Fractions are one of the most significant concepts in mathematics. The word fraction is derived from the Latin word “fractus” which means “broken”. The word fraction represents a chunk of a whole. A simple fraction consists of a numerator that represents the upper part above the line and the denominator that denotes below the line. In the case of positive common fractions, both the values of the numerator and denominator represent natural numbers.
The value of the numerator represents a number of equal parts whereas the denominator indicates how many of those chunks constitute a whole. The value of the denominator cannot be zero. Common fractions are classified into proper fractions and improper fractions. The fractions where the numerator is lower than the denominator are called proper fractions. The fractions where the numerator is greater than the denominator are termed improper fractions. The worth of a proper fraction is always lower than the number one.
Adding Proper Fractions
In the case of like fractions, the numerators are being added and the sum of the two fractions are obtained. Consider an example of adding (5 / 8) + (7 / 8). The numerators 5 and 7 can be added to get 12 as the denominators are the same. So, the answer is 12 / 8 which can be simplified to 3 / 2. When the two given fractions are unlike fractions, that is when the denominators are not the same, the least common multiple (LCM) is taken. When the denominators become the same through LCM, the numerators can be added. Take two fractions (2 / 3) + (4 / 5). LCM of the denominators need to be taken in order to obtain the sum. The least common multiple of 3 and 5 is 15. A number is multiplied by both the fractions such that the denominator is equalized. So, (2 / 3) * (5 / 5) + (4 / 5) * (3 / 3)
= (10 / 15) + (12 / 15)
= (10 + 12) / 15
= 22 / 15
Subtracting Proper Fractions
The process of subtracting proper fractions is similar to that of addition. If the fractions given are like fractions, then the difference of numerators is obtained by retaining the denominator. For example, (9 / 8) – (5 / 8) = 3 / 8. When the two given fractions are unlike fractions, that is when the denominators are not the same, the least common multiple (LCM) is taken. When the denominators become the same through LCM, the numerators can be subtracted. Take two fractions (8 / 9) – (3 / 4). LCM of the denominators need to be taken in order to obtain the sum. The least common multiple of 9 and 4 is 36. A number is multiplied by both the fractions such that the denominator is equalized. So, (8 / 9) * (4 / 4) – (3 / 4) * (9 / 9)
= (32 / 36) – (27 / 36)
= (32 – 27) / 36
= 5 / 36
Multiplying Proper Fractions
In the case of obtaining the product of two proper fractions, the numerators and denominators are multiplied individually. The two fractions are (2 / 6) * (5 / 4). On multiplication of numerators, we get 2 * 5 = 10 and denominators we get, (6 * 4) = 24. The answer is 10 / 24 which can be simplified to 5 /12.
Dividing Proper Fractions
In the process of dividing two fractions that are proper in nature, the division sign is replaced by the multiplication sign and the first fraction is multiplied by the reciprocal of the second fraction. The concept of proper fractions is briefly described above. For more information on different topics such as trigonometry, calculus, set theory, mathematical logic, topology, partial fractions, differential equations etc refer to the BYJU’S website. It contains detailed explanations along with solved examples.