A fraction represents a portion of a whole, which could be a region or a collection. The term "fraction" originates from the Latin word 'fractio', meaning 'to break'. The ancient Egyptians, credited as the earliest civilization to explore fractions, employed them to tackle various mathematical challenges, such as dividing food, supplies, and dealing with the lack of a standardized currency.
A fraction is a numerical representation of a part of a whole. Typically, fractions can denote a portion of any quantity relative to the whole, which can encompass various specific entities or values.
Fundamentally, fractions are composed of two numbers: the numerator and the denominator. The numerator signifies the quantity of selected or shaded parts of the whole, while the denominator represents the total number of parts comprising the whole.
Fractions, denoted as `x/4` when a number is divided into four equal parts, represent a fundamental concept wherein `1/4` signifies one part out of four equal divisions. Commonly referred to as fractions, they play a significant role in various aspects of our daily routines. In practical scenarios, fractions appear abundantly, representing the division or portioning of entities into equal segments. Imagine you have a whole pizza. If you slice it in half, each half represents a fraction, equal to `1/2`. So, each part of the sliced pizza represents a fraction of the whole.
Fraction parts comprise a numerator and a denominator, with a horizontal bar dividing them.
Let's examine some key characteristics of fractions that resemble those of whole numbers, natural numbers, and so forth.
Property | Explanation | Example |
Commutative Property (Addition & Multiplication) | The result remains the same regardless of whether fractions are added or multiplied in different orders. | Addition: `\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}`
Multiplication: `\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}` |
Associative Property (Addition & Multiplication) | The grouping of fractions in addition or multiplication does not alter the outcome. | Addition: `\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a}{b} + \left(\frac{c}{d} + \frac{e}{f}\right)`
Multiplication: `\left(\frac{a}{b} \times \frac{c}{d}\right) \times \frac{e}{f} = \frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f}\right)` |
Identity Element | In multiplication, the identity element is `1`, as multiplying a fraction by `1` results in the same fraction. However, in addition, adding `0` to a fraction does not alter its value. | Multiplication: `\frac{a}{b} \times 1 = \frac{a}{b}`
Addition: `\frac{a}{b} + 0 = \frac{a}{b}` |
Multiplicative Inverse | The product of a fraction and its reciprocal equals `1`. | `\frac{a}{b} \times \frac{b}{a} = 1` |
When you multiply a fraction by the sum of two other fractions, it's the same as multiplying the fraction by each of the two fractions separately and then adding the results together. | `\frac{a}{b} \times \left(\frac{c}{d} + \frac{e}{f}\right) = \frac{a}{b} \times \frac{c}{d} + \frac{a}{b} \times \frac{e}{f}` |
Fractions are classified into different types based on the characteristics of their numerator and denominator. They are:
Proper fractions
Proper fractions are fractions where the numerator is smaller than the denominator.
For instance, `\frac{4}{9}, \frac{1}{3}, \frac{2}{5}` etc., are proper fractions.
Improper fractions
An improper fraction is a fraction where the numerator exceeds or equals the denominator. It represents a quantity equal to or greater than one whole.
For instance, `\frac{7}{4}`, `\frac{9}{2}`, `\frac{11}{6}`, etc.
Mixed fractions
A mixed fraction combines an integer with a proper fraction.
For instance, `\frac{7}{2}` is represented as `3\frac{1}{2}`, where \(3\) is the whole number and `\frac{1}{2}` is the proper fraction.
Another example would be `4\frac{3}{4}`, `9\frac{2}{5}`, etc.
Like fractions
Fractions with identical denominators are referred to as like fractions.
For instance, `\frac{4}{12}` and `\frac{7}{12}` are like fractions.
Unlike fractions
Fractions with distinct denominators are referred to as unlike fractions.
For instance, `\frac{7}{13}` and `\frac{21}{25}` are unlike fractions.
Equivalent fractions
Two fractions are considered equivalent if, upon simplification, either fraction is equal to the other.
For instance, `\frac{5}{10}` and `\frac{1}{2}` are equivalent fractions.
Since, `\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}`
Unit fractions are fractions where the numerator is `1`.
For instance, `\frac{1}{5}`, `\frac{1}{8}`, `\frac{1}{11}`, and so forth.
Fractions positioned on a number line inhabit the space between two whole numbers. Each fraction denotes a portion of the whole, dividing it into equal segments. The number of segments between the integers corresponds to the denominator of the fraction, while the numerator specifies the exact location of the fraction along the line.
For instance, to represent `\frac{4}{6}` on a number line, the interval would be between \(0\) and \(1\), and since \(6\) is the denominator, the whole is partitioned into \(6\) equal parts, where the first represents `\frac{1}{6}`. Likewise, the second part represents `\frac{2}{6}`, and so forth.
Example `1`: Find two equivalent fractions for `\frac{3}{12}`.
Solution:
Let us write the equivalent fractions for `\frac{3}{12}` using multiplication and division.
a) By multiplying both the numerator and the denominator of `\frac{3}{12}` by `2`, we get:
`\frac{3 \times 2}{12 \times 2} = \frac{6}{24}`
b) By dividing both the numerator and the denominator of `\frac{3}{12}` by `3`, we get:
`\frac{3 \div 3}{12 \div 3} = \frac{1}{4}`
Thus, `\frac{6}{24}` and `\frac{1}{4}` are equivalent to `\frac{3}{12}`. In other words, `\frac{6}{24}`, `\frac{1}{4}`, and `\frac{3}{12}` are equivalent fractions.
Example `2`: Identify the type of fraction.
`\frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \frac{1}{9}, \frac{1}{11}`
Solution:
These fractions are considered unit fractions because they all have a numerator of `1`.
Example `3`: Compute the sum of `\frac{3}{7}` and `\frac{5}{14}`.
Solution:
`\frac{3}{7} + \frac{5}{14}`
LCM of `7` and `14` is `14`.
`= \frac{6 + 5}{14}`
`= \frac{11}{14}`
Example `4`: In a group of `60` people, `\frac{1}{3}` of them prefer tea over coffee. How many people prefer coffee?
Solution:
Total number of people `=` \( 60 \)
Fraction of people who prefer tea `= \frac{1}{3}`
Number of people who prefer tea:
`\frac{1}{3} \times 60 = 20`
Number of people who prefer coffee:
\( 60 - 20 = 40 \)
Therefore, the number of people who prefer coffee is \( 40 \).
Example `5`: Reduce the fraction `\frac{24}{60}` to the simplest form.
Solution:
Given fraction: `\frac{24}{60}`.
Factors of `24: 1, 2, 3, 4, 6, 8, 12,` and `24`.
Factors of `60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,` and `60`.
Hence, the greatest common factor of `24` and `60` is `12`.
i.e. GCF `(24, 60) = 12`
Now, divide both the numerator and the denominator of the given fraction `\frac{24}{60}` by `12`, we get
`\frac{24}{60} = \frac{\frac{24}{12}}{\frac{60}{12}} = \frac{2}{5}`
Hence, the simplest form of the fraction `\frac{24}{60}` is `\frac{2}{5}`.
Q`1`. Compute the sum of `\frac{1}{3}` and `\frac{4}{9}`.
Answer: c
Q`2`. In a garden, `\frac{1}{5}` of the flowers are roses. If there are `80` flowers in total, how many flowers are not roses?
Answer: d
Q`3`. Find two equivalent fractions for `\frac{7}{21}`.
Answer: a
Q`4`. A bakery sells \(120\) cupcakes, and `\frac{1}{6}` of them are chocolate-flavored. How many cupcakes are chocolate-flavored?
Answer: b
Q`5`. Reduce the fraction `\frac{28}{84}` to the simplest form.
Answer: a
Q`1`. How do you add fractions with different denominators?
Answer: Find the least common denominator (LCD) and then add the fractions by converting them to equivalent fractions with the same denominator.
Q`2`. What is a mixed number?
Answer: A mixed number combines a whole number and a proper fraction, representing a value larger than one.
Q`3`. How do you subtract fractions?
Answer: Subtract fractions by finding a common denominator, then subtracting the numerators while keeping the denominator the same.
Q`4`. What is a fraction?
Answer: A fraction represents a part of a whole, consisting of a numerator (top number) and a denominator (bottom number).
Q`5`. How do you simplify fractions?
Answer: Simplify fractions by dividing the numerator and denominator by their greatest common divisor (GCD) to reduce them to the smallest form.