What is a Fraction?
A fraction is used to represent a whole number that is divided into equal parts. The numbers that exist between whole numbers like 0, 1, 2, 3, 4, and so on, can be represented using fractions.
A fraction is generally expressed as \( \frac{a}{b} \). Since a fraction is the same as division, \( \frac{a}{b} \) is the same as \( a\div b \). \( \frac{1}{2},\frac{2}{3} \), and \( \frac{4}{5} \) are some examples of fractions, as well as numbers being divided.
What is a Unit Fraction?
A fraction whose numerator is 1 is known as a unit fraction. In the case of unit fractions, the value of a in \( \frac{a}{b} \) will always be 1. The denominator b can be any whole number. A few examples of unit fractions are \( \frac{1}{2},\frac{1}{4},\frac{1}{7} \) and \( \frac{1}{16} \).
Representing a Non-unit Fraction Using Unit Fractions
We can represent any non-unit fraction as a sum of unit fractions. For example, \( \frac{4}{5} \) can be written as \( \frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5} \). Here, \( \frac{4}{5} \) is a non-unit fraction, and \( \frac{1}{5} \) is a unit fraction.
We have represented the non-unit fraction as a sum of a group of unit fractions.
Representing a Non-unit Fraction as a Multiple of a Unit Fraction
Just like we used addition to represent a non-unit fraction in terms of a unit fraction, we can also use multiplication to do the same.
The sum of unit fractions has repeated terms. Whenever we have a summation of repeated terms, we can use multiplication to simplify the process.
So, we can replace the sum with the product of the unit fraction and the number of times the fraction is added. For example, \( \frac{5}{8} \) is expressed as \( \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8} \), when using the summation method.
Here, \( \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8} \) can also be replaced with \( 5\times \frac{1}{8} \).
In simple terms, we are taking out the a from \( \frac{a}{b} \) and writing it as \( a\times \frac{1}{b} \).
Multiplying a Fraction by a Whole Number
A fraction can be multiplied by a whole number with the help of unit fractions. To make it easy, we can convert the fraction into a unit fraction and then multiply the whole number to get the product.
For example, \( 7\times \frac{2}{3}=7\times (2\times \frac{1}{3}) \)
\( =(7\times 2)\times \frac{1}{3} \)
\( =14\times \frac{1}{3}=\frac{14}{3} \)
So, \( 7\times \frac{2}{3}=\frac{14}{3} \)
Solved Examples
Example 1: Write \( \frac{7}{4} \) as a sum and a multiple of unit fractions.
Solution:
To write \( \frac{7}{4} \) as a sum of a unit fraction, we can represent it as 7 parts that are \( \frac{1}{4} \) of the whole.
\( \frac{7}{4}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4} \)
Since we are adding \( \frac{1}{4} \) seven times, we can express the sum using multiplication.
\( \frac{7}{4}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=7\times \frac{1}{4} \)
Example 2: Write the \( \frac{12}{7} \) as a multiple of a unit fraction.
Solution:
\( \frac{12}{7}=\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7} \)
\( =12\times \frac{1}{7}\)
Therefore, \( \frac{12}{7}=12\times \frac{1}{7}\)
Example 3: Write the product \(6 \times \frac{3}{5}\) as a multiple of a unit fraction.
Solution:
\( 6\times \frac{3}{5}=6\times (3\times \frac{1}{5}) \)
\( =(6\times 3)\times \frac{1}{5} \)
\( =18\times \frac{1}{5}=\frac{18}{5} \)
So, \( 6\times \frac{3}{5}=\frac{18}{5} \)
Example 4: Complete the table.
Solution:
\( 5\times \frac{1}{6}=\frac{5}{6} \)
\( 5\times \frac{2}{6}=5\times (2\times \frac{1}{6})=(5\times 2)\times \frac{1}{6}=10\times \frac{1}{6}=\frac{10}{6} \)
\( 5\times \frac{3}{6}=5\times (3\times \frac{1}{6})=(5\times 3)\times \frac{1}{6}=15\times \frac{1}{6}=\frac{15}{6} \)
\( 5\times \frac{4}{6}=5\times (4\times \frac{1}{6})=(5\times 4)\times \frac{1}{6}=20\times \frac{1}{6}=\frac{20}{6} \)
\( 5\times \frac{5}{6}=5\times (5\times \frac{1}{6})=(5\times 5)\times \frac{1}{6}=25\times \frac{1}{6}=\frac{25}{6} \)
\( 5\times \frac{6}{6}=5\times (6\times \frac{1}{6})=(5\times 6)\times \frac{1}{6}=30\times \frac{1}{6}=\frac{30}{6} \)
Example 5: You need \( \frac{3}{4} \) cups of fresh strawberries to make a glass of strawberry milkshake. If Kevin used \( \frac{15}{4} \) cups of strawberries to make strawberry milkshake, how many glasses of milkshake did he make?
Solution:
Quantity of strawberries needed to make a glass of strawberry milkshake \( =\frac{3}{4} \) cups
Quantity of strawberries used by Kevin \( =\frac{15}{4} \) cups
We need to find the number of times \( \frac{3}{4} \) is added to get \( \frac{15}{4} \)
\( \frac{15}{4}=\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4} \) Simplify
\( =5\times \frac{3}{4} \)
So, Kevin used \(\frac{3}{4} \) five cups of strawberries to make milkshakes. This shows us that he has enough strawberries to make 5 glasses of milkshake.
Example 6: Nathan has three \( \frac{15}{2} \) gallon jerry cans full of gas. Write an equation to find the total amount of gas that Nathan has.
Solution:
Capacity of 1 jerry can \( =\frac{15}{2} \)
Number of jerry cans = 3
Total amount of gas he has \( =\frac{15}{2}+\frac{15}{2}+\frac{15}{2}=3\times \frac{15}{2} \)
\( =\frac{45}{2}=22.5 \) gallons.
A fraction whose numerator is 1 is known as a unit fraction. A unit fraction can have any whole number in its denominator, but the numerator will always be 1.
A fraction can be expressed as follows: \( \frac{a}{b}=a\times \frac{1}{b} \). Here, the numerator is multiplied with a unit fraction having the same denominator.
A mixed number can be expressed using a unit fraction by converting the mixed number into an improper fraction. The improper fraction can be expressed as a sum or product of the corresponding unit fraction.