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 August 28th, 2017, 05:36 PM #1 Member   Joined: May 2015 From: U.S.A. Posts: 44 Thanks: 0 Changing a proportion by adding more The question is: "A manufacturer of soft drinks advertises their OJ as 'naturally flavored' although it only contains 5% OJ. A new regulation calls for 10% juice. How much pure OJ must be added to 900 gal. of OJ to meet the regulation?" I tried a couple ideas: $\displaystyle 900 + x = .1$ because you need to add an unknown amount to get a 10% juice. Definitely not it, though. Then I tried $\displaystyle 900 + x = 900(1.1)$ because you need to add an unknown amount to get a 10% juice, and I tried the 900(1.1) because the final amount would have to be greater than 900, due to adding the juice to an already established amount. But the last one gives $\displaystyle x = 90$, and that's 10% of the initial amount, which isn't what the problem is looking for. It seems to be a problem of perpetually increasing dilution that I can't figure out how to account for: if you add 90 gallons of pure juice, you now have 990 gallons of OJ that needs to be at 10% juice, in which case you need 99 gallons of pure juice, but if you add 9 more gallons, you're at 999 gallons.... and so on. There seems to be something incredibly basic I'm missing. Any insights?
 August 28th, 2017, 07:16 PM #2 Senior Member     Joined: Sep 2015 From: Southern California, USA Posts: 1,413 Thanks: 717 we add $j$ gal of 100% juice and end up with $900+j$ gallons of total liquid that is 10% juice. Let's look at the amount of pure juice on each side. $900(0.05) + j = (900+j)(0.1)$ $45 + j = 90+ 0.1 j$ $0.9j = 45$ $j = 50$ so we must add 50 gallons of pure juice to obtain 950 gallons of 10% juice
August 28th, 2017, 08:19 PM   #3
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Quote:
 Originally Posted by Rexan I tried . . . $\displaystyle 900 + x = .1$
That equation should be $0.1(900 + x) = 0.05(900) + x$, which implies $900 + x = 450 + 10x$.
Hence $x = (900 - 450)/(10 - 1) = 50$.

Alternatively, use the method shown below.

There are currently 45 gallons of pure OJ with a remaining 855 gallons.
As 10/(100 - 10) = 1/9, the amount of pure OJ needs to be 1/9 of 855 gallons, i.e. 95 gallons,
so 50 gallons of pure OJ must be added.

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