# CLASS-7COMPARISON OF RATIONAL NUMBER

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

1. Find the least common denominator (LCD) of the two fractions. The denominators are 4 and 6.
2. The LCD of 4 and 6 is 12.
3. Convert each fraction to an equivalent fraction with the LCD as the new denominator:-

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

• --------- has a numerator of 9.

12

10

• ---------​ has a numerator of 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

1. Cross-multiply the numerators and denominators:- Multiply the numerator of the first fraction by the denominator of the second fraction:-

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

1. Convert each fraction to a decimal by dividing the numerator by the denominator:-

2

------- = 0.4

5

3

------- ≈ 0.4286

7

2. Compare the decimal values:-

2          3

0.4 < 0.4286, so ------ < -------

5          7

• Common Denominator Method:- Convert fractions to a common denominator and compare numerators.
• Cross Multiplication Method:- Cross-multiply and compare the products.
• Decimal Conversion Method:- Convert fractions to decimals and compare the decimal values.

Using these methods, you can accurately compare any pair of rational numbers.