LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

COMPARISON 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**

**Find the least common denominator (LCD) of the two fractions. The denominators are 4 and 6.****The LCD of 4 and 6 is 12.****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**

**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**

__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**.