
Find two rational numbers between the following using average method:
a) -2 and 3
b) $\dfrac{2}{3}$and $\dfrac{{13}}{{14}}$
Answer
457.5k+ views
Hint: In order to find the number of rational numbers between any two numbers, add the given two numbers and divide the sum of two numbers by 2. Repeat this again adding the average of previous two rational numbers with the third number.
Complete step-by-step solution:
Here, in first part of question, we have
a) -2 and 3
As we know, the simplest method to find a rational number between any two rational numbers a and b is to divide their sum by 2.
Here, we have a= -2 and b= 3
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{ - 2 + 3}}{2}$= $\dfrac{1}{2}$= 0.5
Hence, we need to find two rational numbers between them,
Now we find rational number between -2 and 0.5 (which is rational numbers between -2 and 3)
Here, we have a= -2 and b= 0.5
Rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{ - 2 + 0.5}}{2}$= $\dfrac{{ - 1.5}}{2}$= -0.75
Therefore, two rational numbers between -2 and 3 are $\dfrac{1}{2}$ and $\dfrac{{ - 1.5}}{2}$
b) $\dfrac{2}{3}$ and $\dfrac{{13}}{{14}}$
Now, In Second part of question-
Here, we have a= $\dfrac{2}{3}$ and b= $\dfrac{{13}}{{14}}$
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{\dfrac{2}{3} + \dfrac{{13}}{{14}}}}{2}$= $\dfrac{{\dfrac{{28 + 39}}{{42}}}}{2}$= $\dfrac{{\dfrac{{67}}{{42}}}}{2}$= $\dfrac{{67}}{{84}}$
Hence, we need to find two rational numbers between them,
Now we find rational number between $\dfrac{2}{3}$ and $\dfrac{{67}}{{84}}$ (which is rational numbers between $\dfrac{2}{3}$ and $\dfrac{{13}}{{14}}$)
Here, we have a= $\dfrac{2}{3}$ and b= $\dfrac{{67}}{{84}}$
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{\dfrac{2}{3} + \dfrac{{67}}{{84}}}}{2} = \dfrac{{\dfrac{{56 + 67}}{{84}}}}{2} = \dfrac{{\dfrac{{123}}{{84}}}}{2} = \dfrac{{123}}{{168}}$
Therefore, two rational numbers between $\dfrac{2}{3}$and $\dfrac{{13}}{{14}}$ is $\dfrac{{67}}{{84}}$ and $\dfrac{{123}}{{168}}$.
Note: Always remember that a rational number is a number which can be written in the form of $\dfrac{{\text{p}}}{{\text{q}}}$(ratio) where the denominator(q) is not equal to 0. This means it can be represented in the form of a fraction. Therefore, we say every rational number has a numerator and a denominator, that is, one integer divided by another integer, where the denominator is not equal to zero.
Complete step-by-step solution:
Here, in first part of question, we have
a) -2 and 3
As we know, the simplest method to find a rational number between any two rational numbers a and b is to divide their sum by 2.
Here, we have a= -2 and b= 3
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{ - 2 + 3}}{2}$= $\dfrac{1}{2}$= 0.5
Hence, we need to find two rational numbers between them,
Now we find rational number between -2 and 0.5 (which is rational numbers between -2 and 3)
Here, we have a= -2 and b= 0.5
Rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{ - 2 + 0.5}}{2}$= $\dfrac{{ - 1.5}}{2}$= -0.75
Therefore, two rational numbers between -2 and 3 are $\dfrac{1}{2}$ and $\dfrac{{ - 1.5}}{2}$
b) $\dfrac{2}{3}$ and $\dfrac{{13}}{{14}}$
Now, In Second part of question-
Here, we have a= $\dfrac{2}{3}$ and b= $\dfrac{{13}}{{14}}$
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{\dfrac{2}{3} + \dfrac{{13}}{{14}}}}{2}$= $\dfrac{{\dfrac{{28 + 39}}{{42}}}}{2}$= $\dfrac{{\dfrac{{67}}{{42}}}}{2}$= $\dfrac{{67}}{{84}}$
Hence, we need to find two rational numbers between them,
Now we find rational number between $\dfrac{2}{3}$ and $\dfrac{{67}}{{84}}$ (which is rational numbers between $\dfrac{2}{3}$ and $\dfrac{{13}}{{14}}$)
Here, we have a= $\dfrac{2}{3}$ and b= $\dfrac{{67}}{{84}}$
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{\dfrac{2}{3} + \dfrac{{67}}{{84}}}}{2} = \dfrac{{\dfrac{{56 + 67}}{{84}}}}{2} = \dfrac{{\dfrac{{123}}{{84}}}}{2} = \dfrac{{123}}{{168}}$
Therefore, two rational numbers between $\dfrac{2}{3}$and $\dfrac{{13}}{{14}}$ is $\dfrac{{67}}{{84}}$ and $\dfrac{{123}}{{168}}$.
Note: Always remember that a rational number is a number which can be written in the form of $\dfrac{{\text{p}}}{{\text{q}}}$(ratio) where the denominator(q) is not equal to 0. This means it can be represented in the form of a fraction. Therefore, we say every rational number has a numerator and a denominator, that is, one integer divided by another integer, where the denominator is not equal to zero.
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
When people say No pun intended what does that mea class 8 english CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How many ounces are in 500 mL class 8 maths CBSE

Which king started the organization of the Kumbh fair class 8 social science CBSE

Advantages and disadvantages of science

What is BLO What is the full form of BLO class 8 social science CBSE
