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August 28th, 2017, 05:36 PM   #1
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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?
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August 28th, 2017, 07:16 PM   #2
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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
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August 28th, 2017, 08:19 PM   #3
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Quote:
Originally Posted by Rexan View Post
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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