Before you understand this concept, you are advice to read:
How to calculate H.C.F ?
Definition: A rational number is in standard or simplest or lowest form when following two conditions are fulfilled:
Numerator and denominator have only 1 as its highest common factor.
Denominator is a positive integer.
Lets study some examples on this -
Example 1: Is rational number 6/7 is in standard form ?
Solution : This proceeds in the following ways:
First find the HCF of denominator and numerator i.e. 6 & 7 and we get:
HCF of 6 & 7 = 1
Since, Numerator and denominator have only 1 as its highest common factor; so as explained above first condition is fulfilled.
Now check second condition ?
Here we can see that denominator is 7, which is a positive integer. So this fulfills second condition also.
Now, since both the conditions are fulfilled, we can say that given rational number 6/7 is in standard form.
Example 2: Is rational number 25/45 is in standard form ?
Solution : This proceeds in the following ways:
First find the HCF of denominator and numerator i.e. 25 & 45 and we get:
HCF of 25 & 45 = 5
Since, Numerator and denominator have 5 as its highest common factors; so this does not fulfils above explained first condition.
Hence, 25/45 is not in simplest form.
Example 3: Is rational number 6/-7 is in standard form ?
Solution : This proceeds in the following ways:
First find the HCF of denominator and numerator i.e. 6 & 7 and we get:
(Note: here we find hcf and ignores the negative operator)
HCF of 6 & 7 = 1
Since, Numerator and denominator have only 1 as its highest common factor; so as explained above first condition is fulfilled.
Now check second condition ?
Here we can see that denominator is (-7), which is a negative integer. So this does not fulfils second condition.
Now, since only one condition is fulfilled here, so we can say that given rational number 6/-7 is not in lowest form.
In above example 2 and example 3, you had understood that given rational numbers are not in standard form. Therefore now let's study
How we can convert these rational numbers into standard form
Example 4: Convert 25/45 in standard form.
Solution : This proceeds in the following steps:
Step 1: Find the HCF of denominator and numerator i.e. 25 & 45 and we get:
HCF of 25 & 45 = 5
Step 2: Divide both numerator and denominator with HCF calculated in step 2 and we get:
(25 / 5) / (45 / 5)
= 5 / 7
Now, if we further find the HCF of denominator and numerator above calculated rational number i.e. 5 and 7; we will get 1 as it highest common factor.
So, this fulfils first condition (explained above)
Now check second condition ?
Here we can see that denominator is 7, which is a positive integer. So this fulfils second condition also.
Now, since both conditions are fulfilled, we can say that given rational number 25/45 after conversion gives resultant rational number 5/7 and this resultant rational number 5/7 is in standard form.
Example 5: Convert 6/-7 in simplest form ?
Solution : This proceeds in the following ways:
First find the HCF of denominator and numerator i.e. 6 & 7 and we get:
(Note: here we find hcf and ignores the negative operator)
HCF of 6 & 7 = 1
Since, Numerator and denominator have only 1 as its highest common factor; so as explained above first condition is fulfilled.
Now, we multiply both numerator and denominator with (-1) and we get:
6 X (-1) / -7 X (-1)
= -6 / 7
Now check second condition ?
Here we can see that denominator is 7, which is a positive integer. So this fulfils second condition too.
Now, since both conditions are fulfilled, we can say that given rational number -6 / 7 after conversion gives resultant rational number -6 / 7 and this resultant rational number -6 / 7 is in simplest form.
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