A Complete Guide To Fractions

For many math students, fractions are a tough thing. But, Fractions loom large on the ACCUPLACER. 

Fraction basics
Fractions are one of a variety of ways to represent positive values that are less than 1 — that is, parts of a larger whole.

Identifying the numerator and the denominator
Every fraction consists of two whole numbers placed one above the other. The top number is called the numerator and the bottom number is called the denominator.
For example, in the fraction 2/9 , the numerator is 2 and the denominator is 9.

Understanding reciprocals
The reciprocal of a fraction is simply that fraction turned upside-down, so that the numerator (top number) and denominator (bottom number) are exchanged. For example, the reciprocal of 3/5 is 5/3 .
When a fraction has a 1 in the numerator, the reciprocal of that fraction is the denominator. For example, the reciprocal of 1/7 is 7/1 , which simplifies to 7.
You can turn any whole number into a fraction by placing it in the numerator of a fraction with 1 in the denominator. For example, 14 is equal to the fraction 14/1 , so its reciprocal is 1/14.
On the ACCUPLACER, you’ll use the reciprocal of a fraction when dividing fractions.

Simplifying and increasing the terms of fractions
You can express the value of a single fraction in a variety 

SIMPLIFYING FRACTIONS
Consider the fraction 6/8 . You can probably see that both 6 and 8 are divisible by 2. You can simplify (or reduce) this fraction by dividing both of these numbers by this common factor: (6/2) / (8/2) = 3/4
The simplified form of a fraction is usually the preferred form, and most ACCUPLACER multiple-choice answers involving fractions include only simplified fractions.
To simplify a fraction, divide both the numerator and denominator by a common factor. You may need to repeat this process until the fraction is completely simplified. For example, consider the fraction 60/72. You can begin simplifying this fraction by repeatedly dividing by 2 as many times as possible: (60/2)/(72/2) = (30/2)/(36/2) = (15/2)/(18/2) =  15/18
Now, notice that both 15 and 18 are divisible by 3, so divide both by 3: (15/3)/(18/3) = 5/6
This fraction is fully simplified because 5 and 6 have no common factor greater than 1.

INCREASING THE TERMS OF FRACTIONS
To solve some problems that involve fractions, you may need to increase the terms of both the numerator (top number) and denominator (bottom number) without changing the value of the fraction. To do this, multiply both the numerator and denominator by the same number.
For example, you can increase the terms of 1/4 by multiplying its numerator and denominator by the same number: (1*2)/(4*2) = 2/8    
The resulting fractions all look different, but are all equal to 1/4 .
Often, the goal of changing the terms of a fraction is to change the denominator to a specific number that is a multiple of the current denominator. For example, suppose you want to change the terms of 6/25 so that the denominator equals 100. To do this, consider what value you need to multiply the denominator by to achieve this goal: (6 * ?)/(25 * ?) = ?/100

As you can see, to change the denominator of 25 to 100, you need to multiply the denominator by 4. And to preserve the value of the fraction, you also need to multiply the numerator by 4: (6 * 4) / (25 * 4) = 24/100
Therefore, 24/100 is an alternative way to express the value of  6 /25.

Working with improper fractions and mixed numbers
When the numerator (top number) of a fraction is less than its denominator, that fraction is called a proper fraction. For example, 2/3, 4/13, 81/244
Every positive proper fraction expresses a value that is between 0 and 1.
In contrast, when a fraction’s numerator is greater than or equal to its denominator, it’s called an improper fraction. For example, 3/2, 63/29, 100/100.
Every positive improper fraction expresses a value that is greater than or equal to 1.
Like improper fractions, a mixed number also expresses a value of 1 or greater. A mixed number includes both a whole number and a proper fraction: for example,  1 1/2, 12 7/10, 139 47/83
Every mixed number has an equivalent value that can be expressed as an improper fraction.

Converting mixed numbers to improper fractions
Improper fractions are often easier to calculate with, especially when multiplying and dividing fractions. So on the ACCUPLACER, you may need to change a mixed number to an improper fraction at the beginning of a problem in order to solve it.
To change a mixed number to an improper fraction:
Multiply the denominator by the whole number, then add the numerator, and copy this value as the numerator of your answer.
Copy the denominator.

For example, to change 3 2/5 to an improper fraction, multiply 3 * 5 + 2 to find the numerator of the answer, then copy the denominator of 5: 3 2/5 = (3 * 5 + 2) / 5 - 17/5

Converting improper fractions to mixed numbers
Mixed numbers tend to be easier to read and understand than improper fractions. For example, if Jack puts 13 1/2 gallons of gasoline in the tank, you can easily estimate that Jack bought between 13 and 14 gallons. In contrast, if Jack expresses this same value as 27/2 gallons, you’re likely to be confused. 
For this reason, multiple-choice answers on the ACCUPLACER are usually expressed as mixed numbers rather than improper fractions.
You can change an improper fraction to a mixed number in two ways, depending on whether the improper fraction is small or large.

CONVERTING SMALLER IMPROPER FRACTIONS
When you perform basic operations on fractions, the result is often an improper fraction in which the numerator of the improper fraction is not much larger than its denominator. Here is a fast and easy way to change that improper fraction to a mixed number:
1. Write the number 1 (or add 1 to the previous whole number value).
2. Copy the denominator.
3. Subtract the numerator from the denominator to find the new numerator.

Here’s how to convert the improper fraction 8/5 to an equivalent mixed number:8/5 = 1 3/5
Here, you wrote the number 1, copied the denominator of 5, and subtracted 8-5 = 3 to find the numerator.
Notice that the denominator of the original improper fraction and its mixed-number equivalent remain the same. This is always true when converting an improper fraction to a mixed number.
As another example, change 13/4 to a mixed number by repeating this process: 13/4 = 1 9/4 = 2 5/4 = 3 1/4
Here, you repeatedly increased the value of the whole number by 1, and repeatedly subtracted the denominator of 4 from the numerator. The process is complete when the fractional portion of the result is a proper fraction — in this case, 1/4.

CONVERTING LARGER IMPROPER FRACTIONS
When the numerator of an improper fraction is much larger than its denominator, the method in “Converting smaller improper fractions” can become time-consuming.
For example, suppose you need to change 81/2 to a mixed number. You may not want to go through this process: 81/2 = 1 79/2 = 2 77/2 = 
A better method in this case is to divide the numerator by the denominator. When you complete the division,
The quotient (result of the division) is the whole number value.
The remainder is the numerator.
The denominator stays the same.
For example, to convert 81/2 to a mixed number, you divide 81/2 = 40rl  to get — that is, 40 with a remainder of 1 — so, 81/2 = 40 1/2

Adding and subtracting fractions that have the same denominator
Just as with whole numbers, you can apply the Big Four operations — addition, subtraction, multiplication, and division — to fractions.
When adding and subtracting fractions, the main thing to notice first is whether the fractions have the same denominator (bottom number). 

Adding fractions with the same denominator
To add fractions that have the same denominator:
1. Add the numerators.
2. Keep the denominator the same.

For example,  1/5 + 3/5 = 4/5
This rule works equally well for adding more than two fractions:  1/7 + 2/7 + 3/7 = 6/7

In some cases when adding fractions, you may need to simplify the result in one or both of the following ways:

1. Simplify the fraction.
2. Change an improper fraction to a mixed number .

For example: 7/12 + 1/12 = 8/12
You can reduce this fraction by dividing both the numerator and the denominator by 4: (8/4)/(12/4) = 2/3

As another example: 7/10 + 9/10 = 16/10
Convert this improper fraction to a mixed number, and then simplify the result: 1 6/10 = 1 3/5

Subtracting fractions with the same denominator
Subtracting fractions that have the same denominator is as simple as adding them:
1. Subtract the numerators.
2. Keep the denominator the same.

For example, 5/7 - 2/7 = 3/7

As with whole-number subtraction, when you subtract a smaller fraction minus a greater one, your result is a negative number. For example, 2/9 - 7/9 = - 5/9
Here, subtracting the numerators 1-7 results in the negative fraction -5/9.

As when adding fractions, in some cases you will need to simplify your result. For example: 11/12 - 5/12 = 6/12
Here, to simplify the result, divide both the numerator and the denominator by 6: (6/6)/(12/6) = 1/2

Adding and subtracting fractions that have different denominators
Adding and subtracting fractions that have different denominators can be a little complicated, and may seem absolutely daunting if you got lost in the shuffle in the third or fourth grade.
The reason for this difficulty is that most students learn to add or subtract fractions by finding a common denominator — that is, by getting both fractions to have the same denominator before you add or subtract. And this step is often the most confusing part of adding and subtracting fractions.

Two simple methods for finding a common denominator:
1. Cross-multiplication — This method always works, but sometimes results in large numbers that are hard to work with.
2. Increasing the terms of the smaller fraction — This method only works in some cases, but it can help keep the numbers small.

Look how straightforward it seems in comparison with your memories from grade school.

Method #1 — Using cross-multiplication
When a pair of fractions have different denominators, you can always add or subtract them using cross-multiplication — that is, multiplying the numerator of each fraction by the denominator of the other. This step gets the denominators equal to each other, which allows you to add or subtract the fractions easily, as in “Adding and subtracting fractions that have the same denominator.”

ADDING FRACTIONS WITH CROSS-MULTIPLICATION
To add a pair of fractions using cross-multiplication, follow these steps:
1. Cross-multiply the two fractions to find the new numerators.
2. Multiply the denominators to find the new denominator.

For example, to add 2/5 + 1/4 , cross-multiply the two fractions 2 *4 = 8 — and 5 * 1 = 5 — placing these results in the numerators to add in the next step; then multiply 5 * 4 = 20 to get the denominator of both fractions.
2/5 + 1/4 = 8/20 + 5 /20
The result is a pair of fractions with the same denominator. Now you can add these fractions by adding numerators and keeping the denominator the same:  13/20

In some cases, after you finish adding a pair of fractions, the result can be simplified. 

For example, consider the addition problem 1/6 + 3/4 . You can solve this problem using cross-multiplication, as follows: 1/6 + 3/4 = 4 /24 + 18/24 = 22/24
As it stands, this problem is still incomplete, because the answer can be simplified by dividing both the numerator and the denominator by 2: (22/2) / (24/2) = 11/12
This simplified version of the fraction is most likely the one that will appear as a multiple-choice answer on the ACCUPLACER.

As a final example, here’s how you add : 6/7 + 3/5 
6/7 + 3/5 = 30/35 + 21/35 = 51/35
This time, the result is an improper fraction, so you need to change it to a mixed number (as in “Converting improper fractions to mixed numbers”):  1 16/25

SUBTRACTING FRACTIONS WITH CROSS-MULTIPLICATION
Cross-multiplying works equally well for subtracting fractions with different denominators:
1. Cross-multiply the two fractions to find the new numerators.
2. Multiply the denominators to find the new denominator.

For example, to subtract 7/9 - 3/5 , cross-multiply the two fractions 7 * 5 = 35 — and 9 * 3 = 27— and place these result in the numerator to subtract in the next step. Then multiply 9 *5 = 45 .

7/9 - 3/5 = 35/45 -  27/25
Again, the result is two fractions with the same denominator. Now subtract the numerators and leave the denominator the same: = 8/45

As when subtracting fractions with the same denominator, when you subtract a smaller fraction from a greater one, the result is a negative fraction. For example: 1/4 - 5/7  = 7/28 - 20/28 = -13/28
Here, the subtraction results in the negative fraction .

In some cases when subtracting, you may need to reduce your answer. For example: 3/8 - 1/10 = 30/80 - 8/80 = 22/80
To complete this problem, reduce this fraction by a factor of 2: (22/2)/(80/2) =11/40

Method #2 — Increasing the terms of one fraction
When the denominator of one fraction is divisible by the denominator of another fraction, you can get the denominators the same by increasing the terms of the smaller denominator. In this section, I show you how this method works, and why it can be preferable to cross-multiplication.

ADDING FRACTIONS BY INCREASING THE TERMS OF ONE FRACTION
Consider the problem 3/8 + 9/16 . If you try to add it using cross-multiplication, this is the result: 3/8 + 9/16 = 48/128 + 72/128 = 120/128
These numbers grew large very quickly, which increases your chance of making a calculation error. Worse yet, you’re still not done, because you need to reduce this final fraction.

However, there’s an easier way to do this problem. Notice that the denominator 8 is a factor of the denominator 16. This fact allows you to get the denominators the same in the following way:
1. Increase the terms of the fraction with the lower denominator so that this denominator matches that of the other fraction (see “Increasing the terms of fractions”).
2. Add the fractions using the rules for fractions with the same denominator (see “Adding fractions with the same denominator”).

Use this method to add 3/8 + 9/16, increasing the terms of 3/8 so that the denominator becomes 16, and then adding the results: 3/8 + 9/16 = 6/16 + 9/16 = 15/16

Remember that when adding fractions by any method, you still may have to change an improper fraction to a mixed number.
For example, to add 7/10 + 13/20 , increase the terms of the first fraction by a factor of 2: 7/10 + 13/20 = 14/20 + 13/20 = 27/20
Now, change the resulting improper fraction to a mixed number: 1 7/20

Here’s a quick trick that you can use in some cases when adding fractions: When a pair of fractions both have 1 in the numerator, you can add them quickly and easily as follows:
1. Add the denominators to find the numerator of the answer.
2. Multiply the denominators to find the denominator.

For example, 1/2 + 1/3 = 5/6
Remember, in some cases, you’ll need to reduce the result. For example:  1/4 + 1/10 = 14/40 = 7/20

SUBTRACTING FRACTIONS BY INCREASING THE TERMS OF ONE FRACTION
As when adding fractions, in some cases when subtracting fractions, you can avoid the large numbers that cross-multiplication sometimes produces.
Increase the terms of the fraction with the lower denominator so that this denominator matches that of the other fraction (see “Increasing the terms of fractions”).
Subtract the fractions using the rules for fractions with the same denominator (see “Subtracting fractions with the same denominator”).

For example, you could calculate 17/18 - 7/9 using cross-multiplication, but an easier way is to change the terms of 7/9, changing the denominator to 18.
In this case, the result can be simplified: 17/18 - 7/9 = 17/18 - 14/18 = 3/18
This method works equally when subtracting a smaller fraction minus a larger one.
For example, to subtract 3/14 - 5/7 , increase the terms of the second fraction, changing its denominator to 14: 3/14 - 5/7 = 3/14 - 10/14 = -7/14
This time, simplify by dividing both the numerator and the denominator by 7: - 1/2

Multiplying and dividing fractions
Here’s some good news: Multiplying and dividing fractions is usually a lot easier than adding or subtracting them. 

Multiplying fractions
You can multiply any pair of fractions without worrying whether their denominators are the same or different. And to sweeten the deal, you can often simplify the problem before you multiply to make the numbers smaller and easier to calculate.

To multiply a pair of fractions, multiply the two numerators to get the numerator of the answer, and multiply the two denominators to get the denominator. For example,

2/5 * 1/3 = 2/15
In some cases, you can cancel factors in the numerator of one fraction and the denominator of the other fraction before you multiply. 
For example, to multiply 8/9 * 9/14 , begin by canceling out a factor of 9 in the numerator of the second fraction and the denominator of the first fraction, replacing both with the number 1:  8/9 * 9/14 = 8 / 14 (9 is cancelled)
Next, divide both 8 and 14 by a factor of 2.
Now, finish up by multiplying across as usual, 4 * 1 = 4 and  1 * 7 = 7:

Dividing fractions
To divide one fraction by another, turn the problem into fraction multiplication using the mnemonic (memory trick) Keep-Change-Flip:

1. Keep the first fraction just as it is.
2. Change the division sign () to a multiplication sign ().
3. Flip the second fraction — that is, change it to its reciprocal (for more on reciprocals, see “Understanding reciprocals”.)

From there, find the answer by multiplying the two resulting fractions. An example will help make this idea clear. 
Suppose you want to divide (5/8) / (2/3). Turn the problem into multiplication by applying Keep-Change-Flip, then multiply fractions across. (For more on fraction multiplication, see “Multiplying fractions” earlier in this section.)

 (5/8) / (2/3) =  (5/8) * (3/2) = 15/16

In some cases when dividing fractions, you will have an opportunity to simplify the problem before you multiply by canceling factors. For example, (7/10) / (4/5) = (7/10) * (5/4)
Now, cancel a factor of 5 in both the numerator and denominator, and complete the multiplication: = 7/8

Be careful not to cancel factors before changing the problem to multiplication. 

For example, suppose you want to divide (1/6)/(6/11) . Here, you may be tempted to cancel factors of 6. However, watch what happens when you change the division to multiplication: (1/6)/(6/11) = (1/6) * (11/6) = 11/66
As you can see, when you change 6/11 to 11/6 , the opportunity to cancel a factor of 6 disappears. Never try to cancel factors in a fraction division problem until you have changed it to multiplication.

Operations on mixed numbers
Working with mixed numbers is probably not your favorite weekend activity when compared with, say, snowboarding or beach volleyball. This section probably won’t change your mind, but I hope it will convince you that these problems now feel at least a little easier than they did when you were in middle school.

Adding mixed numbers — without carrying
Adding mixed numbers isn’t much more difficult than adding fractions. The method I outline next includes three steps that allow you to avoid carrying, which many students find confusing.

To add a pair of mixed numbers, follow these steps:

1. Add the whole number values.
2. Add the fractional values.
3. Add the results of Steps 1 and 2.

The only difficult step here is Step 2, where you add fractions (see “Adding fractions”).
For example, to add 6 3/4 + 2 4/5, begin by adding the whole number parts of the mixed numbers.: 6 + 2 = 8

Next, add the two fractional parts:
3/4 + 4/5 = 15/20 + 16/20 = 31/20

This value is an improper fraction, so change it to a mixed number: 1 11/20
To complete the problem, add the results from the first two steps:

8 + 1 11/20 = 9 11/20

Subtracting mixed numbers — without borrowing
Generally, subtracting mixed numbers is the most difficult calculation with fractions. This is a shame, because it doesn’t have to be.
The problem here is usually because of borrowing, which involves a bunch of difficult, tedious steps. The method here allows you to subtract mixed numbers in three steps without borrowing:

1. Subtract the whole number parts.
2. Subtract the fractional parts. (The result may be positive or negative.)
3. Add the results from Steps 1 and 2.

For example, suppose you want to subtract 4 2/3 - 1 8/15. To begin, subtract 4-1 = 3 .
Next, subtract 2/3 - 8/15 . You can use cross-multiplication here, but to avoid big numbers, try increasing the terms of 2/3, changing the denominator to 15, before you subtract: 2/3 - 8/15 = 10/15 - 8/15 = 2/15
To complete the problem, you need to add the results from the first two steps: .
3 + 2/15 = 3 2/15

A hard problem that would normally require you to borrow in order to solve it: 6 1/3 - 2 3/4 .
To begin, subtract 6 - 2 = 4 . 

Next, subtract 1/3 - 3/4 . This step involves some work:1/3 - 3/4  = 4/12 - 9/12 = - 5/12
The result is a negative number.

Now, to complete the problem, you need to add the results from the first two steps. But because your result from Step 2 is negative, this addition looks more like subtraction: 4 - 5/12

Here’s a quick way to subtract a whole number minus a fraction:
1. Subtract 1 from the whole number: 4 -1 = 3.
2. Copy the denominator: 12.
3. Subtract the denominator minus the numerator to find the numerator of the answer: 12 - 5 = 7.

4 - 5/12 = 3 7/12

This method for subtracting mixed numbers without borrowing always works. And if you’ve struggled to master the skill of subtracting mixed numbers, or simply given up on it, I think you’ll agree that each of these steps is a lot easier than the usual way that mixed number subtraction is taught.

Multiplying mixed numbers
To multiply mixed numbers, first turn them into improper fractions (see “Converting mixed numbers to improper fractions”). Then, multiply as you normally would (see “Multiplying fractions”).

For example, to multiply 3 1/2 * 1 3/5 , begin by changing the mixed numbers to their equivalent improper fractions: 3 1/2 * 1 3/5 =  7/2 * 8/5
Now, simplify the problem by canceling a factor of 2, then multiply: =28/5
Complete the problem by converting this improper fraction to a mixed number (see “Converting mixed numbers to improper fractions”). To do so, divide , so: 28/5 = 5r3 = 5 3/5

Dividing mixed numbers
You can divide mixed numbers by changing them to improper fractions (see “Converting mixed numbers to improper fractions”) and performing the division as you normally would (see “Dividing fractions”). 
For example, to divide 2 1/4 by 1 5/7, begin by converting these two mixed numbers to improper fractions: 2 1/4 by 1 5/7 = 9/4 * 7/12
Next, change the problem to multiplication using Keep-Change-Flip:
You can make this problem easier by canceling a factor of 3 before you multiply: = 21/16
To complete the problem, convert the resulting improper fraction to a mixed number as shown in “Converting smaller improper fractions.” To do this, write down the number 1, then copy the denominator of 16, and subtract 21 - 15  = 5 to find the numerator: 1 5/16.