Fractions
These topics are covered in the Mathematics syllabus in preparation for the University of Malta MATSEC O level exam. However a strong knowledge of such topics will prove to be extremely important and useful for the workings in the Intermediate level and Advanced level topics. These topics are intended as a revision and brush-up for students starting their A level Pure Mathematics in Malta.
What are the 3 types of fractions?
There are 3 types of fractions, namely:
- Proper Fractions
- Improper Fractions
- Mixed Fractions
Proper Fraction
A proper fraction is a fraction in which the numerator is less than the denominator.
Examples:
(a) \frac{2}{3}
(b) \frac{3}{9}
(c) \frac{2}{100}
Improper Fractions
An improper fraction is a fraction in which the numerator is greater or equal to the denominator.
Examples:
(a) \frac{5}{3}
(b) \frac{10}{8}
(c) \frac{9}{9}
Mixed Fractions
A mixed fraction is made up of two parts: one part is an integer and the other part is a proper fraction. An improper fraction can be expressed as a mixed fraction and vice versa.
Examples:
(a) 1\frac{3}{5}
(b) 2\frac{1}{3}
(c) 4\frac{1}{2}
Converting an improper fraction into a mixed fraction
Suppose that you are given an improper fraction \frac{p}{q}. Let n be the number of times q divides p (that is n is the quotient), and let r be the remainder after dividing p by q. Then:
\frac{p}{q}=n\frac{r}{q}
Example 1: Convert the following improper fractions into mixed fractions:
(a) \frac{13}{3}
Hence 3 divides 13 for 4 times, and leaves a remainder of 1 (13-12)
Therefore, \frac{13}{3}=4\frac{1}{3}
(b) \frac{14}{5}
Hence 5 divides 14 for 2 times, and leaves a remainder of 4 (14-10)
Therefore, \frac{14}{5}=2\frac{4}{5}
(c) \frac{5}{2}
Hence 2 divides 5 for 2 times, and leaves a remainder of 1 (5-4)
Therefore, \frac{5}{2}=2\frac{1}{2}
(d) \frac{5}{3}
Hence 3 divides 5 just 1 time, and leaves a remainder of 2 (5-3)
Therefore, \frac{5}{3}=1\frac{2}{3}
(e) \frac{22}{6}
The fraction is not in its lowest terms (since the numerator and the denominator have a common factor). Let us express the given fraction in its lowest terms, by dividing by 2:
\frac{22}{6}=\frac{11}{3}3 \times 3 = 9
Hence 3 divides 11 for 3 times, and leaves a remainder of 2 (11-9)
Therefore, \frac{22}{6}=\frac{11}{3}=3\frac{2}{3}
(f) \frac{30}{9}
The fraction is not in its lowest terms (since the numerator and the denominator have a common factor). Let us express the given fraction in its lowest terms, by dividing by 3:
\frac{30}{9}=\frac{10}{3}3 \times 3 = 9
Hence 3 divides 10 for 3 times, and leaves a remainder of 1 (10-9)
Therefore, \frac{30}{9}=\frac{10}{3}=3\frac{1}{3}
Converting a mixed fraction into an improper fraction
Suppose that we are given a mixed fraction, let us say, n\frac{r}{q}. We know that such a mixed fraction is made up of two parts: the integer (n in this case) and the proper fraction (\frac{r}{q} in this case).
Let us obtain the associated improper fraction. The numerator of the improper fraction is obtained by multiplying n by q and adding r. The denominator of the improper fraction is q
n\frac{r}{q}=\frac{nq+r}{q}
Example 2: Convert the following mixed fractions into improper fractions:
(a) 4\frac{1}{3}
4\frac{1}{3}=\frac{4(3)+1}{3}=\frac{13}{3}
(b) 2\frac{4}{5}
2\frac{4}{5}=\frac{2(5)+4}{5}=\frac{14}{5}
(c) 2\frac{1}{2}
2\frac{1}{2}=\frac{2(2)+1}{2}=\frac{5}{2}
(d) 1\frac{2}{3}
1\frac{2}{3}=\frac{1(3)+2}{3}=\frac{5}{3}
Negative numbers in fractions
Consider fractions that have a negative number in their numerator or in their denominator or in both.
Case 1: Negative number only in the numerator
In this case, the negative sign is taken out of the numerator and put in front of the fraction. This results in a negative fraction. Example:
\frac{-1}{3}=-\frac{1}{3}
Case 2: Negative number only in the denominator
This is similar to the previous case. Here the negative sign is taken out of the denominator and put in front of the fraction. This results in a negative fraction. Example:
\frac{1}{-3}=-\frac{1}{3}
Case 3: Negative number in the both the numerator and the denominator
The negative signs in the numerator and the denominator cancel out. This results in a positive fraction. Example:
\frac{-1}{-3}=\frac{1}{3}
Addition and Subtraction of two fractions (proper or improper)
In addition, when the two proper fraction have the same denominator, the numerators are added, and denominator is left unchanged.
Similarly, in subtraction, when the two proper fraction have the same denominator, the numerator of second fraction is subtracted from the numerator of the first fraction, and denominator is left unchanged.
Example 3: Perform the following additions and subtractions:
(a) \frac{1}{5}+\frac{2}{5}
\begin{aligned}&\frac{1}{5}+\frac{2}{5}\\=&\frac{3}{5}\end{aligned}
(b) \frac{2}{3}+\frac{1}{3}
\begin{aligned}&\frac{2}{3}+\frac{1}{3}\\=&\frac{3}{3}\\=&1\end{aligned}
(c) \frac{3}{5}+\frac{3}{5}
\begin{aligned}&\frac{3}{5}+\frac{3}{5}\\=&\frac{6}{5}\\=&1\frac{1}{5}\end{aligned}
(d) \frac{2}{3}-\frac{1}{3}
\begin{aligned}&\frac{2}{3}-\frac{1}{3}\\=&\frac{1}{3}\end{aligned}
(e) \frac{2}{5}-\frac{3}{5}
\begin{aligned}&\frac{2}{5}-\frac{3}{5}\\=&\frac{-1}{5}\\=&-\frac{1}{5}\end{aligned}
(f) -\frac{2}{5}-\frac{1}{5}
\begin{aligned}&-\frac{2}{5}-\frac{1}{5}\\=&\frac{-2}{5}-\frac{1}{5}\\=&\frac{-3}{5}\\=&-\frac{3}{5}\end{aligned}
Now consider addition and subtraction in which the fractions have a different denominator. In this case, the first step is to obtain a common denominator for the two fractions.
Example 4: Perform the following additions and subtractions:
(a) \frac{1}{2}+\frac{2}{3}
\begin{aligned}&\frac{1}{2}+\frac{2}{3}\\=&\frac{3}{6}+\frac{4}{6}\\=&\frac{7}{6}\\=&1\frac{1}{6}\end{aligned}
(b) \frac{1}{3}+\frac{1}{6}
\begin{aligned}&\frac{1}{3}+\frac{1}{6}\\=&\frac{2}{6}+\frac{1}{6}\\=&\frac{3}{6}\\=&\frac{1}{2}\end{aligned}
(c) \frac{1}{5}-\frac{2}{3}
\begin{aligned}&\frac{1}{5}-\frac{2}{3}\\=&\frac{3}{15}-\frac{10}{15}\\=&-\frac{7}{15}\end{aligned}
(d) -\frac{4}{7}+\frac{4}{9}
\begin{aligned}&-\frac{4}{7}+\frac{4}{9}\\=&\frac{-4}{7}+\frac{4}{9}\\=&\frac{-36}{63}+\frac{28}{63}\\=&\frac{-8}{63}\\=&-\frac{8}{63}\end{aligned}
Addition and Subtraction involving mixed fractions
Addition and subtraction involving mixed fractions can easily be done by converting mixed fractions into improper fractions.
Example 5: Perform the following additions and subtractions:
(a) 1\frac{1}{3}+2\frac{1}{3}
\begin{aligned}&1\frac{1}{3}+2\frac{1}{3}\\=&\frac{4}{3}+\frac{7}{3}\\=&\frac{11}{3}\\=&3\frac{2}{3}\end{aligned}
(b) 1\frac{1}{3}-2\frac{1}{3}
\begin{aligned}&1\frac{1}{3}-2\frac{1}{3}\\=&\frac{4}{3}-\frac{7}{3}\\=&\frac{-3}{3}\\=&-\frac{3}{3}\\=&-1\end{aligned}
(c) 1\frac{2}{5}-2\frac{1}{4}
\begin{aligned}&1\frac{2}{5}-2\frac{1}{4}\\=&\frac{7}{5}-\frac{9}{4}\\=&\frac{28}{20}-\frac{45}{20}\\=&\frac{-17}{20}\\=&-\frac{17}{20}\end{aligned}
(d) -\frac{2}{7}-2\frac{2}{3}
\begin{aligned}&-\frac{2}{7}-2\frac{2}{3}\\=&\frac{-2}{7}-\frac{8}{3}\\=&\frac{-6}{21}-\frac{56}{21}\\=&\frac{-62}{21}\\=&-2\frac{20}{21}\end{aligned}
Multiplication of proper or improper fractions
This is done in two steps:
- The new numerator is obtained by multiplying the two numerators.
- The new denominator is obtained by multiplying the two denominators.
Example 6: Perform the following multiplications:
(a) \frac{2}{3}\times\frac{4}{5}
\begin{aligned}&\frac{2}{3}\times\frac{4}{5}\\=&\frac{2\times 4}{3 \times 5}\\=&\frac{8}{15}\end{aligned}
(b) \frac{5}{3}\times\frac{4}{3}
\begin{aligned}&\frac{5}{3}\times\frac{4}{3}\\=&\frac{5\times 4}{3 \times 3}\\=&\frac{20}{9}\\=&2\frac{2}{9}\end{aligned}
(c) (\frac{1}{2})(\frac{3}{2})
\begin{aligned}&(\frac{1}{2})(\frac{3}{2})\\=&\frac{1\times 3}{2 \times 2}\\=&\frac{3}{4}\end{aligned}
(d) (\frac{1}{3})(\frac{1}{6})
\begin{aligned}&(\frac{1}{3})(\frac{1}{6})\\=&\frac{1\times 1}{3 \times 6}\\=&\frac{1}{18}\end{aligned}
Multiplication involving negative fractions
When exactly one fraction is negative, the multiplication results in a negative fraction.
When both fractions are negative, the two negative signs cancel out and the multiplication results in a negative fraction.
Examples:
(a) -\frac{2}{5}\times\frac{1}{3}
\begin{aligned}&-\frac{2}{5}\times\frac{1}{3}\\=&-\frac{2}{15}\end{aligned}
(b) \frac{2}{3}\times-\frac{1}{3}
\begin{aligned}&\frac{2}{3}\times -\frac{1}{3}\\=&-\frac{2}{3}\times\frac{1}{3}\\=&-\frac{2}{9}\end{aligned}
(c) -\frac{1}{2}\times -\frac{3}{2}
\begin{aligned}&-\frac{1}{2}\times -\frac{3}{2}\\=&\frac{1}{2}\times\frac{3}{2}\\=&\frac{3}{4}\end{aligned}
Multiplication of mixed fractions
Here the mixed fractions should be converted into improper fractions and multiplication is carried out as already shown for proper or improper fractions.
Example 7: Perform the following multiplications:
(a) 1\frac{2}{3}\times 2\frac{4}{5}
\begin{aligned}&1\frac{2}{3}\times 2\frac{4}{5}\\=&\frac{5}{3}\times\frac{14}{5}\\=&\frac{\cancel{5}}{3}\times\frac{14}{\cancel{5}}\\=&\frac{14}{3}\end{aligned}
(b) 2\frac{5}{3}\times 2\frac{4}{3}
\begin{aligned}&2\frac{5}{3}\times 2\frac{4}{3}\\=&\frac{11}{3}\times\frac{10}{3}\\=&\frac{110}{9}\end{aligned}
(c) 3\frac{2}{3}\times -\frac{3}{5}
\begin{aligned}&3\frac{2}{3}\times -\frac{3}{5}\\=&-3\frac{2}{3}\times \frac{3}{5}\\=&-\frac{11}{3}\times\frac{3}{5}\\=&-\frac{11}{\cancel{3}}\times\frac{\cancel{3}}{5}\\=&-\frac{11}{5}\end{aligned}
(d) -1\frac{1}{4}\times - 1\frac{2}{3}
\begin{aligned}&-1\frac{1}{4}\times – 1\frac{2}{3}\\=&1\frac{1}{4}\times 1\frac{2}{3}\\=&\frac{5}{4}\times \frac{5}{3}\\=&\frac{25}{12}\end{aligned}
Division of proper or improper fractions
Division of fractions is done by transforming the division operation into the multiplication operation by replacing the second fraction by its reciprocal.
Note that the reciprocal of a proper or improper fraction is obtained by replacing the numerator with the denominator and vice-versa.
Example 8: Perform the following divisions:
(a) \frac{2}{3}\div\frac{4}{5}
\begin{aligned}&\frac{2}{3}\div\frac{4}{5}\\=&\frac{2}{3}\times\frac{5}{4}\\=&\frac{\cancel{2}^1}{3}\times\frac{5}{\cancel{4}^2}\\=&\frac{5}{6}\end{aligned}
(b) \frac{5}{3}\div -\frac{4}{3}
(c) -\frac{5}{3}\div -\frac{6}{4}
\begin{aligned}&-\frac{5}{3}\div -\frac{6}{4}\\=&\frac{5}{3}\div \frac{6}{4}\\=&\frac{5}{3}\div \frac{3}{2}\\=&\frac{5}{3}\times \frac{2}{3}\\=&\frac{10}{9}\end{aligned}
(d) 2\div \frac{4}{3}
\begin{aligned}&2\div\frac{4}{3}\\=&2\times\frac{3}{4}\\=&\frac{2}{1}\times\frac{3}{4}\\=&\frac{\cancel{2}^1}{1}\times\frac{3}{\cancel{4}^2}\\=&\frac{3}{2}\end{aligned}
(e) \frac{4}{3}\div 2
\begin{aligned}&\frac{4}{3}\div 2\\&=\frac{4}{3}\times \frac{1}{2}\\&=\frac{\cancel{4}^2}{3}\times \frac{1}{\cancel{2}^1}\\&=\frac{2}{3}\end{aligned}
(f) \frac{\frac{5}{7}}{\frac{2}{3}}
\begin{aligned}&\frac{\frac{5}{7}}{\frac{2}{3}}\\=&\bigg(\frac{5}{7}\bigg)\bigg(\frac{3}{2}\bigg)\\=&\frac{15}{14}\end{aligned}
(g) \frac{\frac{5}{3}}{\frac{2}{3}}
\begin{aligned}&\frac{\frac{5}{3}}{\frac{2}{3}}\\=&\bigg(\frac{5}{3}\bigg)\bigg(\frac{3}{2}\bigg)\\=&\bigg(\frac{5}{\cancel{3}}\bigg)\bigg(\frac{\cancel{3}}{2}\bigg)\\=&\frac{5}{2}\end{aligned}
Alternatively:
\begin{aligned}&\frac{\frac{5}{3}}{\frac{2}{3}}\\=&\frac{\frac{5}{\cancel{3}}}{\frac{2}{\cancel{3}}}\\=&\frac{5}{2}\end{aligned}
(h) \frac{\frac{5}{6}}{-\frac{7}{3}}
\begin{aligned}&\frac{\frac{5}{6}}{-\frac{7}{3}}\\=&\bigg(\frac{5}{6}\bigg)\bigg(-\frac{3}{7}\bigg)\\=&-\bigg(\frac{5}{6}\bigg)\bigg(\frac{3}{7}\bigg)\\=&-\bigg(\frac{5}{\cancel{6}^2}\bigg)\bigg(\frac{\cancel{3}^1}{7}\bigg)\\=&-\frac{5}{7}\end{aligned}