MATH 27

FRACTIONS - ADDITION & SUBTRACTION

 

FRACTION BASICS:

 

Fractions are part of a whole. Consider a full pallet consisting of 24 cases of paper. If seven cases are removed, then the pallet has 17 cases remaining out of a total of 24. Only a fraction of a whole pallet remains. A fraction may also be written as .

 

The top number (17) in the fraction is called the numerator, and the bottom number (24) is called the denominator.

 

If two more whole pallets are added to the partial pallet, how many cases are there? How many whole pallets are there?

 

Since each pallet has 24 cases, there are 2 times 24, or 48 cases in the two added pallets. Add to this the 17 origional cases for a total of 65 cases.

 

There are between two and three whole pallets. Since a whole pallet is 24/24, there is 65/24 cases. This may be expressed as a mixed fraction also: pallets. This is found by dividing the numerator by the denominator, and the remainder is the numerator of the fractional part: with a remainder of 17/24

 

ADDING FRACTIONS - WITH COMMON DENOMINATORS:

 

As in the above example, when the wholes are divided into the same number of parts (denominators are equal), add the numerators over the common denominator:

 

 

If three partial pallets containing 20, 13, and 18 cases are consolidated, how many whole cases are there? First: You cannot have more than __________ whole pallets (Hint: there are 3 partial pallets). Express answer as an improper fraction, and as a mixed number. Show work.

 

 

 

 

ADDING FRACTIONS - WITH DIFFERENT DENOMINATORS:

 

Fractions with different denominators, such as 2/3 & 6/5, cannot be added directly. They must be first converted to fractions that have the same denominator.

 

For example: Two partially filled pallets (one that contains six packs and the other holds individual bottles) must be inventoried. One can hold a total of 60 six packs of juice (360 bottles), but it only contains 37 six packs (37/60 full), and the other pallet can contain 360 bottles, but it only contains 223 bottles (223/360 full). You cannot add 37 and 223 and get a meaningful number. Either everything must be expressed in six packs or in bottles. If expressed in bottles, 37 six packs is: 37 x 6 = 222 bottles. The total number of bottles is 222 + 223 = 445, or 445/360 pallets. This fraction can be reduced (by removing factors common to the numerator and denominator) to 89/72 or pallets

 

REDUCING FRACTIONS TO LOWER TERMS:

 

A fraction may be reduced to lower terms if there is a common factor(s) to both the numerator and denominator.

 

For example:

 

The common factor was 8, and since 8/8 = 1 they can be eliminated from the fraction.

 

Examples: Always remove the largest common factor to the numerator & denominator.

 

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RAISING FRACTIONS TO HIGHER TERMS (building up fractions):

 

A fraction may be built to higher terms by multiplying both its denominator and numerator by the same quantity, thus multiplying it by "1". Because it is multiplied by "1" it is equivalent to the origional fraction.

 

For example:

 

 

 

Examples: Always multiply the numerator and denominator by the same quantity to build fractions.

 

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CONVERTING FRACTIONS TO ONES WITH COMMON DENOMINATORS:

 

To convert fractions to ones with common denominators, one must determine the lowest common denominator (LCD). To do this factor each denominator. The LCD must contain each factor, the greatest number of times that it occurs in any single factored grouping.

 

Example: Find the LCD for:

 

The LCD must contain 2· 2· 3· 3· 5 = 180

 

Use the process of building up fractions to convert each fraction to one with the same LCD.

 

 

Example: Find the LCD for:

 

The LCD must contain: 22 · 32 · x3 · y2 Þ 36x3y2

 

Use the process of building up fractions to convert each fraction to one with the common LCD.