# Rational Numbers

## Exercise 9.1

Question 5: The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.  Answer: P=2\1/3=7/3

Q=2\2/3=8/3

R=-1\1/3=-4/3

S=-1\2/3=-5/3

Question 6: Which of the following pairs represent the same rational number?

(a) -(7)/(21) and 3/9

Answer: Since one of the numbers is negative and another is positive, hence they cannot be same.

(b) -(16)/(20) and (20)/(25)

Answer: Since one of the numbers is negative and another is positive, they cannot be same.

(c) (-2)/(-3) and 2/3

Answer: Both the numbers are positive and have same numerator and denominator, so they are same.

(d) (-3)/(5) and (-12)/(20)

Answer: Let us first convert the given numbers into standard form:
First number has co-prime numerator and denominator, so it is in standard form. Second number can be converted to standard form as follows:

(-12)/(20)=(-3xx4)/(5xx4)=-3/5

So, both numbers are same.

(e) (8)/(-5) and (-24)/(15)

Answer: First number has co-prime numerator and denominator.
Let us convert the second number into standard form:

(-24)/(15)=(-8xx3)/(5xx3)=-8/5

So, both numbers are same.

(f) 1/3 and -1/9

Answer: One of the numbers is negative and another is positive, so they cannot be same.

(g) (-5)/(-9) and (5)/(-9)

Answer: One of the numbers is negative and another is positive, so they cannot be same.

Question 7: Rewrite the following rational numbers in the simplest form:

(a) (-8)/(6)

Answer: (-8)/(6)=(-2xx4)/(2xx3)=(-4)/(3)

(b) (25)/(45)

Answer: (25)/(45)=(5xx5)/(5xx9)=5/9

(c) (-44)/(72)

Answer: (-44)/(72)=(-4xx11)/(4xx18)=(-11)/(18)

(d) (-8)/(10)

Answer: (-8)/(10)=(-2xx4)/(2xx5)=-4/5

Question 8: Fill in the boxes with the correct symbol out of >, < and =.

(a) (-7)/(5)[ ]2/3

Answer: In this case, the negative number is less than the positive number.

(-7)/(5)<2/3

(b) (-4)/(5)[ ](-5)/(7)

Answer: Let us first find the LCM of denominators and then convert the numbers with LCM as denominator. LCM of 5 and 7 is 35.

(-4)/(5)=(-28)/(35) and (-5)/(7)=(-25)/(35)

Hence, (-4)/(5)<(-5)/(7)

(c) (-7)/(8)[ ](14)/(-16)

Answer: LCM of 8 and 16 is 16.

(-7)/(8)=(-14)/(16)

Hence, (-7)/(8)=(14)/(-16)

(d) (-8)/(5)[ ](-7)/(4)

Answer: LCM of 5 and 4 is 20.

(-8)/(5)=(-32)/(20) and (-7)/(4)=(-35)/(20)

Hence, (-8)/(5)>(-7)/(4)

(e) (1)/(-3)[ ](-1)/(4)

Answer: LCM of 3 and 4 is 12.

(-1)/(3)=(-4)/(12) and (-1)/(4)=(-3)/(12)

Hence, (1)/(-3)<(-1)/(4)

(f) (5)/(-11)[ ](-5)/(11)

Answer: (5)/(-11)=(-5)/(11)

(g) 0[ ](-7)/(6)

Answer: Any negative number is less than zero.

Hence, 0>(-7)/(6)

Question 9:

(a) 2/3, 5/2

Answer: LCM of 3 and 2 is 6.

2/3=4/6 and 5/2=(15)/(6)

Hence, 5/2>2/3

(b) (-5)/(6), (-4)/(3)

Answer: LCM of 3 and 6 is 6.

(-4)/(3)=(-8)/(6)

Hence, (-5)/(6)>(-4)/(3)

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

Answer: LCM of 4 and 3 is 12.

(-3)/(4)=(-9)/(12) and (-2)/(3)=(-8)/(12)

Hence, (-2)/(3)>(-3)/(4)

(d) (-1)/(4), 1/4

Answer: Any negative number is always less than any positive number.

Hence, 1/4> -1/4

(e) -3\2/7, -3\4/5

Answer: LCM of 7 and 5 is 35.

-3\2/7=(-22)/(7) and -3\4/5=(-19)/(5)

Now, (-22)/(7)=(-110)/(35) and (-19)/(5)=(-133)/(35)

Hence, -3\2/7> -3\4/5

Question 10: Write the following rational numbers in ascending order:

(a) -3/5, -2/5, -1/5

Answer: In this case, denominator is same in all numbers:

Hence, -3/5< -2/5< -1/5

(b) 1/3, -2/9, -4/3

Answer: LCM of 3 and 9 is 9. Let us convert these numbers with denominator as 9.

1/3=3/9 and -4/3=(-12)/(9)

Hence, -4/3< -2/9<1/3

(c) -3/7, -3/2, -3/4

Answer: LCM of 2, 4 and 7 is 28. Let us convert these numbers with denominator as 28.

-3/7=-(12)/(28), -3/2=-(42)/(28), -3/4=-(21)/(28)

Hence, -3/2< -3/4< -3/7