There are 350 students in a school who like Orange, Grapes, and Apple, and students can like more than one fruit as well. 6% of the total students like only Orange, 18% like only Apple, and 12% like only Grapes. $X\%$ of the total students like Orange and Grapes but not Apple, $Y\%$ like A

Question

There are 350 students in a school who like Orange, Grapes, and Apple, and students can like more than one fruit as well. 6% of the total students like only Orange, 18% like only Apple, and 12% like only Grapes. $X\%$ of the total students like Orange and Grapes but not Apple, $Y\%$ like Apple and Orange but not Grapes, and $Z\%$ like Apple and Grapes but not Orange. The values of $X$, $Y$, and $Z$ are multiples of 10, and no two values are the same. How many students like all three fruits?

Accepted Answer

14. 6% of 350 = 21, 18% of 350 = 63, and 12% of 350 = 42. Let the pairwise-only groups be 10%, 20%, and 30% in some order, i.e. 35, 70, and 105 students; their sum is 210. Adding the single-fruit counts gives 21 + 63 + 42 = 126, so the remaining students are 350 - 126 - 210 = 14, who like all three fruits.

Solution

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