Number Properties Section 4: Fractions

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Note: This is an introductory section for beginners or intermediates with the exception of the final cross multiplication section.

A fraction is one number divided by another number. It represents an unsolved division, such as 3/5.

• The numerator is the top number that represents the number of parts that are selected.
  
• The bottom number is the denominator. The denominator represents the number of equal parts into which an entity has been divided.

For example, if a garden is divided into 5 equal plots, 3 of the plots is 3/5 of the garden.

• A fraction that has 0 as its denominator (e.g., 5/0) is infinitely large and undefined (not a real number or integer).
  
• If the numerator is zero, then the fraction equals zero (e.g. 0/5 = 0).
  
• If the fraction has a numerator equal to the denominator (e.g., 5/5), the fraction is equal to 1.

Mixed Numbers are numbers that are equal to an integer plus a fraction.

The number 4 2/3 is the integer 4, plus the fraction 2/3. Any mixed number can be written as a fraction, and any fraction greater than 1 can be written as a mixed number.

• To express 4 2/3 as a fraction:
  1) multiply 4 × 3
  2) add the numerator to this product, 12 + 2 = 14
  3) divide by the denominator: 4 2/3 = 14/3
  
• To convert the fraction 17/5 into a mixed number:
  1) divide by the denominator (17 divided by 5 is 3 with 2 remaining)
  2) add the remainder over the denominator: 17/5 = 3 2/5
  

Convert 4 5/7 into a fraction.

Convert 79/9 into a mixed number.

A fraction that has a common factor in both numerator and denominator is equal to the fraction with the common factor canceled. The fraction 6/10 is equivalent to the fraction 3/5 since the common factor 2 occurs in both numerator and denominator of 6/10.

In fact, the following fractions are all equivalent:

3/5 = 6/10 = 9/15 = 12/20

A fraction that has no common factors in the numerator and denominator is said to be expressed in lowest terms.

A fraction with a negative numerator or denominator is equivalent to a negative fraction, that is, -3/5 = 3/-5 = - (3/5). If both numerator and denominator are negative, the fraction is positive, that is, -3/-5 = 3/5.

Reducing Fractions to Lowest Possible Terms

	Always solve fractions in the lowest terms The GMAT always puts fractions in the lowest possible form. This means that with any fraction question, try to put it in the lowest form before going to the answer choices. The reason for this is obvious; if they didn't use the lowest form it would create confusion about the right answer because there could be two answers on a multiple choice test that are the same value. (1/2 and 2/4).

Use the Greatest Common Factor (GCF) to bring fractions down to the lowest possible form. Divide the numerator and denominator by the GCF to get the lowest possible amount.

Reduce 275/525 to lowest possible terms.

275 = 25 × 11
525 = 25 × 21

Cancel out the 25's to get 11/21

Express 26/16 as a mixed number in lowest terms.

To multiply fractions:

1. Cancel out any common factors that appear in both numerators and denominators.
   
2. Then, multiply all numerators to form one numerator and all denominators to form one denominator. This final fraction may then be written as a mixed number, if desired.
   

To divide fractions, say (x/y)/(a/b):

1. Invert the divisor (the fraction a/b).
   
2. Then, multiply the two fractions, i.e., (x/y)/(a/b) = (x/y) × (b/a).

Example:

(5/6) × (7/2) = (7× 5)/(6 × 2) = 35/12

	Shortcut: Canceling Fractions You can speed up multiplying fractions by canceling out any common factors in the numerator and the denominator. Look at this example: 4/5 × 5/8 = The fives cancel out and the 4 cancels out and you are left with 1/2.

To subtract one fraction from another, we simply add a negative fraction to a second fraction. Consequently, the rules for adding and subtracting are the same. The first step is to write the fractions such that each fraction has the same denominator. Then add or subtract the numerators. Then simplify.

1/3 + 1/4 = 4/12 + 3/12 = 7/12

To write all fractions with the same denominator, a quick choice is to multiply all denominators together. However, this may give a rather large denominator. To avoid a large denominator, we could find the least common denominator (LCD); it is the least common multiple (LCM) of all the denominators.

NOTE: Splitting the numerator is OK

(8 + 10) 3

can be split into

8 3 + 10 3

(8 +10)/3 can be split into 8/3 + 10/3

(3x + y)/2x = 3x/2x + y/2x

You cannot, however, split the denominator:

WRONG:

3 (8 + 10)

does not equal

3 8 + 3 10

x y 2x z

=

x y × z 2x

Cross multiplication is a simplified method of solving for two fractions that are set equal to each other. Given the problem:

x 9

×

4 6

1) Multiply x diagonally down by 6 to get 6x

2) Multiply 9 diagonally up by 4 to get 36

3) Then we solve for x and get 6x = 36 or x = 6.

Then we solve for x and get 6x = 36 or x = 6. Although this is one way to solve such an equation, we can solve it more easily by using cross multiplication. In cross multiplication we multiply the numbers diagonally across from each other.

In the previous problem, we would simply multiply (9 × 4) and set it equal to 6 times x.

So we have:

(9 × 4) = 6x
36 = 6x
6 = x

If 17/x = 85/15, then what is x equal to?

	Shortcut: Fractions Comparison If you want to determine which fraction is bigger 2/3 or 400/700? Multiply the numerator of one by the denominator of the other 2 × 700 = 1400 3 × 400 = 1200 1400 is larger, which means 2/3 is greatest.

Contents of Number Properties Chapter: Table of Contents   Section 1: Number Rules   Section 2: Consecutive Numbers   Section 3: Divisibility   Section 4: Fractions   Section 5: Decimals   Section 6: Exponents   Section 7: Roots & Radicals   Section 8: Sample Questions   Decimals

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