Forms are (120/168), (63/168), (108/168) and (160/168) having same denominators.
(iii) (132/(-428)) HCF of 132 and 428 is 4
Divide the given number by its HCF
(132 ÷ (4/(-428)) ÷ 4) = (33/(-107))
Hence, (33/(-107)) is the simplest form of (132/(-428))
(iv) ((-32)/(-56)) HCF of 32 and 56 is 8
Divide the given number by its HCF
(-32 ÷ (8/(-56)) ÷ 8) = (4/7)
Hence, (4/7) is the simplest form of ((-32)/(-56))
Question :3. Fill in the blanks:
(i) ((-5)/7) = (…/35) = (…/49)
(ii) ((-4)/(-9)) = (…/18) = (12/…)
(iii) (6/(-13)) = ((-12)/…) = (24/…)
(iv) ((-6)/…) = (3/11) = (…/(-55))
Solution 3:
(i) ((-5)/7) = (…/35) = (…/49)= ((-5)/7) = ((-25)/35) = ((-35)/49)
Description:
((-5)/7) = (…/35) = (…/49)
On multiplying by 5:
((-5)/7) × (5/5) = ((-25)/35)
On multiplying by 7
((-5)/7) × (7/7) = ((-35)/49)
(ii) ((-4)/(-9)) = (…/18) = (12/…)= ((-4)/(-9)) = (8/18) = (12/27)
Description:
((-4)/(-9)) = (…/18) = (12/…)
On multiplying by -2
((-4)/(-9)) × ((-2)/(-2)) = (8/18)
on multiplying by -3
((-4)/(-9)) × ((-3)/(-3)) = (12/27)
(iii)(6/(-13)) = ((-12)/…) = (24/…)= (6/(-13)) = ((-12)/26) = (24/(-52))
Description:
(6/(-13)) = ((-12)/…) = (24/…)
On multiplying by -2
(6/(-13)) × ((-2)/(-2)) = ((-12)/26)
On multiplying by 4
(6/(-13)) × (4/4) = (24/(-52))
(iv) ((-6)/…) = (3/11) = (…/(-55))= ((-6)/(-22)) = (3/11) = ((-15)/(-55))
Description:
((-7)/…) = (3/11) = (…/(-55))
0n multiplying by -2
(3/11) × ((-2)/(-2)) = ((-6)/(-22))
On multiplying by -5
(3/11) × ((-5)/(-5)) = ((-15)/(-55))
Exercise 4.4
Question 1: Write each of the following rational numbers in the standard form:
(i) (2/10)
(ii) ((-8)/36)
(iii) (4/(-16))
(iv) ((-15)/(-35))
(v) (299/(-161))
(vi) ((-63)/(-210))
(vii) (68/(-119))
(viii) ((-195)/275)
Solution 1:
(i) (2/10) HCF of 2 and 10 is 2
Dividing the fraction by HCF i.e. 2:
(2/10) ÷ (2/2) = (1/5)
Hence, (1/5) is the standard form of (2/10).
(ii) ((-8)/36) HCF of 8 and 36 is 4
Dividing the fraction by HCF i.e. 4:
((-8)/36) ÷ (4/4) = ((-2)/9)
Hence ((-2)/9) is the standard form of ((-8)/36).
(iii) (4/(-16))
Denominator is negative here, so we multiply fraction by -1
(4/(-16)) × ((-1)/(-1)) = (4/(-16))
HCF of 4 and 16 is 4
Dividing the fraction by HCF i.e. 4:
(4/(-16)) ÷ (4/4) = ((-1)/4)
Hence ((-1)/4) is the standard form of (4/(-16)).
(iv) ((-15)/(-35))
Denominator is negative here, so we multiply fraction by -1
((-15)/(-35)) × ((-1)/(-1)) = (15/35)
HCF of 15 and 35 is 4
Dividing the numerator and denominator by HCF i.e. 5:
((-15)/(-35)) ÷ (5/5) = (3/7)
Hence (3/7) is the standard form of ((-15)/(-35))
(v) (299/(-161))
Denominator is negative here, so we multiply fraction by -1
(299/(-161)) × ((-1)/(-1)) = ((-299)/161)
The HCF of 299 and 161 is 23
Dividing the numerator and denominator by HCF i.e. 23:
(299/(-161)) ÷ (23/23) = ((-13)/7)
Hence ((-13)/7) is the standard form of (299/(-161))
(vi) ((-63)/(-210))
HCF of 63 and 210 is 21
Dividing the numerator and denominator by HCF i.e. 21:
((-63)/(-210)) ÷ (21/21) = ((-3)/(-10)) = (3/10)
Hence (3/10) is the standard form of ((-63)/(-210))
(vii) (68/(-119))
Here denominator is negative so we have multiply both fraction by -1
(68/(-119)) × ((-1)/(-1)) = ((-68)/119)
The HCF of 68 and 119 is 17
Dividing the fraction by HCF i.e. 17:
((-68)/119) ÷ (17/17) = ((-4)/7)
Hence, ((-4)/7) is the standard form of (68/(-119))
(viii) ((-195)/275)
The HCF of 195 and 257 is 5
Dividing the fraction by HCF i.e. 5:
((-195)/275) ÷ (5/5) = ((-39)/5)
Hence, ((-39)/5) is the standard form of ((-195)/275)
Exercise 4.5
Question :1. Which of the following rational numbers are equal?
(i) ((-9)/12) and (8/(-12))
(ii) ((-16)/20) and (20/(-25))
(iii) ((-7)/21) and (3/(-9))
(iv) ((-8)/(-14)) and (13/21)
Solution 1:
(i) ((-9)/12) and (8/(-12))
By dividing the fraction by their HCF i.e. 3
Then the standard form of ((-9)/12) is ((-3)/4)
Dividing the fraction of given number by their HCF i.e. by 4
The standard form of (8/(-12)) = ((-2)/3)
Since, the standard forms of two rational numbers are not same. Hence, they are not equal.
(ii) ((-16)/20) and (20/(-25))
Multiplying fraction of ((-16)/20) by the denominator of (20/(-25)) i.e. -25.
((-16)/20) × ((-25)/(-25)) = (400/(-500))
Multiplying the fraction of (20/(-25)) by the denominator of
((-16)/20) i.e. 20
(20/(-25)) × (20/20) = (400/(-500))
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) ((-7)/21) and (3/(-9))
Multiplying fraction of ((-7)/21) by the denominator of (3/(-9))
i.e. -9.
((-7)/21) × ((-9)/(-9)) = (63/(-189))
Now multiply the fraction of (3/(-9)) by the denominator of
((-7)/21) i.e. 21
(3/(-9)) × (21/21) = (63/(-189))
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iv) ((-8)/(-14)) and (13/21)
Multiplying fraction of ((-8)/(-14)) by the denominator of (13/21)
i.e. 21
((-8)/(-14)) × (21/21) = ((-168)/(-294))
Now multiply the fraction of (13/21) by the denominator of
((-8)/(-14)) i.e. -14
(13/21) × ((-14)/(-14)) = ((-182)/(-294))
Clearly, the numerators of the above obtained rational numbers are not equal.
Hence, the given rational numbers are also not equal
Question 2: In each of the following pairs represent a pair of equivalent rational numbers, find the values of x.
(i) (2/3) and (5/x)
(ii) ((-3)/7) and (x/4)
(iii) (3/5) and (x/(-25))
(iv) (13/6) and ((-65)/x)
Solution 2:
(i) (2/3) and (5/x)
They are equivalent rational number so (2/3) = (5/x)
x = ((5 × 3))/2
x = (15/2)
(ii) ((-3)/7) and (x/4)
they are equivalent rational number so ((-3)/7) = (x/4)
x = ((-3 × 4))/7
x = ((-12)/7)
(iii) Given (3/5) and (x/(-25))
They are equivalent rational number so (3/5) = (x/(-25))
x = ((3 × -25))/5
x = ((-75)/5)
x = -15
(iv) (13/6) and ((-65)/x)
They are equivalent rational number so (13/6) = ((-65)/x)
x = 6/13 x (- 65)
x = 6 x (-5)
x = -30
Question 3: In each of the following, fill in the blanks so as to make the statement true:
(i) A number which can be expressed in the form p/q, where p and q are integers and q is not equal to zero, is called a _______.
(ii) If the integers p and q have no common divisor other than 1 and q is positive, then the rational number (p/q) is said to be in the _______.
(iii) Two rational numbers are said to be equal, if they have the same _______ form
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/ _______.
(v) If p and q are positive Integers, then p/q is a _______ rational number and (p/(-q)) is a ______ rational number.
(vi) The standard form of -1 is _______.
(vii) If (p/q) is a rational number, then q cannot _______.
(viii) Two rational numbers with different numerators are equal, if their numerators are in the same _______ as their denominators.
Solution 3:
(i) A number which can be expressed in the form p/q, where p and q are integers and q is not equal to zero, is called a Rational number.
(ii) If the integers p and q have no common divisor other than 1 and q is positive, then the rational number (p/q) is said to be in the Standard form.
(iii) Two rational numbers are said to be equal, if they have the same Standard form
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/ b ÷ m.
(v) If p and q are positive Integers, then p/q is a Positive rational number and (p/(-q)) is a negative rational number.
(vi) The standard form of -1 is((-1)/1) .
(vii) If (p/q) is a rational number, then q cannot Zero .
(viii) Two rational numbers with different numerators are equal, if their numerators are in the same Ratio as their denominators.
Question 4: In each of the following state if the statement is true (T) or false (F):
(i) The quotient of two integers is always an integer.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every traction is a rational number.
(v) Every rational number is a fraction.
(vi) If (a/b) is a rational number and m any integer, then (a/b) = ((a x m))/((b x m))
(vii) Two rational numbers with different numerators cannot be equal.
(viii) 8 can be written as a rational number with any integer as denominator.
(ix) 8 can be written as a rational number with any integer as numerator.
(x) (2/3) is equal to (4/6).
Solution 4:
(i) The quotient of two integers is always an integer.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every traction is a rational number.
(v) Every rational number is a fraction.
(vi) If (a/b) is a rational number and m any integer, then (a/b) = ((a x m))/((b x m))
(vii) Two rational numbers with different numerators cannot be equal.
(viii) 8 can be written as a rational number with any integer as denominator.
(ix) 8 can be written as a rational number with any integer as numerator.
(x) (2/3) is equal to (4/6).
(i) False
Description:
The quotient of two integers is not necessary to be an integer
(ii) True
Description:
Every integer can be expressed in the form of p/q, where q is not zero.
(iii) False
Description:
Every rational number is not necessary to be an integer
(iv) True
Description:
According to definition of rational number i.e. every integer can be expressed in the form of p/q, where q is not zero.
(v) False
Description:
It is not necessary that every rational number is a fraction.
(vi) True
Description:
If (a/b) is a rational number and m any integer, then (a/b) = ((a x m))/((b x m)) is one of the rule of
rational numbers
(vii) False
Description:
They can be equal, when simplified further.
(viii) False
Description:
8 can be written as a rational number but we can’t write 8 with any integer as denominator.
(ix) False
Description:
8 can be written as a rational number but we can’t with any integer as numerator.
(x) True
Description:
When convert it into standard form they are equal
Exercise 4.6
Question :1. Draw the number line and represent following rational number on it:
(i) (2/3)
(ii) (3/4)
(iii) (3/8)
(iv) ((-5)/8)
(v) ((-3)/16)
(vi) ((-7)/3)
(vii) (22/(-7))
(viii) ((-31)/3) (-31/3)
Solution 1:
(i) We know that (2/3) is greater than 2 and less than 3.
∴ it lies between 2 and 3. It can be represented on number line as,
Question :2. Which of the two rational numbers in each of the following pairs of rational number is greater?
(i) ((-3)/8), 0
(ii) (5/2), 0
(iii) ((-4)/11), (3/11)
(iv) ((-7)/12), (5/(-8))
(v) (4/(-9)), ((-3)/(-7))
(vi) ((-5)/8), (3/(-4))
(vii) (5/9), ((-3)/(-8))
(viii) (5/(-8)), ((-7)/12)
Solution 2:
(i) Given ((-3)/8), 0
We know that every positive rational number is greater than zero and every negative
rational number is smaller than zero. Thus, – ((-3)/8) > 0
(ii) Given (5/2), 0
We know that every positive rational number is greater than zero and every negative rational number is smaller than zero. Thus, (5/2) > 0
(iii) Given ((-4)/11), (3/11)
We know that every positive rational number is greater than zero and every negative rational number is smaller than zero, also the denominator is same in given question now we have to compare the numerator, thus((-4)/11) < (3/11).
Question :3. Which of the two rational numbers in each of the following pairs of rational numbers is smaller?
(i) ((-6)/(-13)), (7/13)
(ii) (16/(-5)), 3
(iii) ((-4)/3), (8/(-7))
(iv) ((-12)/5), (-3)
Solution 3:
(i) Given ((-6)/(-13)), (7/13)
Here denominator is same Hence compare the numerator,
Thus ((-6)/(-13)) < (7/13)
(ii) Given (16/(-5)), 3
We know that 3 is a whole number with positive sign
Hence (16/(-5)) < 3
(iii) Given ((-4)/3), (8/(-7))
Consider ((-4)/3)
Multiply both numerator and denominator by 7 then we get
((-4)/3) × (7/7) = ((-28)/21)…… (1)
Now consider (8/(-7))
Multiply both numerator and denominator by 3 we get
(8/(-7)) × (3/3) = ((-24)/21)…… (2)
The denominator is same in equation (1) and (2) now we have to compare the numerator, thus ((-4)/3) < (8/(-7))
(iv) Given ((-12)/5), (-3)
Now consider ((-3)/1)
Multiply both numerator and denominator by 5 we get
((-3)/1) × (5/5) = ((-15)/5)
The denominator is same in above equation, now we have to compare the numerator, thus ((-12)/5) > (-3)
Question :4. Fill in the blanks by the correct symbol out of >, =, or <:
(i) ((-6)/7) …. (7/13)
(ii) ((-3)/5) …. ((-5)/6)
(iii) ((-2)/3) …. (5/(-8))
(iv) 0 …. ((-2)/5)
Solution 4:
(i) ((-6)/7) < (7/13)
Description:
Because every positive number is greater than a negative number.
(ii) ((-3)/5) > ((-5)/6)
Description:
Consider ((-3)/5)
Multiply both numerator and denominator by 6 then we get
((-3)/5) × (6/6) = ((-18)/30)…… (1)
Now consider (-5/6)
Multiply both numerator and denominator by 5 we get
((-5)/6) × (5/5) = ((-25)/30)…… (2)
The denominator is same in equation (1) and (2) now we have to compare the numerator, thus ((-3)/5) >((-5)/6)
(iii) ((-2)/3) < (5/(-8))
Description:
Consider ((-2)/3)
Multiply both numerator and denominator by 8 then we get
((-2)/3) × (8/8) = ((-16)/24)…… (1)
Now consider (5/(-8))
Multiply both numerator and denominator by 3 we get
(5/(-8)) × (3/3) = (15/(-24))…… (2)
The denominator is same in equation (1) and (2) now we have to compare the numerator, thus ((-2)/3) < (5/(-8))
(iv) 0 > ((-2)/5)
Description:
Because every positive number is greater than a negative number
Question :5. Arrange the following rational numbers in ascending order:
(i) (3/5), ((-17)/(-30)), (8/(-15)), ((-7)/10)
(ii) ((-4)/9), (5/(-12)), (7/(-18)), (2/(-3))
Solution 5:
(i) Given (3/5), ((-17)/(-30)), (8/(-15)), ((-7)/10)
The LCM of 5, 30, 15 and 10 is 30
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 30
Consider (3/5)
Multiply both numerator and denominator by 6, then we get
(3/5) × (6/6) = (18/30) ….. (1)
Consider (8/(-15))
Multiply both numerator and denominator by 2, then we get
(8/(-15)) × (2/2) = (16/(-30)) ….. (2)
Consider ((-7)/10)
Multiply both numerator and denominator by 3, then we get
((-7)/10) × (3/3) = ((-21)/30) ….. (3)
In the above equation, denominators are same
Now on comparing the ascending order is:
((-7)/10) < (8/(-15)) < ((-17)/(-30)) < (3/5)
(ii) Given ((-4)/9), (5/(-12)), (7/(-18)), (2/(-3))
The LCM of 9, 12, 18 and 3 is 36
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 36
Consider ((-4)/9)
Multiply both numerator and denominator by 4, then we get
((-4)/9) × (4/4) = ((-16)/36) ….. (1)
Consider (5/(-12))
Multiply both numerator and denominator by 3, then we get
(5/(-12)) × (3/3) = (15/(-36)) ….. (2)
Consider (7/(-18))
Multiply both numerator and denominator by 2, then we get
(7/(-18)) × (2/2) = (14/(-36)) ….. (3)
Consider (2/(-3))
Multiply both numerator and denominator by 12, then we get
(2/(-3)) × (12/12) = (24/(-36)) ….. (4)
In the above equation, denominators are same
Now on comparing the ascending order is:
(2/(-3)) < ((-4)/9) < (5/(-12)) < (7/(-18))
Question :6. Arrange the following rational numbers in descending order:
(i) (7/8), (64/16), (39/(-12)), (5/(-4)), (140/28)
(ii) ((-3)/10),(17/(-30)),(7/(-15)), ((-11)/20)
Solution 6:
(i) Given (7/8), (64/16), (39/(-12)), (5/(-4)), (140/28)
The LCM of 8, 16, 12, 4 and 28 is 336
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 336
Consider (7/8)
Multiply both numerator and denominator by 42, then we get
(7/8) × (42/42) = (294/336) ….. (1)
Consider (64/16)
Multiply both numerator and denominator by 21, then we get
(64/16) × (21/21) = (1344/336) ….. (2)
Consider (39/(-12))
Multiply both numerator and denominator by 28, then we get
(39/(-12)) × (28/28) = ((-1008)/336) (-1008/336) ….. (3)
Consider (5/(-4))
Multiply both numerator and denominator by 84, then we get
(5/(-4)) × (84/84) = ((-420)/336) ….. (4)
In the above equation, denominators are same
Now on comparing the descending order is:
(140/28) > (64/16) > (7/8) > (5/(-4)) > (39/(-12))
(ii) Given ((-3)/10),(17/(-30)),(7/(-15)), ((-11)/20)
The LCM of 10, 30, 15 and 20 is 60
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 60
Consider ((-3)/10)
Multiply both numerator and denominator by 6, then we get
((-3)/10) × (6/6) = ((-18)/60) ….. (1)
Consider (17/(-30))
Multiply both numerator and denominator by 2, then we get
(17/(-30)) × (2/2) = (34/(-60)) ….. (2)
Consider (7/(-15))
Multiply both numerator and denominator by 4, then we get
(7/(-15)) × (4/4) = (28/(-60)) ….. (3)
In the above equation, denominators are same
Now on comparing the descending order is:
((-3)/10) > (7/(-15)) > ((-11)/20) > (17/(-30))
Question :7. Which of the following statements are true:
(i) The rational number (29/23) lies to the left of zero on the number line.
(ii) The rational number ((-12)/(-17)) lies to the left of zero on the number line.
(iii) The rational number (3/4) lies to the right of zero on the number line.
(iv) The rational number ((-12)/(-5)) and ((-7)/17) are on the opposite side of zero on the number line.
(v) The rational number ((-2)/15) and (7/(-31)) are on the opposite side of zero on the number line.
(vi) The rational number ((-3)/(-5)) is on the right of ((-4)/7) on the number line.
Solution 7:
(i) False
Description:
It lies to the right of zero because it is a positive number.
(ii) False
Description:
It lies to the right of zero because it is a positive number.
(iii) True
Description:
Always positive number lie on the right of zero
(iv) True
Description:
Because they are of opposite sign
(v) False
Description:
Because they both are of same sign
(vi) True
Description:
They both are of opposite signs and positive number is greater than the negative number. Thus, it is on the right of the negative number.