SOLUTION: Two kinds of fruits are displayed in a fruit stand. One is avocado fruit whose number is no more than thrice the other, which is melon. How many avocado and melons are there if the

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Question 1177099: Two kinds of fruits are displayed in a fruit stand. One is avocado fruit whose number is no more than thrice the other, which is melon. How many avocado and melons are there if there are atleast 20 total of fruits displayed.
Found 3 solutions by mananth, ikleyn, greenestamps:
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
Two kinds of fruits are displayed in a fruit stand. One is avocado fruit whose number is no more than thrice the other, which is melon. How many avocado and melons are there if there are atleast 20 total of fruits displayed
Let the number of melons be is x.
According to the problem, the number of avocados is no more than thrice the number of melons.
avocado ≤ 3 * melon
at least 20 total fruits displayed.
avocado + melon ≥ 20
melon ≥ avocado / 3
Substituting this into the second equation
avocado + avocado/3 ≥ 20
Multiply by 3
3avocado + avocado ≥ 60
4avocado ≥ 60
avocado ≥ 15
avocado has to be a natural number and it is <= thrice the number of melons, the only sensible value for avocado is 15.
We Know
avocado ≤ 3 * melon
15 ≤ 3 * melon
5 ≤ melon
There are at least 5 melons .
So the possiblities
- 15 avocados and 5 melons
- 12 avocados and 4 melons
- 9 avocados and 3 melons


Answer by ikleyn(52873)   (Show Source): You can put this solution on YOUR website!
.
Two kinds of fruits are displayed in a fruit stand.
One is avocado fruit whose number is no more than thrice the other,
which is melon. How many avocado and melons are there
if there are at least 20 total of fruits displayed.
~~~~~~~~~~~~~~~~~~~~


        @mananth incorrectly interprets the problems restrictions;
        THEREFORE,  his answer,  his solution and his teaching  ALL  are  WRONG.

        So,  I came to bring you a correct solution and to teach you in a right way. 


Let x be the number of melons, and
let y be the number of avocados.


From the condition, you have these inequalities

    y <= 3x        (1)   (the number of avocados is no more than thrice the melons)

    x + y >= 20    (2)   (there are at least 20 total fruits displayed)


Make a plot of the lines 

    y = 3x         (3)

    x + y = 20     (4)


These plots are shown below

    


       Plot  y = 3x (red),  y = 20-x (green)


The domain which is interesting to you is the part of the 1st quadrant, where x >= 0, y >= 0,
which lies under the red line, but above the green line.


The integer points of the quadratic grid represent possible numbers 
of fruits. For example, these pairs are possible

    (x,y) = (melons,avocado) = (5,15), (6,15), (7,15), . . . (horizontal line in the plot y= 15)

                             = (6,18), (7,18), (8,18), . . . (horizontal line in the plot y= 18)

                             = (8,12), (8,13), (8,14), . . . , (8,24) (vertical   line in the plot x = 8)

So, your problem is reduced to the system of inequalities,
and you can solve this system GRAPHICALLY to get better understanding of the solution set.

--------------

Again,  note the @mananth incorrectly interpret the problems restrictions;
THEREFORE,  his answer,  his solution and his teaching  ALL  are  WRONG.

So,  you better  IGNORE  his post for your safety.



Answer by greenestamps(13208)   (Show Source): You can put this solution on YOUR website!


The response from the other tutor starts out with the stated requirement that the total number of melons and avocados is at least 20; then she ends up giving 3 possibilities for the answer, in all of which the total number of fruits is AT MOST 20.

So somewhere in her solution she got mixed up....

However, the problem is absurd, and it doesn't matter whether the total number of fruits is less than 20 or greater than 20; it is easy to see that for either case the number of possibilities is large.

So the problem as stated is severely faulty.

For total number of fruits at least 20 (as the problem states)....

Suppose there are 20 melons. The number of avocados can be no more than three times the number of melons; that means the number of avocados can be any number from 0 to 60, because each of those numbers is "no more than thrice" the number of melons. And of course for any of those possible numbers of avocados the total number of fruits is at least 20.

And of course if the number of melons is greater than 20, then the number of avocados can be anything from 0 to 3 times the number of melons.

So it is obvious that the problem makes no sense (there are an infinite number of solutions) if the total number of fruits is "at least 20".

Now suppose the problem was supposed to say the total number of fruits is AT MOST 20.

Again it is possible that there are 20 melons, because that would mean the number of avocados would have to be 0; and 0 is less than 3 times 20.

And we could have, say, 10 melons; then, to keep the total number of fruits at most 20, the number of avocados could be anything from 0 to 10, and each of those numbers is no more than three times the number of melons.

In this case, because the total number of fruits has to be at most 20, the number of solutions is again large, but not infinite.

So this is another example of the author of the problem trying to be "different" in how he presents the problem, but ends up with a problem that is nonsense.
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