Numbers & Operations
Percents
If math terms formed cliques, then percents, decimals, and fractions would all hang out together. Percent is one more way to represent a part of a whole. The word itself means “of 100” in Latin, so a percent represents some part of the number 100, which is the whole.
Converting a decimal to a percent is simple: take the decimal and move the decimal point two spaces to the right. Then add the lovely % symbol. The number is now a percentage. For example, the decimal 0.62 becomes 62%, and the decimal 0.087 becomes 8.7%. If you have to convert a percent into a decimal, just go the other way and move the decimal point two spaces to the left. So 78% becomes 0.78. How easy is that?
Flipping between percents and fractions is all about using 100 as the denominator. Recall that percent means “of 100,” so 65% means 65 of 100, or . Taking a percent and placing it over 100 does the trick of converting it into a fraction, although often you still have to reduce:
To change a fraction into a percent, you have to turn the denominator to 100. The numerator then becomes the percent. Let’s use as an example. To get the denominator equal to 100, we have to multiply by 25:
That covers all the different ways to switch between fractions, percents, and decimals. It’s good stuff to know, because many SAT items like to test this conversion process in one way or another.
Percents in Word Problems
SAT word problems often use convoluted wording to ask for percents. Let’s inoculate ourselves against these wicked temptations. The typical percent word problem will say something like:
11. 4 is what percent of 10?
Here’s what each term means in Mathspeak:
Term Mathspeak
is =
what insert a variable
percent divide by 100
of multiply
Using this chart, “4 is what percent of 10?” transforms to:
4 is 40% of 10.
Percents Up and Down
Percents can be tricky. Consider this situation. There’s one person in the room. Another walks in. The percentage increase is 100%, because there are now 100% more people in the room. You can use the following formula to calculate percent change:
Now suppose a person leaves. What’s the percentage decrease? Many people jump and say 100%, but percents don’t work that way. The actual percentage decrease is 50%, because exactly half (one out of two) of the people who were in the room left. The number of people who left the room is the same as the number of people who entered, but the percent change is different because the original amounts are different. Using our formula:
If this were an SAT item, you can bet that 100% would have been a distractor.
Consider the following item:
8. The original factory price of a pencil is $5. The Tall Mouse art store buys the pencil at the original price, then sells it at a profit of 10%. A starving artist in Santa Monica buys the pencil at the art store, then decides to reduce the price at which he bought the pencil by 25% and sell it to his even more starving student. In dollars, approximately how much does the student pay for the pencil?
(A) $4.13
(B) $4.40
(C) $4.50
(D) $5.50
(E) $6.75
For an item like this, you have to calculate the result of two successive price changes. To do this, you need to remember that the final price depends on the previous price change.
So let’s calculate the first price increase. To do this, find 10% of $5:
Now add $0.50 to the original price of the pencil:
$5 + $0.50 = $5.50
Now find 25% of 5.50.
This is the amount the art teacher reduced the pencil’s price. When we subtract 1.375 from 5.5, we should have the final price:
That’s answer A.
Be careful while solving double percent items, because the SAT writers love to set traps. For instance, if you reason that because the first price increase is 10% and the second price decrease is 25%, the net decrease is 15%, you are falling into a classic SAT trap.
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