Fractions - GMAT Math Study Guide

Definitions

• Numerator - The top number of a fraction.
  For example, in the fraction ²/₅, the numerator is 2.
• Denominator - The bottom number of a fraction.
  For example, in the fraction ²/₅, the denominator is 5.
• Proper Fraction - A fraction whose numerator is less than its denominator.
  For example, the fraction ³/₄ is a proper fraction since the numerator is less than the denominator (i.e., 3 < 4). But, the fraction ⁴/₃ is not a proper fraction since the numerator is greater than the denominator (i.e., 4 > 3).
• Improper Fraction - A fraction whose numerator is greater than or equal to its denominator.
  For example, the fraction ⁷/₆ is an improper fraction because the numerator is greater than the denominator (i.e., 7 > 6). But, the fraction ⁶/₇ is not an improper fraction since the numerator is less than the denominator (i.e., 6 < 7).
• Mixed Fraction (aka Mixed Number) - The combination of a whole number and a proper fraction.
  For example, the fraction 1³/₈ is a proper fraction since it combines a whole number (i.e., 1) and a proper fraction (i.e., ³/₈).
• Equivalent Fraction - A fraction that has the same numerical value (i.e., decimal value) as another fraction (although the numerator and denominator of the equivalent fractions may not be the exact same number).
  For example, the following fractions are all equivalent fractions since their values are equal to 0.5: ¹/₂ = ²/₄ = ³/₆ = ⁴/₈ = ⁵/₁₀
• Common Denominator - A denominator that is equivalent between two (or more) fractions.
  For example, the fractions ⁷/₉ and ⁵/₉ share a common denominator (i.e., 9) while the fractions ⁷/₁₀ and ⁷/₁₁ do not share a common denominator since 10 ≠ 11
• Reduced Form Fraction - A fraction whose numerator and denominator share no common factors. Said differently, a fraction whose numerator and denominator are smaller than any other equivalent fraction.
  For example, the fraction ¹/₂ is in reduced form since it cannot be simplified any further while ²/₄ can be reduced further since the number 2 is a common factor.

The Idea of a Fraction

In order to conceptualize the concept of a fraction, it can be helpful to see the same fraction represented in different ways. The following shows some of the ways the fraction ³/₈ can be conceptualized.

Graphical Representation

One way to understand a fraction is as a part of a whole. For example, the fraction ³/₈ can be thought of as representing 3 pieces of pizza from an 8 piece pizza.

Number Line Representation

Since a fraction is a number, it occupies a place on the number line. Most fractions are not integers and as a result, most fractions occupy a place on the number line between integers.

Decimal Representation

A fraction can be represented as a numerical value by dividing the numerator by the denominator. For example, the fraction ³/₈ represents the numerical value 0.375 since 3 divided by 8 is 0.375

Percent Representation

A fraction can be represented as a percent by dividing the numerator by the denominator and then multiplying by 100. For example, ³/₄ = 75% and ¹/₂ = 50%

Equivalent Fractions

Fractions whose numerators and denominators do not have the exact same numerical value are not necessarily unequal. As noted above, ¹/₂ = ²/₄. The reason equivalent fractions exist is that the numerator of one fraction and the denominator of that same fraction share a common factor that, when canceled, cause the fraction to have the same value in both the numerator and denominator as another fraction. Consider the following example:

In this example, the pair of equivalent fractions exists because the numerator and denominator of one fraction (i.e., the right one in this case) share a common factor (i.e., 4). When this common factor is canceled, the values of the numerator and denominator for the two fractions become equal.

In the above example, the values of the numerator and denominator of both fractions equaled each other when the fraction on the right was multiplied by ⁴/₄ = 1. It is entirely legal to multiply by ⁴/₄ since ⁴/₄ = 1 and multiplying any number by 1 does not change its value. In general, the fastest way to determine if two fractions are equivalent is to take out as many common factors from each fraction as possible and then see whether the values of the numerator and denominator of each are equal. This involves the process of simplification.

Simplifying Fractions

The same numerical value can be expressed by an infinite number of fractions. For example:

In each case, the next fraction value is simply the fraction ¹/₄ multiplied by one larger numerator and denominator (i.e., multiply by ²/₂, then multiply by ³/₃). Consider the following table:

The fractions ¹/₄, ²/₈, ³/₁₂, ⁴/₁₆, and ⁵/₂₀ are equal in value because multiplying or dividing both a fraction's numerator and a fraction's denominator by the same value (i.e., 1 in this case since ²/₂ = 1, ³/₃ = 1, ⁴/₄ = 1, etc.) does not change the value of the fraction. The reason that multiplying the above fractions by ²/₂ does not change the value of the fraction is that ²/₂ = 1 and any number multiplied by itself is itself (i.e., 1x = x).

A fraction is simplified by undoing the process undertaken in the above example when we generated an infinite number of equal fractions. In other words, instead of multiplying by ²/₂, ³/₃, or ⁴/₄ to create equivalent fractions, we divide both by ²/₂, ³/₃, or ⁴/₄ since: (1) the numerator and denominator both have a factor in common (e.g., 2, 3, or 4) and (2) dividing by ²/₂, ³/₃, or ⁴/₄ does not change the value of the fraction since ²/₂ = 1.

Reduced Form

A fraction is in reduced form if the numerator and denominator share no common factors. Applying this to the above example, ³/₁₂ is not in reduced form since both the numerator and denominator share a common factor of 3. Likewise, the fraction ⁵/₂₀ is not in reduced form because the numerator and denominator share a common factor of 5.

Consider the following example:

Consider another example:

Comparing Fractions

In order to compare the value of fractions, there are two options:

Method 1: Convert to Decimal

Each fraction can be converted into a decimal through dividing the numerator by the denominator. Once in decimal format, the value of each fraction can be compared.

Method 2: Cross Multiply

Another means to compare fractions is to cross multiply such that there is a common denominator. With a common denominator between the two fractions (i.e., with the same number in the denominator), the numerators can be compared and the fraction with the largest numerator is the largest fraction.

Which fraction is larger: ⁵/₇ or ⁷/₉?

Multiplying & Dividing Fractions

Multiplying Fractions

In order to multiply two fractions, begin by multiplying the numerators together. Then, multiply the denominators together.

Tip: Simplify First

While the method above gives the correct answer, the process can be simplified considerably (especially when multiplying without a calculator) through a technique known as canceling, which is the process of eliminating common factors. This process is effectively the process of simplification, described above. Consider the same example, worked using the new technique:

The 3 in the numerator of the right fraction and the 3 in the denominator of the left fraction cancel each other out.

The 2 in the numerator of the left fraction shares a common factor with 14 (=7*2) in the denominator of the right fraction. Cancel the 2s out.

Dividing Fractions

In order to divide two fractions, invert (i.e., flip) the second fraction (i.e., divisor) and then multiply the two fractions together.

Complex Fraction Equality

Consider the following example:

Adding & Subtracting Fractions

Fractions can be added or subtracted. There are five steps to this process:

1. Check to ensure the fractions have a common denominator. Adding fractions with different denominators produces the wrong answer.
2. Find a common denominator (if necessary). The smallest common denominator can be found through the least common multiple.
3. Alter each fraction so that each fraction shares a common denominator (i.e., ensure each fraction has the same number in the denominator).
4. Add the numerators while leaving the denominator unchanged. When you add fractions, the denominator does not change.
5. Simplify the added fraction.

The above five steps are best understood through an example.

1. The fractions do not have a common denominator (i.e., 4 ≠ 5).
2. A common denominator is 20.
3. The fractions must be altered in such a way that the value does not change but the denominator does change. Through multiplying the left fraction by ⁵/₅ = 1 and through multiplying the right fraction by ⁴/₄ = 1, each fraction will share the common denominator 20.
   
4. Add the numerator (i.e., 15 + 16) while leaving the denominator unchanged (i.e., keep the denominator as 20).
   
5. Simplify the added fraction.
   
   

Common Mistake: Adding Fractions With Different Denominators

In order to add two fractions, they must share a common denominator.

Common Mistake: Adding The Denominators

When adding fractions, do not add the denominators. Instead, add the numerator while keeping the common denominator the same value.

Common Mistake: Splitting The Denominator

Splitting the denominator of a fraction will result in a wrong answer.

However, you can split the numerator.

Types of GMAT Problems

1. Adding and Subtracting Percentages

   To add and subtract fractions, first, find a common denominator. Then, the fractions must be changed to have the common denominator. Finally, the numerators can be added or subtracted while the denominator remains the same.

   

   A)
   B)
   C)
   D)
   E)

   Correct Answer: E

   1. Notice that these are mixed fractions. We can convert them into improper fractions by using addition.
      
      
   2. Add the improper fractions.
      
   3. To convert to a mixed fraction, divide 107 by 8. Since 8(13) = 104 and 107-104 = 3, 8 goes into 107 thirteen times with a remainder of 3. The remainder is placed over the denominator, 8.
      
2. Multiplication and Division

   To multiply fractions, multiply the numerators to get the new numerator and then multiply the denominators. Division of fractions can be treated in a similar manner by flipping the second fraction and then multiplying the fractions. A time saving trick can be used by 'canceling out' numbers that are in both the numerator and denominator.

   

   A)	13/15
   B)	12/13
   C)	11/12
   D)	17/18
   E)	10/13

   Correct Answer: B

   1. Calculate the numerator.
      
   2. Calculate the denominator.
      
      
3. Comparing Fractions

   A trick to comparing fractions is using cross multiplication. Multiply the numerator of the first fraction and the denominator of the second and record the number under the first fraction. Then multiply the second numerator and first denominator and record this number. The number that is larger corresponds to the larger fraction (i.e., if the number under the first fraction is larger, then the first fraction is larger). However, in problems with more than two fractions, finding a common denominator may be easier than doing multiple comparisons using cross multiplication.

   Which of the fractions below is the largest?

   A)	55/100
   B)	2/50
   C)	8/20
   D)	12/25
   E)	5/10

   Correct Answer: A

   1. Since the denominators of the fractions are not the same, you cannot compare them in their present form. In order to compare fractions, they must have the same denominator or they must be converted to decimals.
   2. Compare as Fractions With a Common Denominator
      1. In order to use the same denominator, you need a common denominator. 100 is the lowest common denominator. Convert each fraction to be over 100.
         (55/100) = (55/100)
         (2/50) = (4/100); [multiply by 2/2]
         (8/20) = (40/100); [multiply by 5/5]
         (12/25) = (48/100); [multiply by 4/4]
         (5/10) = (50/100); [multiply by 10/10]
      2. Comparing the numerators reveals that 55/100 is the largest of the fractions.
   3. Compare as Decimals
      1. (55/100) = .55
         (2/50) = .04
         (8/20) = .4
         (12/25) = .48
         (5/10) = .5
      2. Comparing the decimals reveals that .55 or 55/100 is the largest of the fractions.