Bunuel wrote:

Solution P is 20 percent lemonade and 80 percent carbonated water by volume; solution Q is 45 percent lemonade and 55 percent carbonated water by volume. If a mixture of P & Q contains 70 percent carbonated water, what percent of the volume of the mixture is P?

(A) 37.5%

(B) 52.5%

(C) 55%

(D) 60%

(E) 75%

Kudos for a correct solution.

PRINCETON REVIEW OFFICIAL SOLUTION:First of all, guesstimate! If you were to average the two Solutions, making it 50% P and 50% Q, the mixture would be 67.5% carbonated water. (To figure this out, simply average the percentages of carbonated water in the two Solutions: 80% + 55% = 135%, then divide by 2 and you’ve got 67.5% for the average.) This is pretty close to the answer because the question was asking us for a mixture that contains 70% carbonated water. Since we’re just a little bit low when we average the two solutions, we’re going to need a little bit more than 50% of Solution P in order to bring the percentage of carbonated water up (remember, Solution P has the higher percentage of carbonated water). Thus, at this point, we know the answer is more than 50%, but not much more. Eliminate answer (A). Furthermore, guesstimation should tell us that answer (E) is probably too big.

Next, rather than doing any algebra or other complicated math, use the answers! Our guesstimation has told us that we should check answers (B), (C), and (D) first. Which of the 3 choices would be easiest to check first? To determine this, think about which choice converts to the simplest fraction. It’s certainly not (B)—why mess with the .5? You may or may not know that (C) 55% is 11/20, but that’s certainly not as simple as (D) 60%, which converts to 3/5. Therefore, check (D) first.

If the mixture were 3/5 P, that means 3 parts of the mixture would be P and 2 parts would be Q (3 parts P out of 5 total parts). Write out the percentage of carbonated water for each part: For the 3 parts that are P, that’s 80%; for the 2 parts that are Q, that’s 55%. Thus, you should have:

80% + 80% + 80% + 55% + 55% = 350%

(By the way, it’s not really necessary to write out the % signs.)

All we have to do now is find the average percentage of carbonated water in each of the 5 parts: 350% divided by 5 = 70%. Hey, that’s the percentage we were looking for—bingo!

Now, don’t psyche yourself out by continuing to worry about the percentages—remember that we were checking answer (D), and answer (D) has checked out (given us the answer we were looking for). Officially, this means that to get a mixture that’s 70% carbonated water, you need 60% of it to be Solution P.

The correct answer is (D).Our sample question shows that you’re often wasting your time on PS if you’re doing complicated algebra or calculations. Always look to guesstimate and/or use the answers first! In the vast majority of cases, this is all you’ll need to do to get the correct answer, so these strategies are far more applicable and useful than brute-force solving with algebra/calculations (with which you’re more likely to make a mistake anyway, especially given the time pressure on the Math section). Guesstimating and using the answers will greatly increase your accuracy and your efficiency!

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