1/4 of the trees are apple trees, 1/6 are plum trees, and 150 are cherry trees, how many trees are in the orchard?

Question

If a certain orchard has only apple, plum and cherry trees and 1/4 of the trees are apple trees, 1/6 are plum trees, and 150 are cherry trees, how many trees are in the orchard?

Answer 1

Let x=number of trees in the orchard
 
Then,
   (1/4)x are apple trees
   (1/6)x are plum trees
   150 are cherry trees
These are the "only" trees.
How many trees are in the orchard?  The sum of these three groups.
 
      x = (1/4)x + (1/6)x + 150
    12x =  3x    +   2x     + 1800          (multiply everything by 12)
      7x = 1800                           (subtract 10x from both sides)
       x = 257.14    (but that's not a whole number of trees!)
 
Now, what you do next will reveal whether you are a true mathematician or not!
 
If you automatically round (either up or down), you have determined a whole number of trees but ignored the fact that the ratio (or percent) was already rounded in the problem and this is a very serious mistake.
 
Yes, (1/4)*(257) + (1/6)*(257) + 150 =  257.0833
 
But, why can't there be 63 apple trees + 42 plum trees + 150 cherry trees?  (=255 total)
      63/255 =  24.7%  ≅ 1/4
      42/255 = 16.47%  ≅ 16.67% = 1/6
 
The fractions 1/4 and 1/6 are very approximate estimates (already rounded, because who would say 63/257^ths are apple trees?) 
 
Do not round after you have computed the answer; determine the amount of rounding of the percent, ratio, or fraction given in the problem (there actually may be a range of acceptable answers).

 

p.s., So with 257 total trees, how many are apple and how many are plum???

Answer 2

T = total number of trees

1/4 T = Apple Trees

1/6 T = Plum Trees

150 = Cherry Trees

 

1/4 T + 1/6 T + 150 = T

 

Find a common denominator for the T

 

3/12 T + 2/12 T + 150 = T

 

5/12 T + 150 = T

150 = T - 5/12 T

150 = 7/12 T (multiply both sides by 12/7)

(150*12)/7 = T

257.1 = T

Since we can't have a decimal of a tree, there are 257 Trees in the orchard