Fundamentals of Mathematics

Fractions

Frustrations with Fractions

If fractions get you frustrated, you are not alone. Many students are wary when it comes to working with fractions. We are here to help you get more comfortable!

Simply put, a fraction is used to describe a part of a whole. The numerator (top) refers to the part and the denominator (bottom) describes the whole. In the fraction ²⁄₃, the numerator is 2 and the denominator is 3.

Equivalent Fractions

Two fractions that describe the same part of a whole are considered equivalent fractions. If two friends cut a pie in half, they can each take ¹⁄₂ of the whole. What if they both cut their piece of the pie into two pieces? Now, they each have two pieces of the pie, which is in four pieces. So, they each have ²⁄₄ of the pie.²⁄₄ and ¹⁄₂ are equivalent fractions.

Two equivalent fractions will be the same when completely reduced (written in simplest form). When one or both of the fractions contain large numbers, it may not be in your best interest to try to reduce or simplify them, although you certainly could do this. A more efficient way is to cross multiply. For example, say you are asked to compare the following two fractions:

You can cross multiply as follows (denominator multiplied by the opposite numerator):

Since the cross products are the same, the fractions are equivalent. If the product on the left were greater than the product on the right, the fraction on the left would be the larger of the two. Similarly, had the product on the right been larger, the fraction on the right would be the larger of the two.

For example:

Since the product on the left is smaller, the fraction on the left is also smaller.

Reducing Fractions

Reducing and simplifying fractions requires you to be able to identify the common factors between the numerator and the denominator. The divisibility rules come in handy here! You are looking for factors that can divide both the numerator and the denominator. One way to do this is to write out the prime factorization of both. Another is by inspection. Let’s use the following example:

If you wanted to write out the prime factorization and cancel out any common factors, you’d end up with the following:

Alternatively, you can use inspection to look for common factors. Since both 420 and 630 end in zeros, they are both divisible by 10. This leaves you with ⁴²⁄₆₃. Both 42 and 63 are divisible by 3 (the sums of the digits in each number are 6 and 9, respectively),leaving you with ¹⁴⁄_21. From here, it’s fairly easy to see that both are divisible by 7, leaving you with the most reduced form: ²⁄_3.

Improper Fractions

Improper fractions typically make an appearance on the SAT. They are simply fractions that represent a whole number. You probably haven’t encountered them in math class since grade school however. When you see an improper fraction on the test, you will typically see answer choices with mixed numbers, meaning numbers that represent an improper fraction as a whole number and a fractional component. For example, ⁷⁄₃ is an improper fraction. Three (3) divides into 7 twice with a remainder of 1. Thus, ⁷⁄₃ could be written as the mixed number 2¹⁄₃.

Other times, you’ll need to do the same conversion, but in reverse, converting a mixed number into an improper fraction. This happens most often when multiplying or dividing with fractions. To do this, simply multiply the whole number by the denominator, add the result to the numerator, and divide the sum by the denominator. Let’s use our previous example, 2¹⁄₃. Writing 2¹⁄₃ as an improper fraction would look something like:

Common Denominators

A common problem encountered when dealing with fractions involves finding a common denominator, or more importantly, the least common denominator. Before you can add or subtract fractions, you will need to find a common denominator.

For example, let’s say you have the following problem:

The simplest way to find a common denominator is to take the product of the two denominators.

Using this as the common denominator, you would write the fractions, thus:

Then you can add them:

This can be done fairly quickly. However, it may result in answers requiring a significant amount of simplification as the common denominator found will not necessarily be the least common denominator. Finding this requires a couple additional steps on the front end. Let’s take the following problem:

First, we factor both the denominators:

The least common denominator will thus be:

All we need to do now is multiply the numerator and denominator of each fraction by the missing factors:

We are still dealing with fractions with rather large numbers, but this did make things simpler. Had we simply multiplied the two denominators together, we would have gotten a common denominator of 138,600!

Example

Answer and Explanation

1. The correct answer is A. This is a time-consuming problem, but nothing difficult. Check out your answer choices first and you’ll see that every option contains the option in I. Thus, there is no point in checking it, it must be greater! Start with the value in II. You can either use your calculator or the comparison method discussed above. The value in II is greater so you can eliminate choice E which doesn’t include it. Moving on to the value in III, it is also greater, so the correct response must contain III, eliminating choices C and D. You are left with choices A and B, so forget looking at IV, you only need to check the value in V which is also greater. Eliminate choice B.