[fs-toc-h2]Intro to percents
A percent is a relative way to express a part of something. If I have a pie cut into 6 equal slices, and I decide to eat 3 slices, I can say that I ate 3/6 of the pie, which can also be simplified as ½, or converted into a decimal (0.5).
To convert to a percent, simply multiple the decimal amount by 100. 3/6 = 0.5 x 100 = 50%
So in this example, I ate 50% of the pie.
The main concept to understand with percentages (which also shows up in geometry & probability) is part over whole.
If you’re given a whole and a part, you can find a percentage.
If you’re given a whole and a percentage, you can find the part.
If you're given a part and a percentage, you can find the whole.
In ‘math’ form:
\[ \frac{\text{part}}{\text{whole}} \cdot 100 = \text{percent} \]
Let's look at a few examples to really understand what I'm talking about.
[fs-toc-h2]Finding a percentage
William, a pineapple salesman, had 30 pineapples to sell. By the end of the day he had sold 20. What percentage does he have left?
To find a percentage, you have to identify your ‘whole’ and ‘part’. We know that William has a total of 30 pineapples to sell, so that will be our whole. Since a percentage is part over whole, 30 will be our denominator.
Finding the part may be where some students mess up, and it's where you really need to read the question they're asking really carefully. You may see that he sold 20 pineapples, and he had 30, so you put 20/ 30 and think you’re done. But the question isn't asking “What percentage of pineapples did he sell”, it's asking how much he has left.
Since our trusty salesman had sold 20, we can assume that he has 10 pineapples left. So our part over whole is:
\[ \frac{10}{30} = 13 = 0.33 \cdot 100 = 33% \]
William has ~33% of his pineapples left.
[fs-toc-h2]Finding a part
Joe, an amazing apple salesman, has 50 apples to sell. At the end of the day he had sold 40% of his apple stock. How many apples has Joe sold?
Let's look back at our equation for percent.
\[ \frac{\text{part}}{\text{whole}} \cdot 100 = \text{percent} \]
If we plug in what we know. Since we know that he sold 40% of his apples, and were trying to find how many apples he sold, 40% is our percentage. The whole is the amount of apples he had to sell, so it's 50. the equation becomes
\[ \frac{\text{part}}{50} \cdot 100 = 40% \]
Since a percentage is just a decimal multiplied by 100, we can divide both sides by 100 to get rid of it on the left side, and convert the right side to a decimal. Doing this we get:
\[ \frac{\text{part}}{50} \cdot 100 = 0.40 \]
Now our equation has become a simple algebra equation, all you have to do is multiply both sides by 50 to cancel out the denominator, and you’ll get your part.
\[ \cancel{50} \cdot \frac{\text{part}}{cancel{50}} \cdot 100 = 0.40 \cdot 50 \]
\[ \text{part} = 0.40 \cdot 50 \]
\[ \text{part} = 20 \text{apples} \]
Joe sold 20 apples.
What you just saw was the full and thorough way to find the part from a whole and percentage. But on the actual test, you can do this in one step in your calculator.
To quickly convert from a percent to a decimal, take the decimal and move it 2 to the left.
\[ 30% = 0.30\]
\[12% = 0.12\]
\[3.1% = 0.031\]
When you have a decimal, multiply it by your whole to find your part. In other words:
\[ text{part} = text{whole} \cdot \text{decimal} \]
[fs-toc-h2]Finding the whole
Dylan, the great orange salesman, had just sold 60% of his oranges. This comes out to be 72 oranges. How many oranges did Dylan have to sell in the beginning of the day?
From now on, it's a good idea to instantly convert your percentages to decimals. Keeping a percentage in actual percentage form is rarely useful. So the very first step ill take is dividing 60% by 100 to get a decimal form. 60/100 = 0.60
Since we’ve converted into a decimal we can take out the “ x 100” from the original percentage equation. So it becomes:
\[ \frac{\text{part}}{\text{whole}} = \text{decimal} \]
We’re looking for how many oranges he had to sell, and we have the decimal percentage and the part, so lets plug them in:
\[ \frac{72}{\text{whole}} = 0.6 \]
Again, now we have a single variable (whole) and 2 numbers. So all we have to do is solve for our variable using order of operations.
\[ \text{whole} = 120 \text{oranges} \]
Dylan had 120 oranges to sell.
If we solve for the variables first, and feel free to memorize this, we get
\[ \frac{\text{part}}{\text{decimal}} = \text{whole} \]
[fs-toc-h2]Solving summary
If you’ve converted your percent to a decimal, then:
\[ \frac{\text{part}}{\text{whole}} = \text{decimal} \]
\[ \frac{\text{part}}{\text{decimal}} = \text{whole} \]
\[\text{whole} \times \text{decimal} = \text{part}\]
Need a bit more help with the basics? Click here!
[fs-toc-h2]Percent decrease/increase
Once you’ve gotten down the basics of percentages, its time to learn some of the SAT’s more “advanced topics” (they really aren't that hard once you learn a couple of things).
[fs-toc-h3]Percent decrease
On the SAT, you’re most likely to be asked this type of question in the form of some sort of discount on an object. Let’s walk through a quick example to explain percent decrease.
John is going shopping for a jacket. He finds out that all items on the clearance section are discounted by 40%. If the original price of the jacket he likes is $125, how much is the discounted price? (Tax and fees not taken into account)
If you remember from our last section, the first step here is to convert our percent into a decimal. Since the given percentage is 40%, we divide by 100 to get 0.4.
Now, a mistake many people make is multiplying $125 by 0.4 to get their final answer. Doing this, though, gets you 40% of the jacket's value, not 40% off the jacket's value.
When decreasing a value by a certain percentage, you have to multiply that value by 1 minus the percent, not the actual given percentage.
In this case, the jacket decreased by 40% (0.4), meaning that you multiply our original value (125) by (1 - 0.4) or (0.6). This is true since, when 40% of the jacket is discounted, you're actually paying for 60% of it.
So, were left with:
\[ x = 125 \cdot 0.6 \]
\[ x = $75 \]
John would end up paying 75 dollars.
[fs-toc-h3]Percent increase
Percent increase is incredibly similar to percent decrease, instead of 1 minus your percent, its plus,
When increasing a value by a certain percent, you have to multiply that value by 1 plus the percent.
Zinkkloid, the village's humble apple farmer, yielded 220 pounds of fruit last harvest. During this harvest, he is projected to harvest 30% more fruit than last time. How much fruit is this year's harvest projected to yield?
When given a story question, it's always a good idea to put it in ‘math terms’ in your head before trying to solve for actual numbers.
Here, the question is really just asking “what is 220 increased by 30%”.
To increase a number by a percent, add its decimal form to 1. In this case, it’ll be \(1 + 0.3)\, which equals 1.3. The last step is to multiply our percent increase (1.3) by our value that we are increasing (220)
\[ x = 220 \cdot 1.3 \]
\[ x = 286 \]
William is projected to yield 286 pounds of fruit.
[fs-toc-h4]Example 1
The question tells us that Type A produces 20% more than Type B, meaning that
\[(\text{Type B}) \times (1.20) = (\text{Type A})\]
Now that we’ve written it as an equation we can substitute in what we know, and solve for what we don't know.
\[(\text{Type B}) \times (1.20) = (\text{Type A})\]
\[(\text{Type B}) \times (1.20) = (144)\]
\[\text{Type B} = 120 \text{ pears}\]
Making answer choice B correct
[fs-toc-h4]Example 2
We can immediately identify that this is a percent increase question as they literally use the words “percent increase”.
We are told that the percent increase from 2012 from 2013 is going to be DOUBLE that of the percent increase from 2013 to 2014.
Since we’re given the data for 2012 and 2013, our first step will be to find the actual increase between those two years.
\[(\text{2012 sales}) \times (\% \text{ increase}) = (\text{2013 sales})\]
\[(5600) \times (\% \text{ increase}) = (5,880)\]
\[\% \text{ increase} = 1.05\]
So we can say that the sales from 2012 to 2013 increased by 5%. Since this is double the increase from 2013 to 2014, diving the number by two will get us our desired % increase, which would be 2.5%
Now that we know that the percent increase from 2013 to 2014 is 2.5 %, we can plug these back into a percent increase equation to get the sales for 2014
\[(\text{2013 sales}) \times (\% \text{ increase}) = (\text{2014 sales})\]
\[(5,880) \times (1.025) = (\text{2014 sales})\]
\[6,027 = \text{2014 sales}\]
Making answer choice B the correct answer.
There is another way to solve this question involving the relative change formula. You would have gotten the exact same answer if everything was done correctly. The next section will be going over this exact process.
[fs-toc-h2]Relative Change
The relative change formula (sometimes referred to as percent change) is
\[\text{relative change} = \frac{\text{New value} - \text{Old value}}{\text{Old value}} \times 100\]
This formula is basically used to do the opposite of what we just did. We were finding increased/decreased values using a given percentage. Now, were going to find that percentage (percent change) using given increased/decreased values
NOTE: This formula gives you the percent value by multiplying by 100. If you just want the decimal value, leave the 100 off.
Let's look at a simple example that can take advantage of this formula.
Last year, Dylan made $1,400 dollars from his lemonade stand business. This year he was able to make $2,100. By what percent did his income increase by?
We’re given two values and trying to find the relative change between them. This means we have to use the relative change formula. Our old value is 1400, and our new value is 2100. All we have to do is plug in into our formula above.
\[\text{relative change} = \frac{2,100 - 1,400}{1,400} \times 100\]
\[\text{relative change} = \frac{700}{1,400} \times 100\]
\[\text{relative change} = 0.5 \times 100\]
\[\text{relative change} = 50\%\]
Dylan's income from his lemonade stand increased by 50%.
NOTE: If your answer is positive, it denotes a percent increase. If your answer is negative, it denotes a percent decrease.
[fs-toc-h4]Example 1
We’re looking to find percent increase, so we can either use the algebraic method used in the last question or use the relative change formula.
Our new value is 79.86 and our old value is 75.74
NOTE: a way I used to memorize the relative change formula was the acronym NOO \((\frac{\text{new} - \text{old}}{\text{old}})\)
\[\text{Relative change} = \frac{78.86 - 75.74}{75.74} \times 100\]
\[\text{Relative Change} = 4.1\%\]
Making answer choice A the correct answer.
[fs-toc-h4]Multiple percent changes on the same object
When an object or number is increased or decreased by multiple percentages, you cannot add all the percentages together, then multiply it by your value. For example, if there is a $200 jacket that is already 40% off, and then you apply a 15% off coupon, you can’t add 0.4 and 0.15 to get 0.55 and multiply it by (1 - 0.55) to get the total decrease. Doing this gets you
\[$200(1-0.55)\]
\[$90\]
In actuality though, the jacket will NOT cost $90. The reason is because you aren't paying for 45% of the original price (1 -0.55), you’re paying for 85% (1-0.15) of the price of 60% of the jacket (1 - 0.4). So, you have to find the original discounted price, THEN apply the next discount.
\[$200 (0.6) = 120\]
\[120(0.85) = 102\]
So, in this situation, you would be spending a total of $102
Here's a bit more help if you need it: