Linear Equation Headache

[senseimichael]

I normally have no issues with linear equations (compared to those crazy questions for Singapore Primary Math where I cannot use algebra), but this one stumped me.

A gardener needs 6 bushels of a potting medium of 40% peat moss and 60% vermiculite. He decides to add 100% vermiculite to his current potting medium that is 50% peat moss and 50% vermiculite. The gardener has 5 bushels of the 50% peat moss and 50% vermiculite mix. Does he have enough of the 50% peat moss and 50% vermiculite mix to make 6 bushels of the 40% peat moss and 60% vermiculite mix?

My solution with algebra:
Let x be amount of 50/50 mix. Let y be amount of 100 mix.
x + y = 6 [1]
y + 0.5x = 0.6 (6) [2]
Subst x = 6 - y into [2]
y + 3 - 0.5y = 3.6
0.5y = 0.6
y = 1.2
Hence because he can only have up to 1 bushel of vermiculite to be added, and he needs 1.2 bushels to get 60% vermiculite at the end, he does not have enough.

However, when I tried applying the numbers directly using proportion:
He has 2.5p + 2.5v = 5 bushels of potting mixture.
2.5p remains the same, but is now 40% of the mixture. In order for the potting mixture to be 60% V, it has to be 3.75v. Again at 3.75v + 2.5p = 6.25 mixture, it is not enough.
But what stumps me is the fact that with proportion, I need 1.25 of V to be added to get 60% vermiculite! It is different from the answer I got using algebra!

One or both of my mathematical assumptions must have been wrong!

 

[Romsek]

You assume he only has 1 bushel of vermiculite.

If he actually removes a bit of peat moss and adds enough vermiculite, slightly over a bushel, the gardener can come up with the 6 bushels of 40/60 as desired.

 

[JeffM]

First, romsek showed you that you were assuming a constraint that was not in the problem. I once worked for a few years with a very creative engineer who said that creativity was discovering which alleged constraints were false.

40% of 6 bushels is 2.4 bushels. So in our 40/60 mix, we need 2.4 bushels of peat moss. How much peat moss do we have in 5 bushels of 50/50 mix? 2.5 bushels obviously. So we can’t use all 5 bushels of the 50/50 mix or else we will have too much peat moss.

[MATH]\dfrac{2.4}{2.5} * 5 = 4.8 = 4 + \dfrac{4}{5}.[/MATH]
6 - 4.8 = 1.2. Using proportions the right way gives the same answer as algebra.

Let’s check the answer using common sense. Mixing 4.8 bushels of one mix and 1.2 bushels of a second mix certainly gives us 6 bushels of a third mix. The second mix has no peat moss. The first mix is 50% peat moss. 50% of 4.8 bushels of first mix obviously means 2.4 bushels of peat moss. And 2.4 bushels of peat moss in 6 bushels of mix means the mix contains 40% peat moss.

What this problem shows is that algebra reduces the need for complex thought. Many problems solvable by basic algebra could be solved by very careful reasoning and arithmetic, but it is much less mental strain to use algebra.

 

[senseimichael]

Thank you for pointing our my errors!