Problem Solving

jzw wrote:

GMATGuruNY wrote:
Here's the allegation method.

When combining a lower percentage (L) with a higher percentage (H) to achieve a goal percentage (G):

The proportion needed of L = H-G (the positive difference between the higher percentage and the goal percentage)
The proportion needed of H = G-L (the positive difference between the lower percentage and the goal percentage)

Mitch - been feverishly studying this. Can you please clarify if the following more pedantic definition is correct? My issue I think was understanding "the higher percentage of what, that which is there already or that which is to be added?"

The proportion needed of L = H-G (the positive difference between the higher percentage of the solution being added and the goal percentage)
The proportion needed of H = G-L (the positive difference between the lower percentage of the existing solution[/b) and the goal percentage)

Might be easier to think this way:

The proportion needed of each ingredient in the mixture is equal to the distance between the OTHER two percentages.

In the problem at hand:
Percentage of water in the original solution = 20.
Percentage of water in the added water = 100.
Percentage of water in the final mixture = 25.
(Original solution) : (added water) = (100-25) : (25-20) = 75:5 = 15:1.

Also - you made a statement below "Since the amount of the original mixture (the lower percentage L) is not changing from 125..." How do we know this? Ie, if one is adding water to the mixture, won't the final volume increase from 125 gallons to - something more than this? Please advise. Thanks!

None of the original solution is being removed.
No more of the original solution is being added.
Thus, the amount of the original solution does not change from 125.
The TOTAL VOLUME changes -- since water is being added -- but the amount of the original solution WITHIN the final mixture remains 125.

So here is how I did it using the box method. I attached pictures bec typing math isn't pretty.

First step is to write out what we know. Always show yourself what your target goal is.

Next is to use the information we have to figure out what we don't have.

Finally, use the information that we get to calculate your target.

Now, you can see in the third picture that there are a couple of ways to get the final piece of the puzzle, you can subtract the original water amount from the final water amount, but the quicker way is that once you see the total mixture is now 133.3, and we know the original total was 125, and we know that only water was added, so we can subtract 125 from 133.3 and get 8.3.

It probably isn't as quick as the allegation method, but it works. They key to mastering the box method is learning how to adapt the box to accommodate the information given. What is nice about it is that you lay out all the information in a nice chart and it is easier to see what you need and then what you need to do with that new information. You need to be quick in forming the box, always making sure to put the correct info in the correct column. I use it for:

Price x Units = Total Cost
Rate x Time = Distance
Percentage x Mixture = Total Amount

What I like about the allegation method is that it's much quicker if you know how to use it. I'm not there yet, but I want to add it to my arsenal of weapons, as one of the things I learned in the Princeton Review Hard Math Class is that it is good to be able to adapt and use different methods if you see one will work better/quicker for a particular question. So, to that end..

Mitch - what I had written earlier:

The proportion needed of L = H-G (the positive difference between the higher percentage of the solution being added and the goal percentage)

The proportion needed of H = G-L (the positive difference between the lower percentage of the existing solution[/b) and the goal percentage)

Is that correct?