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Two Math
Strategies: Multiple Changes in Percentages
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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