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A fraction is in its lowest terms when the numerator and denominator no longer have a common factor, a number that they share. A factor is a part of a number, for example 3 and 2 are factors of 6 because you can multiply them together to get the number 6. Every fraction has to be in its lowest terms in order to be a correct answer. The fraction is not in its lowest terms because 2 can be divided into both numbers to make them smaller. When you divide 2 into the numerator and denominator you get the lowest term fraction of .
EXAMPLE: Find the lowest fraction (reduce the fraction) of . = There are no other numbers which will divide into both 2 and 5 so your fractions is reduced to lowest terms.
EXAMPLE: Find the lowest fraction (simplify) of . = There are no other numbers which will divide into both 1 and 6 so your fractions is reduced to lowest terms.
Two rules: Divide the numerator and the denominator by the same number.
Get the numerator and denominator as small as possible.
Practice: Reduce: 1. ; 2. ; 3.
2.5 Equivalent Fractions
If the numerator and the denominator are multiplied or divided by the same number, they are called equivalent fractions. Equivalent means that the two fractions have the same value, they are equal. For example: = = x = or ö =
How do you know if two fractions are equivalent? Cross multiplication is a quick way to tell.
Are these fractions equal? = Cross multiply 1 x 13 = 4 x 3; 13 Ø 12, not equal
= Cross multiply 4 x 115 = 23 x 20; 460 = 460, equal.
Practice: 1. Does = ? 2. Are and equivalent?
2.6 Comparing Fractions
Comparing fractions with the same denominator requires comparing just the numerators.
> because 3 > 2
Comparing fractions with different denominators requires finding a common denominator. The lowest common denominator (least common denominator) (LCD) is the smallest multiple of the denominators.
For example, the LCD of and is 36 because the least common multiple.of 12 and 18 is 36.
= = x =
= = x = ; <
Practice: 1. Compare these like denominator fractions: and .
2. Compare these unlike denominator fractions. and .
2.7 Improper Fractions and Mixed Numbers
An improper fraction has a numerator larger than or equal to the denominator, such as or . In other words, it is top-heavy. A n improper fraction or a mixed fraction can show the same amount. For example 1 = , shown here:
In a formula, improper fractions are easier to use instead of mixed fractions (mixed numbers).
Mixed Fraction: What is: 1 + 2 1/4 ÿ ? ÿ
ÿ Is it: 1+2+1/4 ÿ = 3 1/4? û Or is it: 1 + 2 x 1/4 ÿ = 1 1/2? Improper Fraction: What is: 1 + 9/4 ÿ ? ÿ It is: 4/4 + 9/4 = 13/4 ÿ Example: For everyday use, people understand mixed fractions better. But it is easier to say "I ate 2 1/4 sausages", than "I ate 9/4 sausages"
What about when the numerator and denominator are equal? For example 4/4 ? It is the same as a whole number, as one, but it is written as a fraction.
To convert an improper fraction to a mixed fraction, follow these: -
Divide the numerator by the denominator
Write down the whole number answer
Then Write down any remainder above the denominator. ÿ Example: Convert to a mixed fraction.
Divide: 9 ö 4 = 2 with a remainder of 1 Write down the 2 and then write down the remainder 1 above the denominator 4, like this: 2 To convert a mixed fraction to an improper fraction, follow these:
Multiply the whole number by the denominator.
Add that to the numerator
Write the answer on top of the denominator. ÿ Example: Convert 2 to an improper fraction.
Practice: 1. Which one of these fractions is a improper fraction? ( ( (
2. Change this mixed number into an improper fraction. 1 (
2.8 Adding and Subtracting Fractions
To add fractions use these steps:
Step 1: The denominators have to be are the same
Step 2: Add or subtract the numerators and put the answer over the denominator in step 1
Step 3: Simplify (reduce) the fraction (if needed).
Example 1: ( + ( ( - (
Step 1. The bottom numbers are already the same.
Step 2: Add or subtract the numerators and put the answer over the denominator. or or 0.
Step 3. Simplify the fraction: You can divide 2 into both numbers so reduced it is ÿ
Example 2: ( + ( ( - (
Step 1: The bottom numbers are different. We can't add them like this:
ÿ + = ? ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ
We need to have the same denominator. We can multiply the top and bottom of the first fraction = . The denominators are the same now, so we can go to step 2.
Step 2: Add or subtract the top numbers and put them over the same denominator:
ÿ + = orÿ for subtraction ÿ ÿ ÿ ÿ ÿ ÿ Step 3: Simplify the fraction: 3/6 is the same as 1/2:
ÿ 2/6 + 1/6 = 3/6 = 1/2 ÿ ÿ ÿ ÿ ÿ ÿ And we have the answer. Adding we get a reduced and subtracting we get 1/6
Practice 1. Add the same denominator fractions and .
2. Subtract the same denominator fractions minus .
3. Add the fractions and .
4. Subtract the fractions and .
2.9 Adding and Subtracting Mixed Numbers
A mixed fraction (mixed number) is a whole number and a proper fraction added together, such as 1 (. There are one and one fourth red boxes.
To make it easy to add and subtract them, just change to improper fractions. ( Remember an improper fraction is "top-heavy") Our squares above would be : in the solid red box and red in the other box.
The best way to add mixed fractions is:
Change them to improper fractions
Then change back to a mixed fraction.
EXAMPLE: What is 2 + 3?
Change to improper fractions: 2 =
Common denominator of 4: =
Change back to Mixed Fractions: = 6
To subtract follow the same method, but subtract instead of add:
Example: What is 15 - 8 ?
Convert to Improper Fractions: 15 =
Common denominator of 12: =
Now Subtract: - =
Convert back to Mixed Fractions: = 6
PRACTICE: 1. 3 + 1 = ? 2. 3 - 1 = ?
2.10 Multiplying Fractions
There are 3 simple steps to multiplying fractions:
Multiply the numerators.
Multiply the denominators.
Simplify the fraction.
Example 1 x
Step 1. Multiply the top numbers: x = =
Step 2. Multiply the bottom numbers: x = =
Step 3. Simplify the fraction: =
Example 2 2 x 3
Step 1. Change to improper fractions 2 = =
3 = =
Step 2 Multiply the top numbers: x = =
Step 3. Multiply the bottom numbers: x = =
Step 3. Simplify the fraction: = 286 ö 35 = 8
PRACTICE: 1. x = ? 2. 4 X 2 = ?
2.11 Dividing Fractions
It is impossible to divide fractions. Why? When fractions need to be divided you have to take the `reciprocal' of the second fraction and then multiply the fractions.
What is reciprocal? The dictionary says that when there is a fraction and the fraction is turned upside down , that is the reciprocal of the fraction.
EXAMPLE: The reciprocal of is . The fraction is just flipped over.
Division of fractions looks like this: ö = x = = lowest terms
Take the reciprocal of the second fraction and multiply.
Division of mixed fractions (numbers) 4 ö 2
Change to improper fractions 4 = =
2 = =
Take the reciprocal, multiply and reduce to lowest terms
x = = = 2 lowest terms
PRACTICE: 1. ö 16 2. ö
3. 1 ö 3 4. 1 ö
Chapter 2 Fractions Quiz
1. Janey had a whole pie. She cut it into 12 pieces. She ate one piece and gave Karen and Candy 2 pieces. What is the fraction of the pie that is left?
2. My recipe calls for cups of white flour and 6 cups of whole wheat flour. How much flour do I need in total of my recipe?
3. Which fraction is larger or ?
4. What is the improper fraction for 3 ?
5. What is 5 + 4 ?
6. What is the answer to 5 ö 2 ?
7. Sean swam 2 miles at swim team practice. If Becky swam 1 times as far as Sean, then how many miles did Becky swim?
8. Charlie works at a livery and uses a 60 pound bag of oats to feed the horses. If each horse gets pounds of food, then how many horses can Charlie feed with one bag of oats?
9. What is the reciprocal of 3 ?
10. List these fractions in order from smallest to largest.