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Finding Fractional Parts of a Set

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will look at finding the fractional part of a set. We will look at how to do this visually and algebraically, and we will look at how the two processes relate. Lastly, we will examine examples of both processes.

Fractional Part of a Set

Suppose that a record label is deciding whether or not to sign a band called the Quadratics. To make their decision, they have the Quadratics play in front of an audience consisting of 40 music experts. Afterwards, they survey the music experts, and if 3/4 of them liked the performance, then they will sign the Quadratics to their label.

As the Quadratics are getting ready to go on, they are trying to figure out how many of the music experts need to like their performance in order to get signed. They know there are 40 music experts, and they know that 3/4 of them need to like the performance. Hmmm…so how many need to like the performance?

fracpart1

In mathematics, this problem has to do with what is called a fractional part of a set. Basically, the fractional part of a set is a fraction of a set of objects. In this scenario, the set is the music experts in the audience, and the fractional part would be 3/4 of those music experts, or the number of music experts needed to get the Quadratics signed.

Let's help the Quadratics out by finding the fractional part of the set in their scenario!

Finding the Fractional Part of a Set

A fraction, a/b, of something is a portion, or part, of something. In the fraction a/b, we call a the numerator and b the denominator. When finding a fraction, a/b, of a number, the denominator represents how many equal parts we will divide that number into, and the numerator represents how many of those parts we are considering.

Notice that when we are finding the fractional part, a/b, of a set of n objects, we are basically finding a/b of n. Therefore, we divide n into b equal parts and consider a of them.

Let's break this down into steps to clarify. To find the fractional part, a/b, of a set with n objects, we use the following steps:

  1. Divide the number of objects in the set into b equal groups, or parts.
  2. Take a of those groups.
  3. The total number of objects in the a groups that you took in step two is the fractional part, a/b, of the set with n objects.

Consider the Quadratics' scenario again. The fractional part is 3/4, so the denominator is 4, and the numerator is 3. The number of objects in the set, or number of music experts, is 40.

To find the fractional part, 3/4, of 40 music experts, we first divide 40 into 4 equal groups, or parts.


fracpart2


We see that this results in four groups with ten music experts in each. Step 2 tells us to take three of these groups. Then, 3/4 of 40 is equal to how many objects there are total in those three groups.


fracpart3


We see that doing this results in 30 total music experts. Therefore, 3/4 of 40 is 30, and the Quadratics need 30 music experts to like their performance in order to get signed.

We can also do this algebraically. Each step that we just observed corresponds to an algebraic operation. The first step says to divide the total number of objects, n, in to b equal groups. This corresponds to dividing n by b. The second step then says to take a of those parts, and that the total number of objects in those a parts is the fractional part. This corresponds to multiplying the result you got in step 1 by a. In other words, to find the fractional part, a/b, of a set with n objects in it algebraically, we follow these steps.

  1. Divide n by b.
  2. Multiply the result you got in step 1 by a.

Again, consider the Quadratics' example. We would first divide 40 by 4.

  • 40 / 4 = 10

Then, we would multiply that result by 3.

  • 10 × 3 = 30

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