Fractions

You've probably run into a lot of whole numbers when working on math problems, but how many times have you encountered fractions? While a whole number looks like 1, 12, or 675, a fraction is a portion of a whole number and looks like $\frac{1}{4}$ , $\frac{3}{10}$ , or $\frac{16}{19}$ . You will encounter fractions a lot in math, so it's important to understand what they are and how to use them.

What is a fraction?

A fraction is a way of expressing division. The fraction $\frac{a}{b}$ actually means $a \div b$ .

It's important to note that fractions are not whole numbers. In fact, a fraction represents a portion of a whole number.

The number below the bar of a fraction is called the denominator. It tells the number of equal parts into which a whole number has been divided.

The number above the bar, the numerator, tells how many equal parts are being considered.

Here is an example:

The fraction $\frac{2}{5}$ actually means $2 \div 5$ .

In this case, the whole has been divided into 5 parts and 2 are being considered.

In the following figure, you see that the fraction $\frac{2}{5}$ is represented by a shape divided into 5 pieces of equal size, and 2 of them are shaded.

Proper fractions and improper fractions

Let's take a look at two types of fractions you'll see often when solving math problems: proper fractions and improper fractions.

Proper fractions

A proper fraction is a fraction whose numerator is less than its denominator.

An example of a proper fraction is $\frac{3}{7}$ .

Let's take a look at the fraction represented in an image:

The fraction is represented by a shape divided into 7 pieces of equal size, with 3 of them shaded.

Improper fractions

An improper fraction is a fraction whose numerator is greater than the denominator.

An example of an improper fraction is $\frac{14}{5}$ .

Its representation as a shape will be different from a proper fraction because $\frac{14}{5}$ is greater than 1. In order to draw it, we'll need 3 equal shapes, as opposed to one. Each shape is divided into 5 pieces of equal size, and 14 of them are shaded.

Mixed numbers and fractions in lowest terms

Now, we'll go over mixed numbers and fractions that are not in lowest terms.

Mixed numbers

A mixed number is a number that consists of a whole number and a fraction. it can be written as an alternate expression of an improper fraction. Likewise, an improper fraction can be written as a mixed number.

For example, the improper fraction $\frac{14}{5}$ can be written as $2 \frac{4}{5}$ .

Let's look at how to write the mixed number $7 \frac{2}{5}$ as an improper fraction.

First, write the mixed number as a sum of a whole number and a proper fraction.

$7 \frac{2}{5} = \frac{7}{1} + \frac{2}{5}$

Then, we'll write the two parts with a common denominator.

$7 \frac{5}{5} + \frac{2}{5} = \frac{35}{5} + \frac{2}{5}$

Now, add the fractions.

$\frac{35}{5} + \frac{2}{5} = \frac{37}{5}$

The mixed number $7 \frac{2}{5}$ is the improper fraction $\frac{37}{5}$ .

Now, let's write the improper fraction $\frac{11}{7}$ as a mixed number. First, divide the numerator by the denominator.

$\frac{11}{7} = 11 \div 7 = 1 R 4$

The quotient is 1, and the remainder is 4.

The quotient is the whole part of the mixed number, and the remainder is the numerator of the fractional part.

Keep the denominator the same.

$\frac{11}{7} = 1 \frac{4}{7}$

The improper fraction $\frac{11}{7}$ is the mixed number $1 \frac{4}{7}$ .

Fractions in lowest terms

A fraction in lowest terms means that the numerator and denominator have no common factor other than 1.

In order to write a fraction in lowest terms, you'll divide the numerator and denominator by the greatest common factor.

Here's an example:

Write $\frac{45}{75}$ in lowest terms.

45 and 75 have a common factor of 15.

$\frac{45}{75} = \frac{45 \div 15}{75 \div 15} = \frac{3}{5}$

$\frac{45}{75}$ in lowest terms is $\frac{3}{5}$ .

Practice questions on fractions

a. What is the number below the fraction bar called?

Answer: denominator

b. What type of fraction has a numerator that is less than its denominator?

Answer: proper fraction

c. True or false: the improper fraction $\frac{14}{5}$ is greater than 1.

Answer: True

d. Write $6 \frac{1}{3}$ as an improper fraction.

$6 \frac{1}{3} = \frac{6}{1} + \frac{1}{3}$

$= \frac{6 \times 3}{3} + \frac{1}{3}$

$= \frac{18}{3} + \frac{1}{3}$

Answer: $\frac{19}{3}$

e. Write $\frac{25}{4}$ as a mixed number.

$\frac{25}{4} = 25 \div 4 = 6 R 1$

Quotient: 6. Remainder: 1. Denominator: 4.

Answer: $6 \frac{1}{4}$

f. Write $\frac{28}{40}$ in lowest terms.

28 and 40 have a common factor of 4.

$\frac{28}{40} = \frac{28 \div 4}{40 \div 4}$

Answer: $\frac{7}{10}$

Topics related to the Fractions

Lowest Terms

Adding and Subtracting Fractions with Unlike Denominators

Comparing Fractions

Flashcards covering the Fractions

3rd Grade Math Flashcards

Basic Arithmetic Flashcards

Practice tests covering the Fractions

Common Core: 3rd Grade Math Diagnostic Tests

3rd Grade Math Practice Tests

Get a better understanding of fractions

Grasping fractions is not always the easiest task. Your student might understand how proper fractions relate to whole numbers but could have trouble comprehending improper fractions. No matter your student's understanding of fractions, they can benefit from taking part in tutoring sessions. A qualified private instructor can answer your student's questions about mixed numbers and help them prepare for a test or work on assignments covering lowest terms. Get more information about the many ways your student can benefit from tutoring by contacting the Educational Directors at Varsity Tutors today.