4.13 SAT Math - Percents

To understand percents, it's important to understand what they are.  When you're dealing with simple fractions, say, you divide a pizza into 8 slices and you take 3 of them, to set up the fraction, you're going to put the number you're interested in for this problem (3) and put it over the number of pieces the thing is divided into (8) to get the fraction 3/8.  Percents work the same way, except that instead of dividing some things into 8 pieces and some into 2 and some into 47, we just divide everything into 100 pieces so that we're dealing with comparable numbers. 
Remembering that fractions are really a number over 100 is very helpful on this test.

-SAT Math Strategy:  Make Percents Fractions.  Your teachers in school generally have you write percents as decimals.  Percents are also decimals because 40/100 is obviously the same as .40.  But writing percents as decimals makes them more difficult on this test.  They'll try to trick you with questions like "What is .03 percent of 3.03 percent of 33?"  If you write all your percents as fractions, you won't need to worry about where to move the decimal when you're done, you'll just be done!

Many of the percent problems on this test are secretly Translation problems in disguise, so in order to talk about percents, we first need to talk about Translation/Word Problems.

Percents in translation work exactly the same.  Go word for word.  However, with percents, anytime you see the word "percent," you're going to put the number in front of that word on top of 100.  This makes your percent a fraction and, again, means you don't ever have to worry about moving that decimal!

Example

Take a look at these:

• 30 percent of 4 is what number?
• What percent of 16 is 22?
• 5/6  is what percent of 204?

Go word for word.

"30 percent:" 30/100 "of:" • "4:" 4 "is:"= "what number:" x  So:

What percent of 16 is 22?  Since "What" means a variable and "percent" means put the number in front over 100, we have x/100.  "of:"

• "16:" 16 "is:" = "22:" 22

And solve for x.

5/6 is what percent of 204? "5/6:" 5/6  "is:" = "what percent:" x/100 "of:"• "204:" 204.

Example Problem 2

Try this one SAT math question:

60 percent of 30 percent of 7 is 14 percent of what number?

To set this up properly, you not only need to go word for word, you need to understand a basic principle of percents:
-SAT Math Hint: Percents do not exist!  Not really.  At least, they can't exist on their own.  A percent can never be all by his lonesome.  A percent must always be 'of' a number, that is, multiplied by a number.  Never ever ever add or subtract percents together – you may only add or subtract real numbers together after you have multiplied the percent through.

So with that hint in mind, how should you set up the above problem?

Going word for word we get:   But is that correct?  Not entirely, because we have a percent (60/100) all alone.  This means we need parentheses, so that the 60 percent will be taken from a real number when we get to it.  So our equation should look like this:

This keeps the percents as always being taken from real numbers, and not just hanging out there as if they were real.

Example Problem 3

Let's see a problem like that in a word problem.

14. A car is normally priced at $12000.  In anticipation of their Labor Day Sale, the dealership raises the price of the car by 15%, and then puts the car on sale from that price at 15% off.  What is the sale price of the car?

(A) $9930
(B) $11730
(C) $11985
(D) $12000
(E) $13800

If you do not remember that percents aren't real numbers, you might make two mistakes: violating the Ten-Second Rule and falling for an Overly Obvious Answer.  You CANNOT simply add 15% and subtract 15% and end up with $12000 as your final answer.  Always ask yourself, what is my percent of?
So what's the first percent of?  If he's raising the price of the car by 15 percent, that means he's adding fifteen percent to the price – but 15 percent of what?  The only number we have so far is the original price, and does that make sense?  Yes.  He's taking 15% of the original price and adding it to the original price.  So the math that needs to be done is .  The result is 13800.  From there, we go on to the next phrase.  He puts the car on sale from that price.  So what is your percent of in this phrase?  $12000 or $13800?  $13800.  That's the new price, so that's the price we're taking a percentage of.  So your math now is .  So the answer is (B) $11730.

Percent Increase/Decrease

For Percent Increase/Decrease Problems, you only need to memorize one formula:

What this means is the change in value divided by the original value times 100 will give you the percent change, for positive or negative, that you are being asked for.
-SAT Math Tip: The numerator in this formula is the change in value, not the new value.  So to find that number, you may need to subtract the new value from the old or vice versa.

To solve, simply plug what you know into the formula, and solve for what's missing.  Try these:

A clothing store is going out of business.  Last year they made approximately $5600 a month in sales, but this year they are only making $3000 a month in sales.  By approximately what percent has business dropped?

Angel makes $410 a week at her waitressing job.  If she gets a 7 percent raise, how much more will she be making per week?

Looking at the first problem, what's the change?  You'll need to subtract to find it: 5600-3000.  Does it matter that it's a percent decrease?  No, because the question asked "by what percent has the business dropped?"  You don't need to make anything negative because that's implied in the question.  From there, simply put that change over the original number (5600) and multiply by 100 to get your percent. 

For the second problem, we're given the percent change, but what we're missing is the numerical change.  So we fill in what we know to get   Solve for the numerator to find the numerical change:   She'll be making an extra $28.70 a week.

Starting SAT Score of 600 or Above:

Percent More Than/Less Than:

More difficult percent problems will ask you not just for how much more someone is making, but what percent more than before that was.  These questions are more difficult because there's an extra step implied in the question that most students miss.

Take a look at this one:

19. Miguel spends $702 on bills, $204 on utilities, and $1200 on rent every month.   The total amount he spends on these necessities is 35 percent more than he spent 8 years ago.  How much did he spend on necessities 8 years ago?

This problem is difficult because you can't simply add up what he spends and subtract 35 percent of that.   The amount the percent is being taken of is an amount we don't know – it's the amount in the past.  In addition, the amount he made 8 years ago is included in the amount he makes now, so you have to make sure any equation you set up includes that. 

There are two ways to solve these kinds of problems.  Both work equally well, so use whichever one you prefer.

The first way to solve these is to memorize the formulas:  x(1 + %) or x( 1 - %) = new amount for more than/less than respectively, where x represents the original amount.  So in the question above, what is the original amount?  We don't know.  But we can find out the original amount.  And is this more than or less than?  It's more.  So we'll use the + equation.  Plug in what we know:   x(1 + 35/100) = 702 + 204 + 1200 Why 35/100?  Because the question said 35 percent, which means we still need to follow the translation chart and put the preceding number over 100.  Clean it up to get 1(35/100)x = 2106.  And then solve for x: x = 1560..

The other way to do these questions is to remember that percents are really ratios.  So you can set it up as a proportion.  To set a percent up as a ratio, keep in mind that 100 percent basically means the whole thing, or the total amount, right?  Except in more than/less than problems, 'the whole thing' was the number from 8 years ago, so the number know is the whole thing + 35%.  So the ratio we would set up would be .  Then set up that proportion with the real numbers on the other side: 100/135 = y/2106  and cross-multiply to solve: .