Prime Numbers
In order to manipulate fractions proficiently, you must understand how numbers are put together. Thus, we must start our review of fractions by reviewing the prime numbers. Let’s work through an exploratory exercise to discover these basic mathematical building blocks.
Start first with a chart that lists the numbers from 1 to 100.
Cross out number 1. 1 is not considered a prime number. Then, cross out every other number except for 2. These numbers are the even numbers. They are divisible by 2 and are therefore not prime.
Then, cross out every third number except for 3. These numbers are divisible by 3 and are therefore not prime.
Then, cross out every fifth number except for 5. These numbers are divisible by 5 and are therefore not prime.
Then, cross out every seventh number except for 7. These numbers are divisible by 7 and are therefore not prime.
The numbers that remain are the prime numbers. They are only divisible by one and themselves. These numbers can be multiplied together to make other numbers. Numbers that are made from the products of prime numbers are called composite numbers.
Highlighted in this image are the prime numbers less than 100:
Prime Factorization of Composite Numbers
Composite numbers can be broken down into their component primes using the factorization method. Take, for example, the composite number 48. If you recall your multiplication facts, you know that 48 is equal to 6 * 8. Write it like this:
48 = 6 * 8
Neither 6 nor 8 are prime numbers, so we need to break this number down further. 6 is also equal to 2 * 3, while 8 is also equal to 2 * 4:
48 = 2 * 3 * 2 * 4
2 and 3 are prime numbers, so those branches are done. 4, however, is not a prime number, so it needs to be broken down into 2 * 2:
48 = 2 * 3 * 2 * 2 * 2
When all you have are prime numbers, you are done. In our example, we have discovered that 48 = 2 * 2 * 2 * 2 * 3.
Here’s the prime factorization for 54:
54 = 6 * 9
54 = 2 * 3 * 3 * 3.
Greatest Common Factor
You can use the factorization method described above to find the greatest common factor of two numbers. Let’s say, for example, that you wanted to find the greatest common factor of 48 and 54. As we found above, 48 = 2 * 2 * 2 * 2 * 3 and 54 = 2 * 3 * 3 * 3. Line these up like this:
48 = 2 * 2 * 2 * 2 * 3
54 = 2 * 3 * 3 * 3
I’ve bolded the prime factors that are shared by both numbers. If you multiply these shared prime factors together, you will discover the greatest common factor is 6.
Least Common Multiple
You can also use the factor tree method to find the least common multiple of two numbers. For example, what if you were asked to find the least common multiple of 6 and 8?
First, break down each number into its prime factors using the factor tree method. You should find that 6 = 2 * 3 and that 8 = 2 * 2 * 2.
Next, find the greatest common factor:
8 = 2 * 2 * 2
6 = 2 * *3
GCF = 2
Then, list the unshared factors:
8 = 2 * 2 * 2
6 = 2 * *3
Unshared factors = 2, 2, 3
Lastly, multiply the unshared factors and the GCF:
2 * 2 * 2 * 3 = 24
Thus, the least common multiple of 6 and 8 is 24.
Fractions – Vocabulary and Concepts
Consider this fraction:
48
54
The top number – 48 – is called the numerator. It represents the part.
The bottom number – 54 – is called the denominator. It represents the whole.
The bar represents division.
In English, you would read this fraction as “48 out of 54” or “the ratio of 48 and 54.” Here is an example sentence from a word problem for which you would write this fraction:
Out of 54 total fence posts, 48 have been painted.
Simplest Form
Tests like the SAT will always express answer choices in simplest form. Thus, you must know how to reduce fractions. If you have mastered finding greatest common factor, reducing fractions is easy. For example, let’s reduce the following fraction:
48
54
First, we must find the greatest common factor of the numerator and the denominator. Fortunately, in this tutorial, we have already done so. Using the factor tree method, I found that the greatest common factor of 48 and 54 is 6.
Next, we must divide the numerator and the denominator by the greatest common factor:
48/6 = 8
54/6 = 9
In simplest form, 48/54 = 8/9.
Equal Fractions and Least Common Denominator
To add, subtract or compare fractions, you must know how to express two different fractions with a common denominator. Let’s say you wish to add, subtract, or compare the following two fractions:
5
6
7
8
First, you must find the least common multiple of the denominators. Above, we found that the least common multiple of 6 and 8 is 24.
Next, you must express the above two fractions as equivalent fractions with denominators of 24:
5 * 4 = 20
6 * 4 = 24
7 * 3 = 21
8 * 3 = 24
As you can see, the number I multiplied each denominator by to get the new denominator of 24 was also used to change the numerator. You must do the same thing to the numerator that you do to the denominator to get an equivalent fraction.
Mixed Numbers and Improper Fractions
Proper fractions are fractions in which the numerator is smaller than the denominator.
If the numerator is larger than the denominator, we say the fraction is improper. On the SAT, it is acceptable to express your answers as improper fractions in simplest form. However, you should understand the relationship between improper fractions and mixed numbers.
Let us consider the improper fraction 23/4. 23/4 is improper because the numerator – 23 – is larger than the denominator – 4. To express this improper fraction as a mixed number, you should first divide 23 by 4. (Remember, the fraction bar represents division.) 23 divided by 4 is 5 with a remainder of 3.
Next, you should express the quotient – 5 – as the whole number and put the remainder – 3 – over the denominator – 4. 23/4 thus equals 5 ¾.
To change 5 ¾ back into an improper fraction, multiply the whole number – 5 – by the denominator – 4. Then add the numerator – 3 – to the product of 5 and 4. Put this answer over the denominator of 4.
5 * 4 + 3 =
4
23
4
Addition and Subtraction of Fractions and Mixed Numbers
Let’s say you want to add the fractions 5/6 and 7/8.
First, you must express the two fractions with a common denominator. We have already done this above:
5 * 4 = 20
6 * 4 = 24
7 * 3 = 21
8 * 3 = 24
Second, you must add the numerators. The denominator remains the same!
20 + 21 =
24
41 or 1 17/24
24
Subtraction works in a similar fashion. If you want to subtract 5/6 from 7/8, first follow step one above:
7 * 3 = 21
8 * 3 = 24
5 * 4 = 20
6 * 4 = 24
Then subtract the numerators. Again, the denominator remains the same!
21 – 20 = 1/24
24
If you need to add or subtract mixed numbers, the easiest method is to change your mixed numbers into improper fractions and follow the steps above. Thus:
2 ½ – 1 ¾ = (2 * 2 + 1)/2 – (1 * 4 + 3)/4 = 5/2 – 7/4 = (5 * 2)/(2 * 2) – 7/4 =
10/4 – 7/4 = 3/4
Multiplication and Division of Fractions and Mixed Numbers
Multiplying fractions turns out to be a little easier. For one thing, you do not have to find the least common denominator. You will, however, need to know how to reduce fractions. Let’s look at an example:
3 * 8
4 9
The easiest way to multiply these two fractions is to reduce before you multiply. When you are multiplying fractions, you can reduce across the multiplication sign. Notice, for example, that in the problem above, 3 is a factor of 9 (3*3) and that 4 is a factor of 8 (4*2). So, to make this problem easier, I will reduce across the multiplication sign:
1 * 2
1 3
Then I will multiply across the top and across the bottom.
1*2 = 2
1*3 3
To multiply mixed numbers, you only need to perform one extra step: you need to turn the mixed numbers into improper fractions:
2 ½ * 1 ¾ = (2 * 2 + 1)/2 * (1 * 4 + 3)/4 = 5/2 * 7/4 = (5*7)/(2*4) = 35/8 or 4 3/8
Division takes yet another step. You must change the division sign to a multiplication sign and flip the second fraction:
2 ½ ÷ 1 ¾ = (2 * 2 + 1)/2 ÷ (1 * 4 + 3)/4 = 5/2 ÷ 7/4 = 5/2 * 4/7
Now, here, I can reduce across the multiplication sign (4 = 2*2).
5/1 * 2/7 = 10/7 or 1 3/7
*****
If you need more practice working with fractions, there are several excellent resources online. In particular, you may want to try Math Drills.com, which features worksheets – and answer keys – on all of the topics discussed above.
Next week, we will expand upon this topic and discuss rational expressions. In the meantime, see if you can simplify the following fraction:
X2 + 3X + 2
X2 + 4X + 4
