Table of Contents
- Introduction
- Proper vs Improper Fractions
- How to Simplify Fractions
- Fraction Comparison
- Fraction Approximation
- Fraction to Decimals
- Common Fraction Mistakes
Introduction
A fraction is a number that represents a ratio or division between two numbers, a and b, with the corollary that b ≠0.
The number a located on top is called the numerator. The number b located on the bottom is called the denominator.
The denominator indicates the total sum of equal parts that something is divided into, whereas the numerator indicates the number of parts that are taken.
If the numerator and denominator of a fraction are the same number, the fraction is equal to 1.
No fraction can have zero as its denominator because division by zero is not defined. A fraction with zero as its numerator is equal to 0. A fraction with one as its denominator is equal to its numerator.
Proper vs Improper Fractions
A proper fraction is a fraction with the numerator smaller than the denominator; therefore the value of the fraction is less than 1.
An improper fraction is a fraction with the numerator greater than or equal to the denominator; therefore it has a value greater than or equal to 1.
A mixed number is composed of a whole number and a proper fraction. It always has a value greater than 1.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, and add this product to the numerator of the fraction. Place this sum over the denominator to get the finished conversion.
How to Simplify Fractions
Time is a critical constraint during the test. Save time when you perform calculations by reducing fractions to their lowest terms. A fraction has been simplified or reduced to its lowest terms when there are no common factors in its numerator and denominator except for the number 1.
To simplify a fraction, cancel out all common factors in the numerator and denominator.
One trick to verify that a fraction is completely reduced to its simplest form is to write the numerator and denominator out as the product of its primes. Make sure common factors are canceled out.
Fraction Multiplication/Division
Multiplying fractions is straightforward, multiply the numerators to get the numerator of the product, multiply the denominators to get the denominator of the product.
Remember to always simplify. With practice you’ll find that you will be able to solve problems in reduced form much faster, sometimes even able to do all of the necessary computations with mental math.
Dividing a fraction is similar to multiplying. Simply flip the second fraction so you are multiplying by the reciprocal.
Fraction Addition/Subtraction
Fractions can only be added or subtracted if they have the same denominator. Alas, there is no shortcut around this.
In the above example, since the denominators 8 and 3 are not the same, find the least common denominator, which is 24. (8 x 3 = 24 ). Multiply both numerator and denominator by the same factor such that the denominator is equal to the least common denominator, and subtract as normal. The process is identical for fraction addition.
Fraction Comparison
It’s fairly common to come across a step where you must quickly compare two fractions. A neat little shortcut to do this is to cross multiply. This will allow you to quickly determine which fraction is greater and which one is smaller.
Example
The answer is 32/63 is greater. What we are essentially doing here is find the least common denominator. The shortcut lies in dropping the common denominator because we already know it’s the same and cross multiplying to find which fraction has the greater numerator.
Fraction Approximation
But sometimes fractions can be too cumbersome and complex to cross multiply. In these cases, approximation may be the best tool to quickly solve a problem. To approximate fractions, adjust only the numerator in your entire set of fractions or adjust only the denominator in your entire set of fractions. However, if both numerator and denominator need to be moved, try to move one up and one down. This minimizes the chance of error in approximation.
Special tip: if you can’t decide whether to approximate the numerator or denominator, always approximate the larger number because this will introduce less error.
Fractions to Decimals
It will be necessary to quickly convert between fractions and decimals on the GMAT. Memorize the list of common conversions in the table below. This will dramatically increase your speed in performing calculations.
Common Fraction Mistakes
- Adding/Subtracting Fractions with Different Denominators
To add or subtract, fractions must always share the same denominator. Never add denominators. Always add the numerator while keeping the bottom denominator in common
- Splitting Denominators
Sometimes splitting denominators is temping. The GMAT has designed certain problems to look this way. Don’t be tempted. This will give you the wrong answer.
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