The SAT focuses far more closely on fractions, decimals, and percents than on any other arithmetic topic. In fact, almost 14 percent of all SAT math questions require some knowledge of fractions, decimals, or percents. On a single SAT, 7 to 8 questions will deal with this topic. Because fractions encompass such a large part of the test, knowing your stuff here can really help your final score.

A fraction describes a part of a whole. The number on the bottom of the fraction is called the denominator, and it denotes the number of equal parts into which the whole is divided. The number on the top of the fraction is called the numerator, and denotes how many of those equal parts the fraction has. For example, the fraction ³/₄ denotes “3 of 4 equal parts,” 3 being the numerator and 4 being the denominator.

You can also think of fractions as similar to division. In fact, ³/₄ means the same thing as

Two fractions are equivalent if there is a number by which both the numerator and the denominator of one fraction can be multiplied or divided to yield the other fraction. For example, ²/₃ is equivalent to ⁴/₆ because if you multiply the numerator and denominator of ² /₃ by 2, you get ⁴/₆:

Equivalent fractions are equivalent in value. When you multiply or divide both the numerator and denominator of a fraction by the same number, you will not change the overall value of the fraction. Because fractions represent a part of a whole, if you increase both the part and whole by the same multiple, you will not change the relationship between the part and the whole. See how ¹/ ₃ of a pizza is exactly the same as ³/₉?

On the SAT, you will sometimes encounter fractions involving large, unwieldy numbers, such as ¹⁸ /₁₀₂. It would probably be hard (and time consuming) for you to work with ¹⁸ /₁₀₂, just because the numbers in the numerator and denominator are so big. When faced with such cases, it is always a good idea to see if the fraction can be reduced, or simplified.

The fastest way to simplify a fraction is to divide both the numerator and denominator by their Greatest Common Factor (GCF). In the case of ¹⁸/ ₁₀₂, the GCF of 18 and 102 is 6, leaving you with ³/₁₇. With your knowledge of divisibility rules, you should be able to see that both the numerator and denominator are divisible by 6. Had you not immediately seen that 6 was the GCF, you could have divided both numbers by 2 and gotten ⁹/₅₁. From there, it would have been pretty obvious that both numerator and denominator are also divisible by 3, yielding ³/₁₇.

On the SAT, when you encounter a fraction that involves big numbers, very often that fraction can be reduced. And because the SAT is in part a test of speed, any knowledge you have that lessens the time it takes to answer a question is very important. You need to get skilled not only at reducing fractions, but also at recognizing when a fraction can be reduced.

Particularly on quantitative comparison questions, you may be asked to compare two fractions. If either the denominators or the numerators of the two fractions are the same, that comparison is easy. ⁸/₉ is obviously greater than ⁵/ ₉, and ⁵ /₉ is greater than ⁵/₁₇. Just remember, if the denominators are the same, the fraction with the larger numerator is bigger. If the numerators of the two fractions are the same, the fraction with the smaller denominator is bigger.

If the two fractions don’t lend themselves to immediate easy comparison, don’t fret. There is a trick that allows you to compare fractions: cross-multiplication. To cross-multiply, multiply the numerator of each fraction by the denominator of the other. Write the product of each multiplication next to the numerator you used to get it. The greater product will be next to the greater fraction. For example:

35, the greater product, is next to the fraction ⁵/₈, so that is the greater fraction.

There are two different types of fractions that the SAT might ask you to add or subtract. It might ask you to work with two fractions that have the same denominator. Or it might ask you to handle two fractions with different denominators.

If fractions have the same denominator, adding them is extremely easy. All you have to do is add up the numerators:

Subtraction works similarly. If the denominators of the fractions are equal, then you simply subtract one numerator from the other:

If the fractions do not have equal denominators, the process is somewhat more involved. The first step is to make the denominators the same. To set the denominators of two fractions equal, find the Least Common Denominator (LCD), which is simply the Least Common Multiple (LCM) of the two denominators. For example, 18 is the LCD of ¹/₆ and ⁴/₉, since 18 is the smallest multiple of both 6 and 9.

Setting the denominators of two fractions equal to each other is a two-step process. First, find the LCD. Second, write each fraction as an equivalent fraction with the LCD as the new denominator, remembering to multiply the numerator by the same multiple as the denominator. For example, if you wanted to add ⁵/₁₂ and ⁴/₉, you would do the following:

The new fraction is, therefore, ¹⁵ /₃₆.

The new fraction is ¹⁶ /₃₆.

Now that the fractions have the same denominator, you can quickly add the numerators to get the final answer. 15 + 16 = 31, so the answer is ³¹/ ₃₆.

If you think it will take you too long to figure out the LCD, you can always multiply the denominators together to get a common denominator that isn’t the least common denominator. For example, if the two denominators are 6 and 8, you can use 48 as your common denominator just as easily as 24 (the LCD). There are two drawbacks to not using the LCD. First, you will have to work with larger numbers. Second, because the answer choices will appear as reduced fractions, you will have to reduce your answer at the end.

Multiplying fractions is quite easy. Simply multiply the numerators together and the denominators together, as seen in the example below.

You can often make multiplying fractions even easier by canceling out. If the numerator and denominator of any of the fractions you need to multiply share a common factor, you can divide by the common factor to reduce both numerator and denominator. For example, the fraction:

can be rewritten, after canceling out the 4, 8, 5, and 10, as:

then, canceling the 2’s, you get:

Though multiplying fractions is fairly easy mechanically, it is a little tricky intuitively. You are probably used to the product of multiplication being bigger than the numbers that are being multiplied. But when dealing with a fraction, the product of two numbers is smaller. Note that this phenomenon only occurs in reference to fractions smaller than one, in which the numerator is smaller than the denominator.

Multiplication and division are inverse operations. It makes sense, then, that to perform division with fractions, all you have to do is invert (flip over) the dividing fraction and then multiply.

Also note that just as multiplication of fractions that are smaller than one results in an even smaller product, division of fractions smaller than one results in a larger product.

A mixed number is composed of a whole number and a fraction: 6 ²/₃, for example, is a mixed number because 6 is a whole number and ²/₃ is a fraction. For the SAT, it is very important that you be able to convert a mixed number into a fraction. Whenever you are asked to perform an operation of any sort on a mixed number, you will first have to convert it to fraction form. Also, for GI questions you will have to convert any mixed numbers to fractions before gridding them in.

To convert a mixed number into a fraction, multiply the whole number by the denominator and add the result to the numerator. Do not change the denominator.

For example, 6² /₃ can be converted into a fraction in the following way: multiply the whole number, 6, by the denominator, 3, and then add the yielded product to the original numerator, 2. The denominator remains the same, so the answer is ²⁰ /₃.

Decimals are simply another way to express fractions. To get a decimal, divide the numerator of a fraction by the denominator. For example, if you take the fraction ² /₅ and divide 2 by 5 you would get .4.

Normally, numbers get bigger when they involve more numerals. The number 4000, for example, is obviously bigger than 4. However, with decimals, more zeros often means less: .4 is larger than .004. If you remember that decimals are just another way to express fractions, the reason for this difference in size is easy to see. .4 is equivalent to the ⁴/₁₀, while .004 is equivalent to ⁴ /₁₀₀₀.

The SAT will occasionally try to trip you up by asking you to compare a decimal such as .002 with the decimal .0008. Because you aren’t so used to looking at decimals and 8 is obviously a larger integer than 2, you may be tempted to overlook that the second decimal includes an additional 0 and choose it as the larger decimal. To avoid such mistakes, all you have to do is be careful. One way to insure that you’re being careful is to line up the decimal points of the two decimals. While .0008 might seem larger than .002,

To make the situation even more obvious, you can add an extra zero to the bottom decimal to make it just as long as the upper

Now you are comparing ⁸ /₁₀₀₀ to ²⁰/₁₀₀₀, and ²⁰/₁₀₀₀ is clearly the bigger of those two numbers. If numbers are being added to the right of the decimal number, then it’s a different story.

The processes of addition, subtraction, multiplication, and division for decimal numbers are quite similar to the rules of those same operations for integers. However, we’re not going to delve into the specifics of those rules right now for a very simple reason: when dealing with the addition, subtraction, multiplication, or subtraction of decimals, it is almost always faster and more accurate to use a calculator. If you type in the correct decimals to begin with, your calculator will always come out with the right answer.

Percents are just another way to talk about a specific type of fraction (which also means that percents are also just another way to talk about a specific type of decimal). Percent literally means “of 100” in Latin, so when you have 25 percent of all the money in the world, that means you have ²⁵/₁₀₀ (or .25) of the world’s money.

But, sadly, you don’t have that much money, and you have to take the SAT. So let’s look at an example question: 4 is what percent of 20? This question presents you with a whole, 20, and then asks you to determine how much of that whole 4 represents in percentage form. To come to the answer, you have to set up an equation that sets the fraction ⁴/₂₀ to ^x/₁₀₀:

if you then cross-multiply to solve for x, you get 20x = 400, meaning x = 20. Therefore, 4 is 20% of 20. You also might realize that instead of working out all this cross multiplication, you could simply cancel out the 20 and the 100 to get

Converting percents into fractions and decimals will almost surely come up on the SAT. To convert from a percent to a fraction, all you have to do is take the percentage number and place it as a numerator over the denominator 100. If you have 88 percent of something, then you can quickly convert it into the fraction ⁸⁸/₁₀₀.

To convert from a percent to a decimal, you must take a decimal point and insert it into the percent number two spaces from the right. 79% therefore becomes .79, while 350% becomes 3.5.

To convert from either a fraction or decimal back to a percent, perform the processes in reverse: multiply the fraction by 100 or move the decimal point two spaces to the left.

To save time while taking the SAT, you should memorize some of the conversions between common fractions, decimals, and percents.

Percentage problems on the SAT can often be hard but not because the math they use is confusing. Instead, it is the words the problems use that can cause difficulties. To combat this verbal nastiness, we’re going to look at a sample question and explain where and why it is tricky.

When you see a percentage question, your first goal should always be to determine which number represents the whole and which the part. Intuitively, when you see the question above, you will probably think that 2 is the part, since 2 is smaller than 5, and how can you have a part that’s bigger than the whole? But you can calculate the percentage when the part is bigger than the whole: the answer will simply be bigger than 100%. Now, let’s break down the question. What percent of 2 is 5? This question could be written: 5 is what percent of 2? Once we’ve reorganized the question, it should be obvious that the “of 2” marks the 2 as the whole and the 5 as the part. We can then set up the fraction ⁵ /₂ = 250%.

Remember the most important lesson of percents: before beginning a problem, always identify which number represents the whole and which represents the part.

Percent terminology can be a little tricky, so here is a very short dictionary of terms:

Sometimes students see these terms and figure out what the 10% increase or decrease is, but then forget to carry out the necessary addition or subtraction. The SAT writers know about this tendency and will try to use it to trick you:

To answer this question, you should multiply $20 by .15 to see what the change in price was:

Once you know that price change, then you need to subtract it from the original price, since the question asks you to find the reduced price of the shirt:

The answer is (D). But if you only finished the first part of this question and looked at the answers, you might see the $3 at answer (A) like a big affirmation of correctness and be tempted to choose it without finishing the calculation.

Some SAT questions will ask you to determine a percent of a percent. For example, take the question:

In this question, you are being asked to determine the cumulative effect of two percent changes. The key to solving this type of problem is to realize that each percentage change is dependent on the last. In other words, you have to work out the effect of the first percentage change, come up with a value, and then use that value to determine the effect of the second percentage change. When you are working on a percentage problem that involves a series of percentage changes, you should follow the same basic procedure that we explained for one percentage change, except in this case you should follow the procedure twice. For the first percentage change, figure out what is the whole, calculate the percentage of the whole, make sure to perform addition or subtraction if necessary, then take the new value and put it through these same steps for the second percentage change.

To answer the problem, you should first find 25% of the original price:

Now subtract that .50 from the original price:

Then we use $1.50 and increase it by 10%:

Therefore, Sam buys the banana at a price of $1.50 + $.15 = $1.65.

When dealing with double-percent questions, some students are tempted to simply combine the two percentage changes. But you cannot simply add or subtract the two percent changes and then find that percent of the original value. If you tried to answer the question above by reasoning that the first percentage change lowered the price 25% and the second raised the price 10%, meaning that the total change was –15%, you would get the question wrong:

Now subtract that .30 from the original price:

We promise you that when the SAT gives you a double-percent problem they will include this sort of wrong answer among the choices as a distraction. Don’t fall for the trick. Don’t give them the satisfaction.