overlapping sets

by **guest612** Thu Feb 21, 2008 1:33 pm

A farmer has an apple orchard consisting of Fuji and Gala apple trees. Due to high winds this year 10% of his trees cross pollinated. The number of his trees that are pure Fuji plus the cross-pollinated ones totals 187, while 3/4 of all his trees are pure Fuji. How many of his trees are pure Gala?

Answer: 33.

Can this be solved using the grid (versus algebra)?

Thanks so much!

by **StaceyKoprince** Thu Feb 21, 2008 10:53 pm

Please cite the source of this problem; if you don't cite the author, we will have to delete it (and we obviously won't be able to answer it!)

by **Guest** Thu Mar 20, 2008 2:04 am

This problem was seen on the MGMAT Practice Test.

by **StaceyKoprince** Mon Mar 31, 2008 3:37 pm

Please remember to post the problem in its entirety, including answer choices - the format of the answer choices can completely change how we determine the best way to do the problem!

Sure! You just have to notice that you only have three scenarios, not four (Gala only, Fuji only, or hybrid) and adjust accordingly.

...........F...........notF........tot
G.......0.1x.........?...........0.1x+?
notG....3/4x........0..........3/4x
tot......187...........?..........x

(Remember, I don't have any notGala-notFuji's, so put a zero there.) I'm looking for the ? (pure Gala).

So, 0.1x + 0.75x = 187, or 0.85x = 187.
Also 0.1x + ? + 0.75x = x, or ? = 0.15x (this is what I want to find)

Answer choices are 22, 33, 55, 77, and 88. If 187 represents 85%, and I want 15%, I can get rid of some choices just based on estimation. 15% is less than 1/5 of 85%, so divide 187 by 5 (to do this quickly, multiply by 2 and move the decimal one place to the left) to get 37.4 The answer has to be less than that - eliminate C, D, and E. Is the answer a lot less or a little less? Well, 85/5 = 17, so 37.4 represents 17% of the apples. I want 15%, so it's only a little less - answer is 33.

Alternatively, 187 represents 85% and I want 15%. Figure out what to divide 85% by to get 5% (divide by 17). Divide 187 by the same amount - and, hey, that's nice. Notice that 187 fits our pattern for multiples of 11: the outer two digits, 1 and 7, add up to the middle digit, 8. So this is 17 * 11. Therefore, 187 divided by 17 = 11. 11 represents 5% and I want 15% so multiply by 3 to get 33.