Know Your Fractions
Know Your Fractions
The SAT loves fractions. Loves them. The number of questions on the test that cover fractions in some way is nothing short of stupefying. This means you must know fractions inside and out. Know how to compare them, reduce them, add them, and multiply them. Know how to divide them, subtract them, and convert them to mixed numbers. Know them. Love them like the SAT does. Make them your friend on the test, not your enemy.
To begin, here are the basics: A fraction is a part of a whole. It’s composed of two expressions, a numerator and a denominator. The numerator of a fraction is the quantity above the fraction bar, and the denominator is the quantity below the fraction bar. For example, in the fraction 1 /2, 1 is the numerator and 2 is the denominator.
Equivalent Fractions
Fractions represent a part of a whole, so if you increase both the part and whole by the same multiple, you will not change the relationship between the part and the whole.
To determine if two fractions are equivalent, multiply the denominator and numerator of one fraction so that the denominators of the two fractions are equal (this is one place where knowing how to calculate LCM and GCF comes in handy). For example, 1/2 = 3/6 because if you multiply the numerator and denominator of 1/2 by 3, you get:
As long as you multiply or divide both the numerator and denominator of a fraction by the same nonzero number, you will not change the overall value of the fraction.
Reducing Fractions
Reducing fractions makes life simpler. It takes unwieldy monsters like 450/ 600 and makes them into smaller, friendlier critters. To reduce a fraction to its lowest terms, divide the numerator and denominator by their GCF. For example, for 450/600, the GCF of 450 and 600 is 150. So the fraction reduces down to 3 /4, since and .
A fraction is in its simplest, totally reduced form if its numerator and denominator share no further GCF (in other words, their GCF is 1). There is no number but 1, for instance, that can divide into both 3 and 4, so 3/4 is a fraction in its lowest form.
Comparing Fractions
Large positive numbers with lots of digits, like 5,000,000, are greater than numbers with just a few digits, such as 5. But fractions don’t work that way. While 200 /20,000 might seem like a nice, big, impressive fraction, 2 /3 is actually larger, because 2 is a much bigger part of 3 than 200 is of 20,000.
In certain cases, comparing two fractions can be very simple. If the denominators of two fractions are the same, then 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.
However, you’ll most likely have to deal with two fractions that have different numerators and denominators, such as 200/20,000 and 2/3. Don’t worry. There is an easy comparison tool, which we now reveal: cross-multiplication. Just multiply the numerator of each fraction by the denominator of the other, then write the product of each multiplication next to the numerator you used to get it. Here’s the cross-multiplication of 200 /20,000 and 2/3:
Since 40,000 > 600, 2 /3 is the greater fraction.
Adding and Subtracting Fractions
Adding and subtracting fractions that have the same denominator is a snap. If the fractions have different denominators, though, you need an additional step.
Fractions with the Same Denominators
To add fractions with the same denominators, all you have to do is add up the numerators:
Subtraction works similarly. If the denominators of the fractions are equal, just subtract one numerator from the other:
Fractions with Different Denominators
If the fractions don’t have equal denominators, then before you can actually get to the addition and subtraction, you first have to make the denominators the same. Then adding and subtracting will be a piece of cake, as in the example above. The best way to equalize denominators is to find the least common denominator (LCD), which is just the LCM of the two denominators. For example, the LCD of 1 /2 and 2/3 is 6, since 6 is the LCM of 2 and 3.
But because fractions are parts of a whole, if you increase the whole, you also have to increase the part by the same amount. To put it more bluntly, multiply the numerator by the same number you multiplied the denominator. For the example 1 /2 + 2/3, you know you have to get denominators of 6 in order to add them. For the 1/2, this means you have to multiply the denominator by 3. And if you multiply the denominator by 3, you have to multiply the numerator by 3 too:
So, the new fraction is 3 /6. Repeat the same process for the second fraction, 2 /3, except this time you have to multiply both denominator and numerator by 2:
The new fraction is 4 /6. The final step is to perform the addition or subtraction. In this case, 3/6 + 4/6 = 7/6.
Another approach is to skip finding the LCD and simply multiply the denominators together to get a common denominator. In some cases, such as our example, the product of the denominators will actually be the LCD (2 × 3 = 6 = LCD). But, other times, the product of the denominators will be greater than the LCD. For example, if the two denominators are 6 and 8, you could use 6 × 8 = 48 as a denominator instead of 24 (the LCD). There are two drawbacks to this second approach. The first is that you have to work with larger numbers. The second is that you have to take the extra step of reducing your answer. SAT answer choices almost always appear as reduced fractions. Trust us.
Multiplying Fractions
Multiplying fractions is a breeze, whether the denominators are equal or not. The product of two fractions is the product of their numerators over the product of their denominators:
Want an example with numbers? You got one.
Canceling Out
You can 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
To answer this fraction as it is, you have to multiply the numerators and denominators and then reduce. Sure, you could do it, but it would take some time. Canceling out provides a shortcut.
In this case, you can cancel out the numerator 4 with the denominator 8, and the numerator 10 with the denominator 5, which gives you
Then, canceling the 2’s, you get
Canceling out can dramatically cut the amount of time you need to spend working with big numbers. When dealing with fractions, whether they’re filled with numbers or variables, always be on the lookout for chances to cancel out.
Dividing Fractions
Multiplication and division are inverse operations. It makes sense, then, that to perform division with fractions, all you have to do is flip the second fraction and then multiply. Check it out:
Here’s a numerical example:
Mixed Numbers
A mixed number is an integer followed by a fraction, like 1 1/2. But operations such as addition, subtraction, multiplication, and division can’t be performed on mixed numbers, so you have to know how to convert them into fraction form.
Since we already mentioned 11 /2, it seems only right to convert it. The method is easy: Multiply the integer (the big 1) of the mixed number by the denominator, and add that product to the numerator: 1 × 2 + 1 = 3 is the numerator of the improper fraction. Now, put that over the original denominator: 3/2 is the converted fraction.
Here’s another example:
We said it once, we’ll say it again: Converting mixed numbers is particularly important on grid-in questions, since you can’t actually write a mixed number into the grid. If you tried to grid 11/ 2, the computer that scores your test will read it as 11 /2. Ouch!
Complex Fractions
Complex fractions are fractions of fractions.
Here’s what you should be thinking: “Ugh.” Complex fractions are annoying if you try to take them head on. But you don’t have to. Instead, transform them into normal fractions according to this quick step: Multiply the top fraction by the reciprocal of the bottom fraction.
And here’s an example using actual numbers:
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