Number System

Exercise 1.1 Part 2

Question 3: Find five rational numbers between 3/5 and 4/5.

Answer: First method to find rational number between two numbers:
Since we have to find 5 rational numbers between 3/5 and 4/5, so we will mutiply the numerator and denominator of the given numbers by 5+1, i.e. equal to 6.
We get,

`3/5=(3xx6)/(5xx6)=18/30` and `4/5=(4xx6)/(5xx6)=24/30`

Now, we have to find 5 rational numbers between 18/30 and 24/30, which will come as

`(19/30),(20/30),(21/30),(22/30),(23/30)`

`=(19/3),((2xx10)/(3xx10)),((7xx3)/(10xx3)),((11xx2)/(15xx2)), (23/30)`

`=(19/30),(2/3),(7/10),(11/15)` and `(23/30)`

Thus, five rational numbers between 3/5 and 4/5 are `=(19/30),(2/3),(7/10),(11/15)` and `(23/30)`


You can find as many rational numbers between given numbers using method given above.

Second method to find rational number between two numbers:
(1) One rational number between given numbers 3/5 and 4/5 will be the average between them.
Thus, average between 3/5 and 4/5 will be equal to

`(3/5+4/5)/2=((3+4)/5)/2=(7/5)/2=7/(5xx2)=7/10`

(2) Rational number between 3/5 and 7/10 will be the next rational number between 3/5 and 4/5, since 7/10 is a rational number between 3/5 and 4/5. Thus, next rational number between 3/5 and 7/10 can be found by calculating average between them. Calculation is as follows:

`(3/5+7/10)/2=((6+7)/10)/2=(13/10)/2=13/(10xx2)=13/20`

(3) Similarly, rational number between 4/5 and 7/10 will be the another rational number between 3/5 and 4/5. This another rational number between 3/5 and 4/5 can be found by calculating average between 4/5 and 7/10, calculation is as follows:

`(4/5+7/10)/2=((8+7)/10)/2=(15/10)/2=15/(10xx2)=3/4`


(4) Similarly, rational number between 3/5 and 13/20 will be the another rational number between 3/5 and 4/5, which can be found by calculating average between 3/5 and 13/20, calculation is as follows:

`(3/5+13/20)/2=((12+13)/20)/2=(25/20)/2=25/(20xx2)=5/8`

(5) Similarly, rational number between 3/5 and 5/8 will be the another rational number between 3/5 and 4/5, which can be found by calculating average between 3/5 and 5/8, calculation is as follows:

`(3/5+5/8)/2=((24+25)/40)/2=49/(40xx2)=49/80`

Thus, five rational numbers between 3/5 and 4/5 are 7/10, 13/20, 3/4, 5/8 and 49/80
You can find more rational numbers between the given two numbers 3/5 and 4/5 using above method.

Question 4: State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Answer:

(1) True
Reason: Natural numbers are counting numbers, starting from 1. Example – 1, 2, 3, 4, .. and so on. While whole numbers are all natural numbers including zero. Thus, every natural number is a whole number.

(2) False
Reason: Integers can be positive and negative both. Since whole numbers are only positive, thus every integer is not a whole number, rather only positive integers are whole number.

(3) False
Reason: Numbers are in the form of p/q or can be written in the form of p/q are called rational numbers, while whole numbers are counting numbers including zero. For example 1/2 is a rational number and clearly it is not a whole number. Thus, every rational number is not a whole number.



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