Class 7 Maths

Rational Numbers

Rational Number: A number that can be expressed in the form of p/q where p and q are integers and q≠0, is called a rational number.
All integers and fractions are rational numbers.

If the numerator and denominator of a rational number are multiplied or divided by a non-zero integer then we get a rational number which is equivalent to the given rational number. Following is an example;

2/3=(2xx3)/(3xx3)=6/9

When both numerator and denominator of a rational number are positive integers then the given rational number is a positive rational number. When either the numerator or the denominator is a negative integer then the given rational number is a negative rational number.

The number zero is neither a positive nor a negative rational number. But zero is a rational number because it can be written in the form of p/q where q≠0.

Standard form of Rational Number: When the denominator is a positive integer and both the numerator and denominator are co-prime then the rational number is said to be in the standard form.

There are unlimited number of rational numbers between two rational numbers.

Exercise 9.1

Question 1: List five rational numbers between following rational numbers:

(a) – 1 and 0

Answer: Since we need to find five rational numbers between the given rational numbers, let us write the given numbers with denominator 6.

-1=(-6)/(6) and 0=0/6

Rational numbers between these numbers can be as follows:

-5/6, -4/6, -3/6, -2/6, -1/6

(b) – 2 and – 1

Answer: Let us write the given numbers with denominator 6:

-2=(-12)/(6) and -1=(-6)/(6)

Rational numbers between these numbers can be as follows:

-(11)/(6), (-10)/(6), -9/6, -8/6, -7/6

(c)

-4/5 and -2/3

Answer: LCM of denominators (5 and 3) is 15. Let us write the given numbers with 15 as denominator.

-4/5=-(12)/(15) and -2/3=-(10)/(15)

So, the rational number between two given number is as follows:

-(11)/(15)

To find more rational numbers in between, let us change the denominator to 30 (multiply by 2). Then the numbers can be written as follows:

-(24)/(30), -(22)/(30), -(20)/(30)

This will help us in writing following rational numbers in between:

-(23)/(30) and -(21)/(30)

To find more rational numbers in between, let us change the denominator to 60 (multiply by 2). Then the numbers can be written as follows:

-(48)/(60), -(46)/(60), -(44)/(60), -(42)/(60), -(40)/(60)

This will help is in writing following rational numbers in between:

-(47)/(60), -(46)/(60), -(45)/(60), -(44)/(60), -(42)/(60)

(e) 1/2 and 2/3

Answer: LCM of denominators (2 and 3) is 6. Converting the given numbers with 6 as denominator gives us following numbers:

1/2=3/6 and 2/3=4/6

Above pair has no number between integers in numerators, i.e. 3 and 4.
So, let us convert these numbers to get denominator 12:

1/2=(6)/(12) and 2/3=(8)/(12)

Above pair gives scope for one rational number in between. So, let us convert these numbers to get denominator 24:

1/2=(12)/(24) and 2/3=(16)/(24)

Above pair gives scope for three rational numbers in between. So, let us convert these numbers to get denominator 48:

1/2=(24)/(48) and 2/3=(32)/(48)

Now, required rational numbers as are follows:

(25)/(48), (26)/(48), (27)/(48) and (29)/(48)

Question 2: Write four more rational numbers in each of the following patterns:

(a) -3/5, -(6)/(10), -(9)/(15), -(12)/(20)

Answer: The series is increasing by multiplying the numerator and denominator of the first number subsequently by 2, 3, 4, etc. So, next four rational numbers in the series are:

-(15)/(25), -(18)/(30), -(21)/(35), -(24)/(40)

(b) -1/4, -2/8, -(3)/(12)

Answer: Next four rational numbers in the series are:

-(4)/(16), -(5)/(20), -(6)/(24), -(7)/(28)

(c) (-2)/(3), (2)/(-3), (4)/(-6), (6)/(-9)

Answer: (8)/(-12), (10)/(-15), (12)/(-18), (14)/(-21)

Question 3: Give four rational numbers equivalent to following rational numbers:

(a) -2/7

Answer: -(4)/(14), -(6)/(21), -(8)/(28), -(10)/(35)

(b) (5)/(-3)

Answer: (10)/(-6), (15)/(-9), (20)/(-12), (25)/(-15)

(c) 4/9

Answer: (8)/(18), (12)/(27), (16)/(36), (20)/(45)

Question 4: Draw the number line and represent the following rational numbers on it: 3/4, -5/8, -7/4 and 7/8 