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Strategy 7. Another source of confusion (and a great trap for test writers) is a problem that asks students to calculate the overall % increase or % decrease of a price, rate or speed that has changed more than once. In the final calculation for such problems, the correct denominator is the ORIGINAL whole, not the intermediate one. And, of course, the wrong answers you would get if you used the incorrect denominator will likely be included as answer choices.

Example: A software company discounts its old version of web design software to 50% of its original price. Two months later, when the software has still not sold, the company lists it on eBay at a price that has been reduced by an additional 20%. By what overall percentage has the price been reduced?

a. 55%
b. 60%
c. 70%
d. 75%
e. 80%

The most common mistake for this question is to simply add the two % and assume that you have the answer (50% + 20% = 70%). Wrong!

Simply plugging in a few easy numbers will show us the error. Assume that the software originally cost $100. The first 50% discount reduces its price to $50. The second discount is 20% of $50, or $10, which reduces the price to $40. To calculate the overall percentage that the software has been reduced, we must use the original denominator of $100: $60 / $100 = 60%.

Strategy 8. In Strategy 7, we demonstrated the correct way to calculate multiple reductions in percentages. But what if the percentage moves up, then down (or vice versa)? Beware of problems that involve multiple changes in percentages, particularly when they change in opposite directions. The SAT writers love this type for question, which is ripe with potential traps.

Example: Susan purchased a new condo when she moved to Los Angeles. Two years later, after a devastating correction in the housing market, she sold it to her neighbor Nathan for 40% less than she originally paid for it. Nathan did a few quick fixes and re-sold the condo to Janice for 20% more than he paid Susan for it. The price that Janice paid for the condo was what percentage of the original price that Susan paid?

a. 28%
b. 40%
c. 65%
d. 72%
e. 80%

The “trap” that many students fall into is simply subtracting 40% and adding back 20%, which leaves an overall loss of 20%. Accordingly, they choose answer choice e.

This answer is wrong, for the same reasons we discussed in Strategy 7. For questions that deal with multiple changes in percentages, the denominators are different for each step. Why? The first change is a reduction of the original price; the second change is an increase of a smaller amount.

Here is the correct approach to the problem. Because the problem works with percentages, we will use 100 as the original price. (We are free to use any number, but since percentages are involved, 100 is the least confusing choice.)

If Susan paid $100 for her condo, then she sold it to Nathan for $60, which is 40% less. Nathan sold the condo to Janice for 20% more than what HE paid for it, which was $60. Nathan therefore sold the condo for $60 + (0.2)($60) = $72.

The question asked us to determine what percentage of Susan’s original price Janet paid for the condo. In this case, the correct answer is 72/100, or 72%, which is answer choice d.

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