What are Fractions?
Equal parts of a whole or a collection of things are represented by fractions. In other words a fraction is a part or a portion of the whole. When we divide something into equal pieces, each part becomes a fraction of the whole.
For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction \(\frac{1}{2}\).
Similarly, if it is divided into 4 equal parts, then each part is one fourth of the whole, that can be represented by the fraction \(\frac{1}{4}\).
Types of Fractions
Proper fractions
A fraction in which the numerator is less than the denominator value is called a proper fraction.
For example, \(\frac{3}{4}\), \(\frac{5}{7}\), \(\frac{3}{8}\) are proper fractions.
Improper fractions
A fraction with the numerator higher than or equal to the denominator is called an improper fraction.
Eg \(\frac{9}{4}\), \(\frac{8}{8}\), \(\frac{9}{4}\) are examples of improper fractions.
Mixed fractions
A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction.
We express improper fractions as mixed numbers.
For example, 5\(\frac{1}{3}\), 1\(\frac{4}{9}\), 13\(\frac{7}{8}\) are mixed fractions.
Unit fraction
A unit fraction is a fraction with a numerator equal to one. If a whole or a collection is divided into equal parts, then exactly 1 part of the total parts represents a unit fraction.
Fractions with Like and Unlike Denominators
Like fractions are those in which two or more fractions have the same denominator, whereas unlike fractions are those in which the denominators of two or more fractions are different.
For example,
\(\frac{1}{4}\) and \(\frac{3}{4}\) are like fractions as they both have the same denominator, that is, 4.
\(\frac{1}{3}\) and \(\frac{1}{4}\) are unlike fractions as they both have a different denominator.
Operations on Fractions
We can perform addition, subtraction, multiplication and division operations on fractions.
Fractions with unlike denominators can be added or subtracted using equivalent fractions. Equivalent fractions can be obtained by finding a common denominator. And a common denominator is obtained either by determining a common multiple of the denominators or by calculating the product of the denominators.
There is another method to add or subtract mixed numbers, that is, solve the fractional and whole number parts separately, and then, find their sum to get the final answer.
Fractions can be Multiplied by Using:
Division operations on fractions can be performed using a tape diagram and area model. Also, when a fraction is divided by another fraction then we can solve it by multiplying the dividend with the reciprocal of the divisor.
Let’s Take a Look at a Few Examples
Addition and subtraction using common denominator
Example 1:
( \(\frac{1}{6} ~+ ~\frac{2}{5}\) )
We apply the method of equivalent fractions. For this we need a common denominator, or a common multiple of the two denominators 6 and 5, that is, 30.
\(\frac{1}{6} ~+ ~\frac{2}{5}\)
= (\(\frac{5}{30} ~+ ~\frac{12}{30}\)) [Rewrite \(\frac{1}{6} \) as \(\frac{1~\times~5}{6~\times~5} ~= ~\frac{5}{30}\) and \(\frac{2}{5}~\text{as}~\frac{2~\times~6}{5~\times~6} ~= ~\frac{12}{30}\)]
= \(\frac{5~+~12}{30}\)
= \(\frac{17}{30}\)
Example 2:
( \(\frac{5}{2}~-~\frac{1}{6}\) )
=(\(\frac{12}{30}~-~\frac{5}{30}\)) [Rewrite \(\frac{2}{5}~\text{as}~\frac{2~\times~6}{5~\times~6}\)= \(\frac{12}{30}\) and \(\frac{1}{6}\) as \(\frac{1~\times~5}{6~\times~5}~=~\frac{5}{30}\)]
= \(\frac{12~-~5}{30}\)
= \(\frac{7}{30}\)
Examples of Multiplication and Division
Multiplication:
(\(\frac{1}{6}~\times~\frac{2}{5}\))
= (\(\frac{1~\times~2}{6~\times~5}\)) [Multiplying numerator of fractions and multiplying denominator of fractions]
= \(\frac{2}{30}\)
Division:
(\(\frac{2}{5}~÷~\frac{1}{6}\))
= (\(\frac{2 ~\times~ 5}{6~\times~ 1}\)) [Multiplying dividend with the reciprocal of divisor]
= (\(\frac{2 ~\times~ 6}{5 ~\times~ 1}\))
=\(\frac{12}{5}\)
Solved Examples
Example 1: Solve \(\frac{7}{8}\) + \(\frac{2}{3}\)
Solution:
Let’s add \(\frac{7}{8}\) and \(\frac{2}{3}\) using equivalent fractions. For this we need to find a common denominator or a common multiple of the two denominators 8 and 3, which is, 24.
\(\frac{7}{8}\) + \(\frac{2}{3}\)
= \(\frac{21}{24}\) + \(\frac{16}{24}\) [Rewrite \(\frac{7}{8}\) as \(\frac{7 ~\times~3 }{8~\times~3}\) = \(\frac{21}{24}\) and \(\frac{2}{3}\) as \(\frac{2 ~\times~8 }{3~\times~8}\) = \(\frac{16}{24}\)]
= \(\frac{21~+~16}{24}\)
= \(\frac{37}{24}\)
Example 2: Solve \(\frac{11}{13}\) – \(\frac{12}{17}\)
Solution:
Let’s subtract \(\frac{12}{17}\) from \(\frac{11}{13}\) using equivalent fractions. For this we need a common denominator or a common multiple of the two denominators 13 and 17, that is, 221.
\(\frac{11}{13}\) – \(\frac{12}{17}\)
= \(\frac{187}{221}\) – \(\frac{156}{221}\) [Rewrite \(\frac{11}{13}\) as \(\frac{11~\times~17}{13~\times~17}\) = \(\frac{187}{221}\) and \(\frac{12}{17}\) as \(\frac{12~\times~13}{17~\times~13}\) = \(\frac{156}{221}\)]
= \(\frac{187~-~156}{221}\)
= \(\frac{31}{221}\)
Example 3: Solve \(\frac{15}{13} ~\times~\frac{18}{17}\)
Solution:
Multiply the numerators and multiply the denominators of the 2 fractions.
\(\frac{15}{13}~\times~\frac{18}{17}\)
= \(\frac{15~~\times~18}{13~~\times~~17}\)
= \(\frac{270}{221}\)
Example 4: Solve \(\frac{25}{33}~\div~\frac{41}{45}\)
Solution:
Divide by multiplying the dividend with the reciprocal of the divisor.
\(\frac{25}{33}~\div~\frac{41}{45}\)
= \(\frac{25}{33}~\times~\frac{41}{45}\) [Multiply with reciprocal of the divisor \(\frac{41}{45}\), that is, \(\frac{45}{41}\) ]
= \(\frac{25~\times~45}{33~\times~41}\)
= \(\frac{1125}{1353}\)
Example 5:
Sam was left with \(\frac{7}{8}\) slices of chocolate cake and \(\frac{3}{7}\) slices of vanilla cake after he shared the rest with his friends. Find out the total number of slices of cake he had with him. Sam shared \(\frac{10}{11}\) slices from the total number he had with his parents. What is the number of slices he has remaining?
Solution:
To find the total number of slices of cake he had after sharing we need to add the slices of each cake he had,
= \(\frac{7}{8}\) + \(\frac{3}{7}\)
= \(\frac{49~+~24}{56}\)
= \(\frac{73}{56}\)
To find out the remaining number of slices Sam has \(\frac{10}{11}\) slices need to be deducted from the total number,
= \(\frac{73}{56}~-~\frac{10}{11}\)
= \(\frac{803~-~560}{616}\)
= \(\frac{243}{616}\)
Hence, after sharing the cake with his friends, Sam has \(\frac{73}{56}\) slices of cake, and after sharing with his parents he had \(\frac{243}{616}\) slices of cake left with him.
Example 6: Tiffany squeezed oranges to make orange juice for her juice stand. She was able to get 25 ml from one orange. How many oranges does she need to squeeze to fill a jar of \(\frac{15}{8}\) liters? Each cup that she sells carries 200 ml and she sells each cup for 64 cents. How much money does she make at her juice stand?
Solution:
First \(\frac{15}{8}\)l needs to be converted to milliliters.
\(\frac{15}{8}\)l into milliliters = \(\frac{15}{8}\) x 1000 = 1875 ml
To find the number of oranges, divide the total required quantity by the quantity of juice that one orange can give.
The number of oranges required for 1875 ml of juice = \(\frac{1875}{25}\) ml = 75 oranges
To find the number of cups she sells, the total quantity of juice is to be divided by the quantity of juice that 1 cup has
= \(\frac{1875}{200}~=~9\frac{3}{8}\)cups
We know that, the number of cups cannot be a fraction, it has to be a whole number. Also each cup must have 200ml. Hence with the quantity of juice she has she can sell 9 cups, \(\frac{3}{8}\)th of a cup cannot be sold alone.
Money made on selling 9 cups = 9 x 64 = 576 cents
Hence she makes 576 cents from her juice stand.
A mixed fraction is a number that has a whole number and a fractional part. It is used to represent values between whole numbers.
When adding fractions with unlike denominators, take the common multiple of the denominators of both the fractions and then convert them into equivalent fractions.