Fractions are used all the time in daily life. Half a sandwich (1/2), two-thirds of a cup of water (2/3), the third quarter of the football game (3/4) and so on. A fraction is essentially a division problem. For instance, how can I divide ONE pie into SIX pieces? It's simple, each person gets 1/6 of the pie. How can I split 5 dollars between four friends? 5/4, or 1 1/4 dollars each!
Oftentimes, a fraction represents a number less than one. For example, 5/7, 10/11, or 100/999. A fraction that represents a number less than one is called a common fraction. A common fraction has 2 parts: the top, or numerator; and the bottom, or denominator. Remember, if the numerator (top) is less than the denominator (bottom), the fraction's value is less than one and it's a common fraction.
Occasionally, you'll see a fraction where the numerator is greater than the denominator. This is called an improper fraction. Examples: 5/3, 7/4, 171/113.
Finally, the last common type of fractional number, a mixed fraction. A mixed fraction has 2 parts, a whole number and a common fraction. For example, 5 1/7 ("five and one-seventh") or 2 2/9 ("two and two-ninths").
Any mixed number can be written as an improper fraction. The reverse is also true, any improper fraction can be written as a mixed number.
1. Set up a division problem (num = numerator, denom = denominator):
______
denom ) num
2. The whole number is the whole number answer to the division problem. The remainder is the numerator of the new fraction. (The new fraction has the same denominator as before.)
Example:
17/12
__1
12) 17
-12
--
5
The mixed number will then be 1 5/12. 1 was the whole number, 5 was the remainder, and 12 was the denominator (same as before).
1. Take the whole number and multiply it by the denominator of the fraction.
2. The the numerator of the fraction.
3. Add the numbers from (1) and (2) together. This is the numerator of the improper fraction. The denominator is the same as before.
Example:
5 7/9 5 * 9 = 45 45 + 7 = 52
The improper fraction is 52/9. (Same denominator as before, see?)
The rules say that you can only add fractions that have the same denominator. This is why we always "find the common denominator" between two fractions before adding/subtracting.
Let's say we want to add 7/8 and 5/12. The denominators are 8 and 12. We need to think of a number that is a multiple of both 8 and 12. 24 works, so let's call that our common denominator.
Now we want to make both denominators "look like" our common denominator. So how do we turn 8 into 24? Multiply by 3! But we can't just multiply the bottom by 3. The rules say that anything we multiply on the bottom, we have to multiply on the top as well. (Otherwise, we'd change the number.) So let's do it:
7/8 ==> (7*3) / (8*3) ==> 21 / 24
How do we make 12 look like 24? Multiply by 2!
5/12 ==> (5*2) / (12*2) ==> 10 / 24
Now we can add the numerators together (and keep the denominators the same):
21/24 + 10/24 = ( 21 + 10 ) / 24 ==> 31 / 24
The answer is 31/24. This answer is perfectly acceptable, but you may want to write a mixed number instead. In that case, it would be 1 7/24.
Subtraction of fractions is exactly the same, except you subtract on the last step:
5/7 - 1/3 = ???
common denominator: 21 (a multiple of 3 and 7)
(5*3)/(7*3) - (1*7)/(3*7) ==> 15/21 - 7/21 (15 - 7) / 21 = 8/21
The answer is 8/21 !!
Multiplying is usually considered "easier" than addition or subtraction. To multiply two fractions, just multiply straight across.
2/3 * 7/6 = (2*7) / (3*6) = 14/18
Of course, there is one more thing. This fraction is not in "lowest terms." To get a fraction in lowest terms, we can reduce it by dividing the top and bottom by the same number. First, we have to find a number that goes into the numerator and denominator evenly. 2 goes into both 14 and 18 evenly, so let's divide:
(14 / 2) / (18 / 2) = 7/9
The only number that goes into 7 and 9 evenly is 1, so we're done! The fraction is completely reduced, or in lowest terms.
Division is almost the same as multiplication -- with one twist! To divide two fractions, we flip the right side and multiply. "Flip" means switch the numerator and denominator (2/3 would become 3/2).
Example:
(1/3) / (5/6) ==> (1/3) * (6/5) 1/3 * 6/5 = 6/15 = 2/5
It's as simple as that.
What if you have to add, subtract, multiply, or divide using mixed numbers? The answer is, don't use mixed numbers... first convert to improper fractions and then proceed with your calculation.
Example:
2 3/5 + 1 1/10 = ?? 2 3/5 = (2*5 + 3) / 5 = 13/5 1 1/10 = (1*10 + 1) / 10 = 11/10 13/5 + 11/10 (13*2)/(5*2) + 11/10 = 26/10 + 11/10 = (26 + 11)/10 = 37/10
The answer is 37/10, or 3 7/10 as a mixed number. Notice that after we converted to improper fractions, we still had to find the common denominator and all that.