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Subtracting Like and Unlike Denominators

Do unlike denominators disturb you? Go to a quiet place, and letís examine the situation.

I. Subtract Fractions with Like Denominators
II. Fractions with Unlike denominators

Reminder: Denominators must be the same before subtraction can take place.
I. Subtract Fractions with Like Denominators

1) Subtract the numerators (The denominators are NOT subtracted.)
2) The denominator remains the same.
3) Reduce, if necessary

Example 1) 8/16 minus 3/16
1. Subtract numerators: 8 - 3 = 5

2. The denominator remains the same: 16
Therefore: 8/16 - 3/16 = 5/16

3. Reducing is not necessary because 5 and 16 do not have any common factors.

Example 2) 14/25 minus 9/25
1. Subtract numerators: 14 - 9 = 5

2. The denominator remains the same: 25
Therefore, 14/25 - 9/25 = 5/25

3. Reducing is necessary because 5 and 25 have at least one common factor.
5/25 = 1/5 (If necessary, refer to article on reducing at the end of this article.)
Therefore: 14/15 - 9/25 = 5/25 = 1/5

Now, this is probably what you have been waiting for:
II. Fractions with Unlike Denominators
1. Find the least common multiple between the two denominators. Most commonly referred to as the least common denominator (LCD)
2. Find the equivalent fraction for each fraction using the LCD.
3. Subtract the numerators (The denominators are NOT subtracted.) The denominator remains the same.
4. Reduce, if necessary

Example: 2/4 minus 8/20
1. Find the least common denominator (LCD) : To find the LCD, use prime factorization. (refer to LCD article) Another way is to count by 4ís and count by 20ís. The first number they have in common is the LCD. One more way is to simply multiply the denominators. The result is not necessarily the lowest denominator,but it will work. Right now, let us use prime factorization method. The prime factors for 4 are 2 * 2. The prime factors for 20 are 2*2*5. The LCD = 2*2*5 =20. (please refer to the Prime Factorization article for a more detailed explanation.)

2. Find the equivalent fraction for each fraction using the LCD.
Since the new dominator is 20. New numerators are needed. 2/4 = ?/20
Ask yourself, the denominator,4, times what number equals 20. It is 5. Since, the denominator was multiplied by 5, the numerator must be multiplied by 5 too. Thus the equivalent fraction for 2/4 using 20 as the denominator is 10/20. The second fraction in this problem already is 20. Therefore, the problem now looks like this: 10/20 - 8/20

2. Subtract numerators.
10 - 8 = 2

3. The denominator remains the same: 20
Therefore, 10/20 - 8/20 = 2/20

4. Reducing is necessary because 2 and 20 have at least one common factor.
2/20 = 1/10 (If necessary, refer to article on reducing at the end of this article.)

Therefore: 2/4 - 8/20 = 10/20 - 8/20 = 2/20 = 1/10


By the Way: What would change if we found the LCD by simply multiplying the denominators? Well, it would not necessarily be called the LCD. It would only be considered a common denominator. Let us try it. 2/4 minus 8/20

1. Find the common denominator: 4 X 20 = 80

2. Find the equivalent fraction for each fraction using the LCD:
Ask yourself, the denominator,4, times what number equals 80. It is 20. Since, the denominator was multiplied by 20, the numerator must be multiplied by 20. Thus, the equivalent fraction for 2/4 using 80 as the denominator is 40/80. Next, 8/20 = ?/80. Ask yourself, the denominator,20, times what number equals 80. It is 4. Since, the denominator was multiplied by 20, the numerator must be multiplied by 4. Thus, the equivalent fraction for 8/20 using 80 as the denominator is 32/80.
Now, the problem reads: 40/80 - 32/80
Continue with steps 3 and 4.

Therefore: 2/4 - 8/20 = 40/80 - 32/80 = 8/80 = 1/10

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