COMPARISON OF RATIONAL NUMBER -
Just like we compare integers and fractions, we can also compare two rational numbers. We know that every positive integer is greater than zero and every negative integer is less than zero. Also every positive integer is greater than every negative integer.
We will learn the comparison of rational numbers in the current topic.
§ Among the positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare the rational numbers with same denominators.
e.g. 24/30 > 22/30 > 21/30
§ A negative rational number is to the left of zero whereas a positive rational number is to the right of zero on a number line. So, a positive rational number is always greater than a negative rational number.
§ To compare two negative rational numbers with the same denominator, their numerators are compared ignoring the minus sign. The number with the greatest numerator is the smallest.
§ e.g. -7/10 < -3/10; -6/7 < -4/7
§ To compare rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators.
§ There are unlimited number of rational numbers between two rational numbers. To find a rational number between the given rational numbers, they are converted to rational numbers with same denominators.
Other Way Of Understanding -
Comparing rational numbers involves determining which of two or more fractions is larger or smaller. Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero.
Here's a step-by-step guide to comparing rational numbers:-
Step 1:- Common Denominator
To compare two rational numbers, it's often easiest to convert them to a common denominator. This way, you can directly compare the numerators.
Example.1) Compare 3/4, and 5/6
Step 2:- Compare the Numerators
3 3 x 3 9
-------- = ---------- = -------
4 4 x 3 12
5 5 x 2 10
------- = ---------- = -------
6 6 x 2 12
With the fractions converted to a common denominator, compare the numerators directly.
9
12
10
12
3 5
Since 9 < 10, ------ < ------ .
4 6
Step 3:- Cross Multiplication (Alternative Method)
Another method to compare two fractions is cross-multiplication.
Example.2) Compare 7/8, and 5/6
7 × 6 = 42
Multiply the numerator of the second fraction by the denominator of the first fraction:- 5 × 8 = 40
2. Compare the results:
7 5
42 > 40, So ------ > ------
8 6
Step 4:- Decimal Conversion
Another way to compare fractions is by converting them to decimals.
Example.3) Compare 2/5 and 3/7
2
------- = 0.4
5
3
------- ≈ 0.4286
7
2. Compare the decimal values:-
2 3
0.4 < 0.4286, so ------ < -------
5 7
Using these methods, you can accurately compare any pair of rational numbers.