Fractions
What is a Fraction?
Fraction is any number which can be expressed in the form $\frac{p}{q}$.
(where p and q are natural numbers and q is non-zero).
E.g. $\frac{1}{2}$, $\frac{3}{4}$, $\frac{11}{12}$ etc.
A fraction denotes a part or parts of a unit.
Types of fractions
Proper Fractions: The fractions where the numerator is less than the denominator. So, obviously such fractions are always less than 1.
E.g. $\frac{1}{2}$, $\frac{3}{4}$, $\frac{99}{100}$ etc.Improper Fraction: The fractions where the numerator is more than the denominator. They are greater than 1.
E.g. $\frac{3}{2}$, $\frac{5}{3}$, $\frac{101}{100}$ etc.Mixed Fraction: They have two parts, an integer part and a fractional part. All mixed fractions are improper fractions, i.e. we can convert a mixed fraction into an improper fraction and vice-versa.
E.g. 3$\frac{1}{2}$ = $\frac{7}{2}$ etc.Compound Fraction : Fractions whose numerator and denominator themselves are fractions.
Comparison of Fractions
We are often required to compare fractions, be it in arithmetic or data interpretation questions. So, let us learn how to find the largest or the smallest fraction in a given set of fractions.
Fractions with same denominators
If the fractions to be compared have the same denominator, then our task is simplified. We just need to compare the numerators. Bigger the numerator, bigger the fraction.
Q. Which of these fractions is greater?
2/5 and 4/5
Explanation:
These two fractions have the same denominator. So, the fraction with the bigger numerator will be the bigger fraction.
As, 4 > 2
So, 4/5 > 2/5
Fractions with same numerators
If the fractions to be compared have the same numerator, then our task is simple. We just need to compare the denominators. Bigger the denominator, smaller the fraction.
Q. Which of these fractions is greater?
3/5 and 3/7
Explanation:
These two fractions have the same numerator. So, the fraction with the bigger denominator will be the smaller fraction.
As, 7 > 5
So, 3/5 > 3/7
Fractions having the same value difference between numerator and denominator
You may encounter some fractions, wherein the difference between the numerator and the denominator is the same in all of them.
For example, $\frac{4}{7}$, $\frac{5}{8}$, $\frac{7}{10}$ (In all these fractions, denominator - numerator = 3)
Two possibilities arise in such a case. Let’s see them one by one.
Possibility 1: Given fractions are less than 1
In such a case, the fraction with the largest values of numerator and denominator will be the largest.
For example, in all these fractions: $\frac{4}{7}$, $\frac{5}{8}$, $\frac{7}{10}$, the difference between denominator and numerator = 3. And all of them are less than 1.
So, $\frac{7}{10}$ will be the largest fraction.
Possibility 2: Given fractions are more than 1
In such a case, the fraction with the smallest values of numerator and denominator will be the largest.
For example, in all these fractions: $\frac{9}{7}$, $\frac{11}{9}$, $\frac{13}{11}$, the difference between denominator and numerator = 2. And all of them are more than 1.
So, $\frac{9}{7}$ will be the largest fraction.
There’s an easy way to remember this.
Compare $\frac{9}{10}$ = 0.9 and $\frac{99}{100}$ = 0.99. Obviously, $\frac{99}{100}$ is larger.
Compare $\frac{11}{10}$ = 1.1 and $\frac{101}{100}$ = 1.01. Obviously, $\frac{11}{10}$ is larger.