Understanding Fractions
Fractions are used in many ways in daily life.
For example, you may see a sale that offers items for half price.
Reports, statistics, and cooking recipes also use fractions.
So, a report might say that one-third of the population is overweight.
A recipe might call for 3/4 cup of milk.
How to Do Fractions − The Basics
Students sometimes find it hard to know how to do fractions.
So, first of all, we will talk about the basics.
Fractions consist of two numbers.
The number on the top is called the numerator.
The number on the bottom is called the denominator.
Rules for Multiplying Fractions
The size of a fraction does not change when you multiply the numerator and denominator by the same number.
Look at the example below.
\(\frac{1}{4} = \frac{1}{4}\times\frac{2}{2} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}\)
When we take 1 from the top of the original fraction and multiply it by 2, we get 2 in the top of the new fraction.
When we take 4 from the bottom of the original fraction and multiply it by 2, we get 8 in the bottom of the new fraction.
As a result, we know that, one-fourth is equal to two-eighths.
Multiplying Fractions − Exercises
Find the missing numbers in the fractions below. Then check your answers.
\(\text{1)} \frac{1}{3} = \frac{?}{9}\)
\(\text{2)} \frac{1}{6} = \frac{?}{24}\)
\(\text{3)} \frac{1}{4} = \frac{3}{?}\)
\(\text{4)} \frac{2}{5} = \frac{?}{20}\)
\(\text{5)} \frac{4}{7} = \frac{12}{?}\)
Multiplying Fractions − Answers and Explanations:
1) 3
The denominator in the new fraction is 9. We know that 3 from the bottom of the first fraction has been multiplied by 3. So, we multiply the numerator of the first fraction by 3 in order to get the numerator of the new fraction.
\(\frac{1}{3} = \frac{1}{3}\times\frac{3}{3} = \frac{3}{9}\)
2) 4
The denominator in the new fraction is 24, so 6 from the denominator of the first fraction has been multiplied by 4. We take the numerator of the first fraction times 4 in order to get the numerator of the new fraction.
\(\frac{1}{6} = \frac{1}{6}\times\frac{4}{4} = \frac{4}{24}\)
3) 12
The numerator in the new fraction is 3, so 1 from the top of the first fraction has been multiplied by 3. We multiply the denominator of the first fraction by 3 in order to get the denominator of the new fraction.
\(\frac{1}{4} = \frac{1}{4}\times\frac{3}{3} = \frac{3}{12}\)
4) 8
The denominator in the new fraction is 20, so 5 from the denominator of the first fraction has been multiplied by 4. We take the numerator of the first fraction times 4 in order to get the numerator of the new fraction.
\(\frac{2}{5} = \frac{2}{5}\times\frac{4}{4} = \frac{8}{20}\)
5) 21
The numerator in the new fraction is 12, so 4 from the top of the first fraction has been multiplied by 3. We multiply the denominator of the first fraction by 3 in order to get the denominator of the new fraction.
\(\frac{4}{7} = \frac{4}{7}\times\frac{3}{3} = \frac{12}{21}\)
Rules for Dividing Fractions
When you see the division symbol in a fraction problem, you need to invert the second fraction.
"Invert" means to interchange the positions of the numerator and the denominator.
For example, if we invert 2/3, we get 3/2.
Once you have inverted the second fraction, need to multiply the two fractions.
Remember to multiply the numerator of the first fraction by the numerator of the second fraction.
Then multiply the denominator of the first fraction by the denominator of the second fraction.
Now look at the following example.
\(\frac{4}{7} \div \frac{2}{5} = \)
\(\frac{4}{7} \times \frac{5}{2} = \)
\(\frac{4 \times 5}{7 \times 2} = \)
\(\frac{20}{14}\)
We get 20 in the numerator because 4 × 5 = 20.
We get 14 in the denominator because 7 × 2 = 14.
Dividing Fractions − Exercises
Now try these sample problems on dividing fractions.
\(\text{1)} \frac{2}{3} \div \frac{3}{5} = ? \)
\(\text{2)} \frac{3}{8} \div \frac{2}{7} = ? \)
\(\text{3)} \frac{1}{3} \div \frac{2}{3} = ? \)
\(\text{4)} \frac{4}{9} \div \frac{3}{4} = ? \)
\(\text{5)} \frac{3}{7} \div \frac{5}{6} = ? \)
Dividing Fractions − Answers and Explanations:
\(\text{1)} \frac{10}{9} \)
\(\frac{2}{3} \div \frac{3}{5} = \)
\(\frac{2}{3} \times \frac{5}{3} = \)
\(\frac{2 \times 5}{3 \times 3} = \)
\(\frac{10}{9} \)
\(\text{2)} \frac{21}{16} \)
\(\frac{3}{8} \div \frac{2}{7} = \)
\(\frac{3}{8} \times \frac{7}{2} = \)
\(\frac{3 \times 7}{8 \times 2} = \)
\(\frac{21}{16} \)
\(\text{3)} \frac{3}{6} \)
\(\frac{1}{3} \div \frac{2}{3} = \)
\(\frac{1}{3} \times \frac{3}{2} = \)
\(\frac{1 \times 3}{3 \times 2} = \)
\(\frac{3}{6} \)
\(\text{4)} \frac{16}{27} \)
\(\frac{4}{9} \div \frac{3}{4} = \)
\(\frac{4}{9} \times \frac{4}{3} = \)
\(\frac{4 \times 4}{9 \times 3} = \)
\(\frac{16}{27} \)
\(\text{5)} \frac{18}{35} \)
\(\frac{3}{7} \div \frac{5}{6} = \)
\(\frac{3}{7} \times \frac{6}{5} = \)
\(\frac{3 \times 6}{7 \times 5} = \)
\(\frac{18}{35} \)
Improper Fractions
An improper fraction is a fraction whose numerator is larger than its denominator.
So, in our exercise above, answers 1 and 2 are improper fractions.
Improper fractions can be converted to mixed numbers.
A mixed number contains a whole number and a fraction.
To get a mixed number from a fraction, we divide the numerator by the denominator.
Then use the remainder as the numerator of the fraction in the mixed number.
\(\text{1)} \frac{10}{9} = \)
\(10 \div 9 = \)
\(1\frac{1}{9}\)
We can check the result as follows:
\(1\frac{1}{9} = \)
\(\frac{9}{9} + \frac{1}{9} = \frac {10}{9}\)
\(\text{2)} \frac{21}{16} \)
\(21 \div 16 = \)
\(1\frac{5}{16}\)
We can check the result as follows:
\(1\frac{5}{16} = \)
\(\frac{16}{16} + \frac{5}{16} = \frac {21}{16}\)
Equivalent Fractions
Equivalent fractions have the same value, even though they may look different.
For example, the following fractions have the same value:
\(\frac{12}{16} = \frac{6}{8} = \frac{3}{4}\)
Simplifying Fractions
In order to simplify a fraction, we need to divide the top and bottom of the fraction by the highest number possible.
The number needs to divide into both the numerator and the denominator evenly, without having a remainder.
The rule for simplifying equivalent fractions is that you have to divide the numerator and the denominator by the same number.
So, we can prove that the fractions from our previous section are equivalent by doing the following calculations.
\(\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \)
\(\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \)
Inequalities with Fractions
Questions on fractions may ask you to compare fractions within an inequality statement.
Inequality questions may also ask you to perform operations on fractions.
Look at the following example.
Example: Which of the following fractions would correctly complete the inequality below?
1/8 < ? < 5/8
A. 1/12
B. 1/10
C. 1/9
D. 1/7
E. 7/8
Remember the following shortcuts when dealing with fractions.
Comparing Fractions with the same denominators:
If the denominators (the numbers on the bottom) are the same, the fraction with the larger numerator is the greater fraction.
This is the case with answer choice E, so let's have a look at it.
7/8 < 5/8 - This is a false statement, so answer choice E is incorrect.
Comparing Fractions with the same numerators:
If the numerators (the numbers on the top) are the same, the fraction with the smaller denominator is actually the greater fraction.
If any of the inequalities in the answer choices have the same numbers in their numerators, you can then just compare the denominators in the answer choices in order to determine which fraction is greater.
This is the case with answer choices A, B, C, and D, so let’s evaluate them.
We can see that 1/12, 1/10, and 1/9 are not greater than 1/8, so answers A, B, and C are also incorrect.
1/8 < 1/7 < 5/8 is a valid mathematical statement.
So, the correct answer is D.
Rules for Adding and Subtracting Fractions
For rules on adding and subtracting fractions, please see our separate page entitled Finding the Lowest Common Denominator.