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.

rational number rational number

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`



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