]Hence, "apples only" = $21.$

Let $x$ = number in "oranges and mangos only".

The total in the Mangos circle is 50.
[color=beige]. . [/color]Hence, "mangos only" = $35-x$.

The total in the Oranges circle is 46.
[color=beige]. . [/color]Hence, "oranges only" = $32-x.$

If we add all the quantities in the diagram, we get $107\,-\,x$
But we are told that this total is $100.$

$\text{Therefore: }\:107\,-\,x \:=\:100 \;\;\;\Rightarrow\;\;\; \fbox{x \:=\:7}$

 October 15th, 2012, 07:04 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,472 Thanks: 2039 You don't need to be given the number of traders who sell all three fruits! Let N denote the "number of traders who sell" function, then 40 + 46 + 50 = N(any fruit) + N(A and M) + N(A and O) + N(M and O only) = 100 + 15 + 14 + N(M and O only). Hence N(M and O only) = 7.
 October 20th, 2012, 06:45 PM #5 Senior Member     Joined: Jan 2012 Posts: 739 Thanks: 7 Re: Venn diagram Thank you soroban! Thanks to all of you; I have actually seen that: (21+5+10+4)+(32- x)+x (35-x) = 100 40+35-x+32-x+x = 100 -x = 100-107 x = -7/-1 = 7 Then the number trader x that sells oranges and mangoes will be 7.
 October 20th, 2012, 06:59 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Venn diagram I have merged your duplicate topics together. As you can see duplication of effort could have been prevented had you posted the topic only once. This is one reason we ask that you pick the most appropriate forum, then post your topic once. Not only does it prevent unnecessary duplication of effort from our valued contributors, who could be helping someone else instead of laboring to give help when the problem may already have been solved elsewhere, it prevents the cluttering of our forums with redundancy.
 October 21st, 2012, 07:42 AM #7 Senior Member     Joined: Jan 2012 Posts: 739 Thanks: 7 Re: Venn diagram Am sorry for duplicating the post.

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in a market survey, 100 traders sell fruits, 40 sell apples, 46 oranges, 50 mangoes, 14 apples and oranges, 15 apples and mangoes, and 10 sell the three fruits. Each of the 100 traders sells at least one of the three fruits. (i) Represent the information

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