Real world math problem, not homework

[hitchpost83]

This seems fairly straightforward and I take pride in being decent at GENERAL math. If I could figure out how to set up the equation properly, I'd likely be able to figure it out. As stated this is real world, not homework. I buy a sugar water mixture that is 66% desolved sugar. If I want to get it down to 50%, how much water do I need to add? I want to be able to fill a 275 gallon tote with 50% sugar water mixture when I'm done, so how much 66% mixture needs to be in the tote to top it off with water to get to 50% mixture?) Is the proper equation:

.66x=.5
x=.7575
275*.7575=208.33

?
Meaning 208.33 gallons of 66% sugar water, plus 66.66 gallons of water, would result in 275 gallons of 50% sugar water mixture?

Thank you math wizards in advance.

 

[Dr.Peterson]

This is a standard algebra word problem, though it doesn't really require all the machinery of algebra. Here's one way to think about it:

You want to end up with 275 gallons of 50% sugar, which means it has to contain 0.50*275 = 137.5 gallons of sugar. Since 66% of your bought mixture is sugar, x gallons of it will contain 0.66x gallons of sugar; if this is to equal 137.5 gallons, you need 137.5/0.66 = 208.33 gallons of the mixture.

So put that much mixture into the tote, and fill it up with water (that is, add 275 - 208.33 = 66.67 gallons of water).

And that's just what you got. I couldn't follow your thinking because you didn't say what x meant, but you must have been thinking right!

 

[hitchpost83]

Thank you Dr. Peterson for responding. Your time and insight is greatly appreciated.

In my equation, x would be equal to the percentage of 66% solution needed to mix with straight water to end at 50% solution.

With the result of .7576, that will allow me to know that I need 75.76% of the solution to be 66% mixture. Meaning no matter the size of the end container I would know to multiply the end size by .7576 . For example. If I only needed 200 gallons for whatever reason, I could multiply .7576 by 200 and get 151.51 gallons of 66% mixture, top it off to 200 gallons, to end up with 200 gallons of 50% mixture. So doing the equation in that manner gives me an easier equation in the future, for future figuring.

Is my thinking correct?

 

[Dr.Peterson]

Yes, your thinking is exactly right.

There's also a technique used by people who do this a lot: When you mix two solutions in some ratio A:B, the new strength will differ from each of the parts in the same ratio; so if we want to mix 66% and 0% to make 50%, the differences in strengths from the new strength, 66%-50% and 50%-0%, are in the ratio 16:50, so the solutions have to be in that ratio. The water will be 16/66 of the whole, and the sugar solution 50/66 of the whole. Those fractions are 0.2424 and 0.7576, respectively -- the numbers you got.

On the side, I imagine that your 66% may really be 66.66%, or 2/3; if we use that, then the ratio by the same method is 2/3-1/2=1/6 to 1/2, which is a 1:3 ratio. The fractions are 1/4 and 3/4, or 25% and 75%.