Samacheer Kalvi Class 8 Maths Chapter 1 Numbers Solutions
Welcome to NCTB Solutions. Here with this post we are going to help 8th class students by providing Solutions for Samacheer Kalvi Class 8 Maths Chapter 1 Numbers. Here students can easily find all the solutions for Numbers Exercise 1.1, 1.2, 1.3, 1.4, 1.5, 1.6 and 1.7. Also here our Expert Maths Teacher’s solved all the problems with easily understandable methods with proper guidance so that all the students can understand easily. Here in this post students will get chapter 1 solutions. Here all the solutions are based on Tamil Nadu State Board latest syllabus.
Numbers Exercise 1.1 Solutions :
(1) Fill in the blanks :
(i) −19/5 lies between the integers __ and __.
Solution :
– 19/5 lies between the integers,
-19/5 = – 3.8
∴ – 19/5 lies between the integers – 3 and – 4
(ii) The decimal form of the rational number 154/-4 is ___
Solution :
15/- 4
= 15 × (- 1)/ – 4 × (- 1)
= – 15/4
= – 3.45
∴ The decimal form of the rational number 15/-4 = – 3.45
(iii) The rational numbers -8/3 and 8/3 are equidistant from ___
Solution :
The rational numbers -8/5 and 8/3 are equidistant from “Zero”.
(iv) The next rational number in the sequence -15/24, 20/-32, -25/40 is __
Solution :
– 15/24 = – 15 ÷ 3/24 ÷ 3 = – 5/8
20/-32 = 20 ÷ (- 4)/- 32 ÷ (- 4) = – 5/8
– 25/40 = – 25 ÷ 5/40 ÷ 5 = – 5/8
– 5/8 = – 5 × (- 6)/8 × (- 6) = 30/- 48
∴ The next rational number in the sequence – 15/24, 20/-32, -25/40, and 30/-48
(v) The standard form of 58/- 78 is ___,
Solution :
The standard from of –
58/- 78 = 58 ÷ (- 2)/- 78 ÷ (- 2) = -29/39
∴ The standard from of 58/- 78 is – 29/39
(2) Say True or False :
(i) 0 is the smallest rational number.
(ii) −4/5 lies to the left of −3/4.
(iii) −19/5 is greater than 15/-4
(iv) The average of two rational numbers lies between them.
(v) There are an unlimited number of rational numbers between 10 and 11.
Solution :
(i) → False
(ii) → True
(iii) → False
(iv) → True
(v) → True
(3) Find the rational numbers represented by each of the question marks marked on the following number lines.
Solution :
(i) Average number between – 4 and – 3
-4 -3/2 = -7/2
Average number between -4 and -7/2
= (-4 -7/2) / 2
= (-8 -7/2) / 2
= -15/2
= – 7.5
(ii) Average number between -1 and 0
– 1 + 0/2 = -1/2
Average number between -1/2 and 0
(-1/2 + 10) = – 1/4
Average number between -1/2 and -1/4 we get
(-1/2 – 1/4) / 2
= (- 2+1/4) / 2
= -3/4×2
= -3/8
= – 0.375
(iii) Average number between 1 and 2 we get
1+2/2 = 3/2
Average number between 3/2 and 2 we get
(3/2+2) / 2
= (3+2/2) / 2
= 7/4
= 1.75
(4) The points S, Y, N, C, R, A, T, I and O on the number line are such that CN = NY = YS and RA = AT = TI = IO. Find the rational numbers represented by the letters Y, N, A, T and I.
Solution :
Average number between s and c
-2 -1/2 = -3/2
Average number between -3/2 and -1
(-3/2 -1) / 2
= (-3 -2/2) / 2
= -5/4
Average number between – 3/2 and – 5/4
– 3/2 – 5/4/2
= (-6 – 5/4) / 2
= -11/8
= -1.35
(5) Draw a number line and represent the following rational numbers on it
(i) 9/4
(ii) -8/3
(iii) -17/-5
(iv) 15/-4
Solution :
(i) Given, 9/4
= 9 × 15 / 4 × 15
= 135/60
(ii) Given, -8/3
= -8 × 20 / 3 × 20
= -160/60
(iii) Given, -17/-5
= -17 × (-12) / – 5 × (-12)
= 204/60
(iv) Given, 15/-4
= 15 × (-15) / -4 × (-15)
= -225/60
(6) Write the decimal form of the following rational numbers
(i) 1/11
(ii) 13/4
(iii) -18/7
(iv) 1 2/5
(v) -3 1/2
Solution :
(i) 1/11
= 0.090909
(ii) 13/4
= 3.25
(iii) – 18/7
= – 2.571428
(iv) 1 2/5
= 7/5
= 1.4
(v) -3 1/2
= -7/2
= – 3.5
(7) List any five rational numbers between the given rational numbers
(i) -2 and 0
(ii) -1/2 and 3/5
(iii) 1/4 and 3/5
(iv) -6/4 and -23/10
Solution :
(i) Given, – 2 and 0
Multiplying 10 with -2/1 and -1/1 we get
– 20/10 – 10/10 0
Between -20/10 and 0 there are many rational numbers, Some of them are –
-19/10, -18/10, -17/10, -10/10, -8/10
(ii) Given, -1/2 and 3/5
-1/2 = – 1 × 5/2 × 5 = – 5/10
3/5 = 3 × 2 /5 × 2 = 6/10
L.C.M 2,5 is 10
-5/10 0 6/10
Between -5/10 and 6/10 there are rational number –
-4/10, -3/10, -1/10, 1-/10, 3/10
(iii) Given, 1/4 and 7/20
1/4 = 1×5/4×5 = 5/20
7/20 = 7×1/20×1 = 7/20
∴ L.C.M of 4,20 is 20.
Rational number lies between 1/4 and 7/20 are –
51/200, 53/200, 60/200, 62/200, 66/200
(iv) Given, – 64 and -23/10
-6/4 = -6×5/4×5 = -30/20
-25/10 = -23×2/10×2 = -46/20
∴ L.C.M of 4, 10 is 20
rational number lies between -46/20 and -30/20 are –
– 45/20, – 43/20, – 40/20, -37/20, -35/20
(8) Use the method of averages to write 2 rational numbers between 14/5 and 16/3
Solution :
Average between 16/5 and 16/3 is
= (14/5 + 16/3) / 2
= (42+80/15) / 2
= 122/30
And average between 14/5 and 122/30 is
= (14/5+122/30) / 2
= (84+122/30) / 2
= 206/60
∴ 2 rational numbers are lies between 14/5 and 16/3 are 12/30 and 2069/60.
(9) Compare the following pairs of rational numbers
(i) -11/5, -21/8
(ii) 3/-4, -1/2
(iii) 2/3, 4/5
Solution :
(i) Given, -11/5 and -21/8
-11/5 = -11×8/5×8 = – 88/40
-21/8 = -21×5/8×5 = -105/40
∴ L.C.M of 5,8 is 40
The denominators are the same and so just comparing the numerators – 88 and -105
∴ -88 > -105
= -88/40 > -105/40
Consequently we conclude that
-11/5 > -21/8
(ii) Given, 3/-4, -1/8
3/-4 = 3×(-1)/-4×(-1) = -3/4
-1/8 = -1×2/2 = -2/4
∴ L.C.M of 4, 2 is 4
The denominator are the same and so just comparing the numerators -3 and -2
-3 < -2
= -3/4 -3/4
(iii) Given, 2/3, 4/5
2/3 = 2×5/3×5 = 10/15
4/5 = 4×3/5×3 = 12/15
∴ L.C.M of 3 and 5 is 15
The denominator are the same and so just comparing the numerators 10 and 12
12/15 > 10/15
∴ Consequently we conclude that 4/5 > 2/3
(10) Arrange the following rational numbers in ascending and descending order
(i) -5/12, -11/8, -15/24, -7/-9, 12/36
(ii) -17/10, -7/5, 0, -2/4, -19/20
Solution :
(i) -5/12, -11/8, -15/24, -7/-9, 12/36
L.C.M of 12, 8, 24, 9, 36 is 72
-5/12 = -5×6/12×6 = -30/72
-11/8 = -11×9/8×9 = -99/72
-15/24 = 15×3/24×3 = -45/72
-7/-9 = 7×(-8)/-9×(-8) = 56/72
12/36 = 12×2/36×2 = 24/72
Comparing the numerators alone we see that
-99 < -45 < -30 < 24 < 56
∴ -99/72 < -11/72 < -30/72 < 24/72 < 56/72
So that -11/8 < -15/24 < -5/12 < 12/36 < -7/-9
So, the ascending order is
-11/8, -15/24, -5/12, 12/36, -7/-9
and the descending order as
-7/-9, 12/36, -5/12, -15/24, -11/8
(ii) -17/10, -7/5, 0, -2/4, -19/20
L.C.M of 10, 5, 4, 20 is 40
-17/10 = -17×4/10×4 = -68/40
-7/5 = -7×8/5×8 = – 56/40
-2/4 = -2×10/4×10 = -20/40
-19/20 = -19×2/20×2 = -38/40
Comparing the numerators alone we see that
-68 < -56 < -38 < -20 < 0
∴ -68/40 < -56/40 < -38/40 < -20/40 < 0/40
So that = -17/10 < -7/5 < -19/20 < -2/4 < 0
So the ascending order is
-17/10, -7/5, -19/20-2/4,0
and the descending order is
0, -2/4, -19/20, -7/5, -17/10
Objective Type Questions :
(11) The number which is subtracted from -6/11 to get 8/9 is
(A) 34/99
(B) -142/99
(C) 142/99
(D) -34/99
Solution :
= -6/11 – 8/9
= -54 – 88/99
= -142/99
Hence, the correct answer is option (B) -142/99.
(12) Which of the following pairs is equivalent
(A) -20/12, 5/3
(B) 16/-30, -8/15
(C) -18/36, -20/44
(D) 7/-5, -5/7
Solution :
Here, -8/15 is equivalent to 16/-30
Since -8/15 = -8×-2/15×(-2) = 16/-30
Hence, option (B) 16/-30, -8/15 is the correct answer.
(13) -5/4 is a rational number which lies between
(A) 0 and -5/4
(B) -1 and 0
(C) -1 and -2
(D) -4 and -5
Solution :
-5/4 = -1.25
Lies between -1 and -2
Hence, the correct answer is option (C) -1 and -2.
(14) Which of the following rational numbers is the greatest
(A) -17/24
(B) -13/16
(C) 7/-8
(D) -31/32
Solution :
L.C.M of 247, 16, 8, 32 is 96
-17/24 = -17×4/24×4 = -68/96
-13/16 = -13×6/16×6 = -78/72
7/-8 = 7×(-12) /-8×(-12) = -84/72
-31/32 = -31 ×3/32×2 = -93/72
Comparing the numerator we see that -93 < -84 < -78 < -68
So that -93/72 < -84/72 < -78/72 < -68/72
∴ -31/32 < 7/-8 < -13/16 < -17/24
The greatest rational number is -17/24
Hence, Option (A) -17/24 is the correct answer.
(15) The sum of the digits of the denominator in the simplest form of 112/528 is __
(A) 4
(B) 5
(C) 6
(D) 7
Solution :
112/528
= 112 ÷ 8/528 ÷ 8
= 14 ÷ 2/66 ÷ 2
= 7/33
Sum of the digits of the denominator in the simplest form of 112/528 – 3 + 3 = 6
So, option (C) 6 is the correct answer
Numbers Exercise 1.2 Solutions :
(1) Fill in the blanks :
(i) The value of -5/12 + 7/15 = ___
Solution :
= -5/12 + 7/15 = ?
= -5/12 + 7/12
= -5 +7/12
= 2/12
= 1/6
Thus, -5/12 + 7/15 = 1/6
(ii) The value of (-3/6) × (18/-9) is ___
Solution :
= (-3/6) × (18/-9)
= -3/6 × (18×(-1)/(-9)×(-1) )
= -3/6 × -18/9
= 1
Hence, the value of (-3/6) × (18/-9) is 1.
(iii) The value of (-15/23) ÷ (30/-46) is ___
Solution :
= (-15/23) ÷ (30/-46)
= -15/23 × -46/30
= 1
Thus, the value of (-15/23) ÷ (30/-46) is 1.
(iv) The rational number ___ does not have a reciprocal.
Solution :
The rational number 0 does not have a reciprocal.
(v) The multiplicative inverse of –1 is ____.
Solution :
The multiplicative inverse of –1 is -1
Since, (-1) × (-1) = 1
(2) Say True or False :
(i) All rational numbers have an additive inverse.
(ii) The rational numbers that are equal to their additive inverses are 0 and –1.
(iii) The additive inverse of-11/-17 is 11/17
(iv) The rational number which is its own reciprocal is –1.
(v) The multiplicative inverse exists for all rational numbers.
Solution :
(i) → True
(ii) → False
(iii) → False
(iv) → True
(iv) → False
(3) Find the sum :
(i) 7/5 + 3/5
(ii) 7/5 + 5/7
(iii) 6/5 + (-14/15)
(iv) -4 2/3 + 7 5/12
Solution :
(i) Given, 7/5 + 3/5
= 7 + 3/5
= 10/5
= 2
(ii) Given, 7/5 + 5/7
= 49 + 25/35
= 74/35
(ii) Given, 6/5 + (-14/15)
= 18 – 14/15
= 4/15
(iv) Given, -4 2/3 +7 5/12
= – 14/3 + 89/12
= -56 + 89/12
= 33/12
(4) Subtract : -8/44 from -17/11
Solution :
Given, -8/44 from -17/11
= -8/44 – (-17/11)
= -8/44 + 17/11
= -8+68/44
= 60/44
= 15/11
(5) Evaluate :
(i) 9/132 × -11/3
(ii) -7/27 × 24/-35
Solution :
(i) Given, 9/132 × -11/3
= -1/4
(ii) Given, -7/27 × 24/-35
= -7/24 × 24 × (-1)/-35 × (-1)
= -7/27 × -24/35
= 8/45
(6) Divide :
(i) -21/5 by -7/-10
(ii) -3/13 by -3
(iii) -2 by -6/15
Solution :
(i) Given, -21/5 by -7/-10
= -21/5 ÷ -7/-10
= -21/5 ÷ -7×(-1) / -10×(-1)
= -21/5 ÷ 7/10
= -21/5 × 10/7
= -6
(ii) Given, -3/13 by -3
= -3/13 ÷ -3
= -3/13 × 1/-3
= -3/13 × -1/3
= 1/13
(iii) Given, -2 by -6/15
= -2 ÷ -6/15
= -2 × 15/-6
= -2 × -15/6
= 5
(7) Find (a + b) ÷ (a – b) if
(i) a = 1/2, b = 2/3
(ii) a = -3/5, b = 2/15
Solution :
(i) a = 1/2, b = 2/3 (given)
∴ (a + b) ÷ (a – b)
= a + b = 1/2 + 2/3
= 3+4/6
= 7/6
∴ a – b = 1/2 – 2/3
= 3-4/6
= -1/6
Now, (a + b) ÷ (a – b)
= 7/6 ÷ 1/-6
= 7/6 × -6/1
= -7
(8) Simplify 1/2 + (3/2 – 2/5) ÷ 3/10 × 3 and show that it is a rational number between 11 and 12
Solution :
1/2 + (3/2 – 2/5) ÷ 3/10 × 3
= 1/2 + 11/10 ÷ 3/10 × 3
= 1/2 + 114/10 × 10/3 × 3
= 1/2 + 11
= 1 + 22/2
= 23/2
= 11.5
The rational number 23/2 lies between 11 and 12.
(9) Simplify :
(i) [11/8 × (-6/33)] + [1/3 + (3/5 ÷ 9/20)] – [ 4/7 × -7/5]
Solution :
[11/8 × (-6/33)] + [1/3 + (3/5 ÷ 9/20)] – [ 4/7 × -7/5]
= [11/8 × -6/33] + [1/3 + [3/5 × 9/20] – [ 4/7 × -7/5]
= – 1/4 + [1/3 + 4/3] – [ -4/5]
= -1/4 + 5/3 + 4/5
= -15+100+48/60
= 133/60
(10) A student had multiplied a number by 4/3 instead of dividing it by 4/3 and got 70 more than the correct answer. find the number.
Solution :
Let, number is x
The student had to find x/4 = 3x/4
He had found x +4/3 = 4x/3
Now, 4x/3 – 3x/4 = 70 (given)
= 16x -9x/12 = 70
= 7x = 70 × 12
= x = 70 × 12/7
= 120
Hence, the number is 120.
Objective Type Questions :
(11) The standard form of the sum 3/4 + 5/6 + (-7/12) is ___
(A) 1
(B) -1/2
(C) 1/12
(D) 1/22
Solution :
3/4 + 5/6 + (-7/12)
= 3/4 + 5/6 – 7/12
= 9+10-7/12
= 12/12
= 1
So, the correct answer is option (A) 1
∴The standard form of the sum 3/4 + 5/6 + (-7/12) is 1
(12) (3/4 – 5/8) + 1/2 = ___
(A) 15/64
(B) 1
(C) 5/8
(D 1/16
Solution :
Given, (3/4 – 5/8) + 1/2
= (6-5/8) + 1/2
= 1/8 + 1/2
= 1+4/8
= 5/8
Hence, option (C) 5/8 is the correct answer.
(13) 3/4 ÷ (5/8 + 1/2) = ___
(A) 13/10
(B) 2/3
(C) 3/2
(D) 5/8
Solution :
3/4 ÷ (5/8 + 1/2)
= 3/4 ÷ (5+4/8)
= 3/4 ÷ 9/8
= 3/4 × 8/9
= 2/3
Hence, the correct answer is option (B) 2/3.
(14) 3/4 × (5/8 ÷ 1/2) = __
(A) 5/8
(B) 2/3
(C) 15/32
(D) 15/16
Solution :
Given, 3/4 × (5/8 ÷ 1/2)
= 3/4 × (5/8 × 2/1)
= 3/4 × 5/4
= 15/16
Hence, the correct answer is option (C) 15/32.
(15) Which of these rational numbers which have additive inverse
(A) 7
(B) -5/7
(C) 0
(D) All of these
Solution :
The correct answer is option (D) All of these
Numbers Exercise 1.3 Solutions :
Question no – (1)
Verify the closure property for addition and multiplication for the rational numbers -5/7 and 8/9
Solution :
Let, a = -5/7 and b = 8/9
a + b = -5/7 + 8/9 = -45 + 56/63 = 11/63 is rational number.
And a + b = -5/7 × 8/9 = -40/63 is rational number.
∴ Closure property is satisfy for the rational number.
Question no – (2)
Verify the commutative property for addition and multiplication for the rational numbers -10/11 and -8/33
Solution :
Let, a = -10/11 and b = -8/33
a + b = -10/11 + -8/33 = -30-8/33 = -38/33
b + a = -8/33 + -10/11 = -8-30/33 = -38/33
∴ Commutative property satisfied for addition.
Now, ab = -10/11 -8/33 = 80/363
ba = -8/33 -10/11 = 80/363
∴ Commutative property satisfied for multiplication.
Question no – (3)
Verify the associative property for addition and multiplication for the rational numbers -7/9, 5/6 and -4/3
Solution :
Let, a = -7/9 , b = 5/6 and c = -4/3
a + b = -7/9 + 5/6 = -14+15/18 = 1/18
(a + b) + c = 1/18 + -4/3 = 1 – 24/18 = -23/18
b + c = 5/6 + -4/3 = 5-8/6 = -3/6 = -1/2
a + (b + c) = -7/9 + -1/2 = -14-9/18 = -23/18
∴ Associative property satisfied for addition.
Now,
a × (b×c) = -7/9 × (5/6 × -4/3) l= -7/9 × (-10/9) = -70/81
(a×b) × c = (-7/9 × 5/6) × -4/3 = -35/9×6 × -4/3 = -70/81
∴ Associative property satisfied for multiplication.
Question no – (4)
Verify the distributive property a × (b + c) = (a × b) + (a + c) for the rational numbers a = -1/2, b = 2/3 and c = -5/6
Solution :
a = -1/2, b = 2/3 and c = -5/6
Given, a × (b + c) = (a × b) + (a + c)
∴ a × (b + c)
= -1/2 × (2/3 + -5/6)
= -1/2 × (4-5/6)
= -1/2 × (-1/6)
= 1/12
(a × b) + (a × c) = (-1/2 × 2/3) + (-1/2 × -5/6)
= -7/6 + 5/12
= -4 + 5/12
= 1/12
∴ Distributive property satisfied.
Question no – (5)
Verify the identity property for addition and multiplication for the rational numbers 15/19 and -18/25
Solution :
Let, a = 15/19 and b = -18/25
∴ a + 0 = 15/19 + 0 = 15/19 = 0 + 15/19 = 0 + a
And b + 0 = -18/25 + 0 = -18/25 = 0 + -18/25 = 0 + b
∴ ‘0’ is additive identify for a and b
Now,
a × 1 = 15/19 × 1 = 15/19 = 1 × 15/19 = 1 × a
b × 1 = -18/25 × 1 = -18/25 = 1 × -18/25 = 1 × b
∴ ‘1’ is multiplicative identify for a and b.
Question no – (6)
Verify the additive and multiplicative inverse property for the rational numbers -7/17 and 17/27
Solution :
Let, a = -7/17 and b = 17/27
∴ a + (-a) = -7/17 + 7/17 = 0 = 7/17 + -7/17 = (-a) + a
b + (-b) = 17/27 + (-17/27) = 0 = (-17/27) + 17/27 = (-b) + b
∴ Inverse property satisfied for addition.
a × 1/a = -7/17 × (1//7) /17 = -7/17 × -7/17 = 1
1/a × a = (1/-7) / 17 × -7/17 = 17/-7 × -7/17 = 1
b × 1/b = 17/27× (1/17) / 27 = 17/27 × 27/17 = 1
1/b × b = (1/17) / 27 = 27/17 × 17/27 = 1
∴ Inverse property satisfied for multiplication.
Objective Type Questions :
Question no – (7)
Closure property is not true for division of rational numbers because of the number
(A) 1
(B) -1
(C) 0
(D) 1/2
Solution :
Correct option – (D) 1/2
Closure property is not true for division of rational numbers because of the number 1/2.
Question no – (8)
1/2 – (3/4 – 5/6) ≠ (1/2 – 3/4) – 5/6 illustrates that subtraction does not satisfy the ___ property for rational numbers
(A) commutative
(B) closure
(C) distributive
(D) associative
Solution :
Correct option – (A)
1/2 – (3/4 – 5/6) ≠ (1/2 – 3/4) – 5/6 illustrates that subtraction does not satisfy the Commutative property for rational numbers.
Question no – (8)
Which of the following illustrates the inverse property for addition
(A) 1/8 – 1/8 = 0
(B) 1/8 + 1/8 = 1/4
(C) 1/8 + 0 = 1/8
(d) 1/8 – 0 = 1/8
Solution :
Correct option – (A)
1/8 – 1/8 = 0 and (1/8) + 1/8 = 0
= 1/8 + (-1/8) = 0
∴ 1/8 + (- 1/8) = (-1/8 + 1/8) = 0
Question no – (8)
3/4 × (1/2 – 1/4) = 3/4 × 1/2 – 3/4 × 1/4 illustrates that multiplication is distributive over
(A) addition
(B) subtraction
(C) multiplication
(D) division
Solution :
3/4 × (1/2 – 1/4) = 3/4 × 1/2 – 3/4 × 1/4
3/4 × (2 – 1/4)
= 3/4 × 1/4
= 3/169
3/4 × 1/2 – 3/4 × 1/4
= 3/8 – 3/16
= 6 – 3/16
= 3/16
∴ Distributive over multiplication.
Hence, the correct answer is option (C) multiplication.
Numbers Exercise 1.4 Solutions :
(1) Fill in the blanks :
(i) The ones digit in the square of 77 is __
Solution :
The ones digit in the square of 77 is 9
(ii) The number of non-square numbers between 24² and 25² is __
Solution :
Given, 24²
= 576
Given, 25²
= 625
∴ Number of non-sequence number 625 – 576 = 49
The number of non-square numbers between 24² and 25² is 49.
(iii) The number of perfect square numbers between 300 and 500 is __
Solution :
The number of perfect square numbers between 300 and 500 is
324 = 18²
361 = 19²
400 = 20²
441 = 21²
484 = 22²
(iv) If a number has 5 or 6 digits in it, then its square root will have __ digits.
Solution :
If a number has 5 or 6 digits in it, then its square root will have 3 digits.
Example :
101 × 101 = 10201
But 99 × 99 = 9801
(v) The value of √180 lies between integers __ and __
Solution :
The value of √180 lies between integers 13 and 14.
169 < 180 < 196
13² < 180 < 14²
(2) Say True or False :
(i) When a square number ends in 6, its square root will have 6 in the unit’s place.
(ii) A square number will not have odd number of zeros at the end.
(iii) The number of zeros in the square of 91000 is 9.
(iv) The square of 75 is 4925.
(v) The square root of 225 is 15.
Solution :
(i) → True
(ii) → True
(iii) → False
(iv) → False
(v) → True
(3) Find the square of the following numbers.
(i) 17
(ii) 203
(iii) 1098
Solution :
(i) Given number, 17
∴ 17²
= 17 × 17
= 289
(ii) 203
∴ 203²
= 203 × 203
= 41209
(iii) 1098
∴ 1098²
= 1098 × 1098
= 1205604
(4) Examine if each of the following is a perfect square.
(i) 725
(ii) 190
(iii) 841
(iv) 1089
Solution :
(i) Given m number, 725
725 = 5 × 5 × 29
= 5² × 29
∴ 725 is not a perfect square.
(ii) 190
190 = 5 × 2 × 19
∴ 190 is not a perfect square.
(iii) 841
841 = 29 × 29
= 29²
∴ 841 is a perfect square.
(iv) 1089
1089 = 3 × 3 × 11 × 11
= 3² × 11²
∴ 1089 is a perfect square.
(5) Find the square root by prime factorisation method.
(i) 144
(ii) 256
(iii) 784
(iv) 1156
(v) 4761
(vi) 9025
Solution :
(i) 144 = 2 × 2 × 3 × 2 × 2 × 3
= (2 × 2 × 3)²
(ii) 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= (2 × 2 × 2 × 2)²
(iii) 784 = 2 × 2 × 7 × 2 × 2 × 7
= (2 × 2 × 7)²
(iv) 1156 = 2 × 17 × 2 × 17
= (2 × 17)²
(v) 4761 = 3 × 23 × 3 × 23
= (3 × 23)²
(vi) 9025 = 5 × 19 × 5 × 19
= (5 × 19)²
(6) Find the square root by long division method.
(i) 1764
(ii) 6889
(iii) 11025
(iv) 17956
(v) 418609
Solution :
(i) 1764
= √1764
= 42
(ii) 6889
= √6889
= 83
(iii) 11025
= √11025
= 105
(iv) 17956
= √17956
= 134
(v) 418609
= √418609
= 647
(7) Estimate the value of the following square roots to the nearest whole number:
(i) √440
(ii) √800
(iii) √1020
Solution :
(i) √440
400 < 400 < 441
20² < 440 < 21²
∴ The nearest whole number is 21.
(ii) √800
784 < 800 < 841
28² < 800 < 29²
∴ The nearest whole number is 29.
(iii) √1020
961 < 1020 < 1024
31² < 1020 < 32²
∴ The nearest whole number is 32.
(8) Find the square root of the following decimal numbers and fractions
(i) 2.89
(ii) 67.24
(iii) 2.0164
(iv) 144/225
(v) 7 18/49
Solution :
(i) √2.89
= √289/100
= √289/√100
= √17 × 17/√10 × 10
= 17/10
= 1.7
(ii) √67.24
= √6724/100
= √6724/√100
= √82 × 82/√10 × 10
= 82/10
= 8.2
(iii) √2.0164
= √20164/√10000
= √142 × 142/√100 × 100
= 142/100
= 1.42
(iv) √144/225
= √144/√225
= √12 × 12/√15 × 15
= 12/15
= 4 × 2/5 × 2
= 8/10
= 0.8
(v) √7 18/49
= √36149
= √361/√49
= √19 × 19/√7 × 7
= 19/7
= 2 5/7
(10) Find the least number by which 1800 should be multiplied so that it becomes a perfect square. Also, find the square root of the perfect square thus obtained.
Solution :
Given, √1800
∴ 1800 = 2×2×2×5×5×3×3
= (2×5×3)² × 2
1800 should be multiplied by 2 so that it becomes a perfect square.
∴ The perfect square is,
1800 × 2 = (2×5×3)² × 2 × 2
= 3600 = (2×5×3×2)²
Objective Type Questions :
(11) The square of 43 ends with the digit ___
(A) 9
(B) 6
(C) 4
(D) 3
Solution :
Correct option – (A)
The square of 43 ends with the digit 9.
(12) ___ is added to 24² to get 25²
(A) 4²
(B) 5²
(C) 6²
(D) 7²
Solution :
25² = 625 , 24² = 576
∴ 24² – 25² = 625 – 576
= 49
= 7²
Hence, option (D) 7² is the correct answer.
(13) √48 is approximately equal to ____
(A) 5
(B) 6
(C) 7
(D) 8
Solution :
Correct option – (C)
√48 is approximately equal to 7.
(14) √128 – √98 + √18 = ___
(A) √2
(B) √8
(C) √48
(D) √32
Solution :
√128 – √98 + √18
= √2 × 2 × 2 × 2 × 2 × 2 × 2 – √2 × 7 × 7 + √2 × 3 × 3
= 8√2 – 7√2 + 3√2
= 4√2
= √16 × 2
= √32
Hence, the correct answer is option (D) √32
(15) The number of digits in the square root of 123454321 is ____
(A) 4
(B) 5
(C) 6
(D) 7
Solution :
Correct option – (B)
The number of digits in the square root of 123454321 is 5
Numbers Exercise 1.5 Solutions :
(1) Fill in the blanks :
(i) The ones digits in the cube of 73 is ___
Solution :
The ones digits in the cube of 73 is 7.
(ii) The maximum number of digits in the cube of a two digit number is ___
Solution :
The maximum number of digits in the cube of a two digit number is 6.
(iii) The smallest number to be added to 3333 to make it a perfect cube is ___
Solution :
The smallest number to be added to 3333 to make it a perfect cube is 42.
Since,
15³ = 15×15×15 = 3375
∴ 3375 – 3333 = 42
(iv) The cube root of 540 × 50 is ___
Solution :
The cube root of 540 × 50 is 30
(v) The cube root of 0.000004913 is ___
Solution :
The cube root of 0.000004913 is 0.007
(2) Say True or False:
(i) The cube of 24 ends with the digit 4.
(ii) Subtracting 10³ from 1729 gives 9³.
(iii) The cube of 0.0012 is 0.000001728.
(iv) 79570 is not a perfect cube.
(v) The cube root of 250047 is 63.
Solution :
(i) → True
(ii) → False
(iii) → False
(iv) → True
(v) → True
Question no – (3)
Show that 1944 is not a perfect cube.
Solution :
1944 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3
= (2 × 3)³ × 3 × 3
∴ 1944 is not a perfect cube.
Question no – (4)
Find the smallest number by which 10985 should be divided so that the quotient is a perfect cube.
Solution :
10985 = 5 × 13 × 13 × 13
We should divided 10985 by 5 so that the quotient is a perfect cube.
10985/5 = 13 × 13 × 13
= 2197 = (13)³
Question no – (5)
Find the smallest number by which 200 should be multiplied to make it a perfect cube.
Solution :
200 = 2 × 10 × 10
We should multiplied 200 by 2 × 2 × 10 for get a perfect square
200 × 2 × 2 × 10 = 2 × 10 × 10 × 2 × 2 × 10
∴ 8000 = (2 × 10)³
Question no – (6)
Find the cube root of 24 × 36 × 80 × 25
Solution :
24 × 36 × 80 × 25 = 4 × 6 × 6 × 6 × 4 × 4 × 5 × 5 × 5
= (4 × 6 × 5)³
= (120)³
Therefore, the cube root is (120)³
Question no – (7)
Find the cube root of 729 and 6859 by prime factorisation.
Solution :
729 = 3 × 3 × 3 × 3 × 3 × 3
= (3 × 3)³
= 9³
And 6859 = 19 × 19 × 19
= (19)³
Question no – (8)
What is the square root of cube root of 46656?
Solution :
46656 = 2 × 2 × 2 × 2 × 2 × 2 × 9 × 9 × 9
= (2 × 2 × 9)³
And (2 × 2 × 2 × 3)²
Question no – (9)
If the cube of a squared number is 729, find the square root of that number.
Solution :
729 = 3 × 3 × 3 × 3 × 3 × 3
= (3 × 3 × 3)²
Question no – (10)
Find the two smallest perfect square numbers which when multiplied together gives a perfect cube number.
Solution :
4 × 16 = 64
2² × 4² = 4³
Numbers Exercise 1.6 Solutions :
(1) Fill in the blanks :
(i) (−1) even integer is ___
Solution :
(−1) even integer is 1.
(2) Say True or False :
(ii) The simplified form of (256)^-1/4 × 4^2 is 1/4
Solution :
This statement is False.
Because :
(256)^-1/4 × 4^2
= 1/(256)^4-1 × 4^2
= 1/(4^4)^1/4 × 4^2
= 1/4 × 4^2
= 4
(iii) Using the power rule, (3^7)^-2 = 3^5
Solution :
Above statement is False.
Because :
= (3^7)^-2
= 1/(3^7(^2
= 1/3^14
(iv) The standard form of 2 × 10^–4 is 0.0002
Solution :
This statement is True
(v) The scientific form of 123.456 is 1.23456 × 10^-2
Solution :
This statement is True.
(3) Evaluate :
(i) (1/3)^3
(ii) (1/2)^-5
(iii) (-5/6)^-3
(iv) (2^-5 × 2^7) ÷ 2^-2
(v) (2^-1 × 3^-1) ÷ 6^-2
Solution :
(i) (1/3)^3
= 1/2×2×2
= 1/8
(ii) (1/2)^-5
= 1/2^-5
= 2^5
= 32
(iii) (-5/6)^-3
= 1/(-5/6)^3
= (6/-5)^3
= – 216/125
(iv) (2^-5 × 2^7) ÷ 2^-2
= (1/25 × 2^7) ÷ 1/2^2
= 2^(7-5) ÷ 1/2^2
= 2^2 × 2^2
= 4×4 = 16
(v) (2^-1 × 3^-1) ÷ 6^-2
= (1/2 × 1/3) ÷ 1/6^-2
= 1/6 × 6^2
= 6
(4) Evaluate :
(i) (2/5)^4 × (5/2)^-2
(ii) (4/5)^-2 ÷ (4/5)^-3
(iii) 2^7 × (1/2)^-3
Solution :
(i) (2/5)^4 × (5/2)^-2
= 2^4/5^4 × 1/(5/2)^2
= 2^4/5^4 × 2^2/5^2
= 2^4+2/5^4+2
= 26/56
= 64/15625
(ii) (4/5)^-2 ÷ (4/5)^-3
= 1/(4/5)^2 ÷ 1/(4/5)^3
= (5/4)^2 ÷ (5/4)^3
= 5^2/4^2 × 4^3/5^3
= 4/5
(iii) 2^7 × (1/2)^-3
= 2^7 × 1/2^-3
= 2^7 × 2^3
= 2^7+3
= 1024
(6) Simplify :
(i) (3^2)^3 × (2×3^5)^-2 × (18)^2
(ii) 9^2 × 7^3 × 2^5/84^3
(iii) 2^8 × 2187/3^5 × 32
Solution :
(i) (3^2)^3 × (2 × 3^5)^-2 × (18)^2
= (3^2)^3 × 1/(2 × 3^5) × (18)^2
= 3^65 × 1/2^2 × (3^5)^2 × (3 × 3 × 2)^2
= 3^6 × 1/2^2 × 3^10 × 3^2 × 3^2 × 2^2
= 1
(ii) 9^2 × 7^3 × 2^5/84^3
= 9^2 × 7^3 × 2^5/(2 × 2 × 3 × 7)^3
= 3^2 × 3^27^3 × 2^5/2^3 × 2^3 × 3^3 × 7^3
= 3/2
(iii) 2^8 × 2187/3^5 × 32
= 2^8 × 3^7/3^5 × 2^
= 2^(8-5) × 3^(7-5)
= 2^3 × 3^2
= 8 × 9
= 72
(7) Solve for x :
(i) 2^2x-1/2^x+2 = 4,
(ii) 5^5 × 5^-4 × 5^x/5^12 = 5^-5
Solution :
(i) 2^2x-1/2^x+2 = 4
= 2^(2x-1)-(x+2)= 4
= 2^x-3 = 2^2
∴ Comparing the powers we get,
x-3 = 2
∴ x = 5
(ii) 5^5×5^-4×5^x/5^12 = 5^-5
= 5^(5+(-4)+x-12) = 5^-5
= 5(x-11) = 5^-5
∴ comparing the powers we get,
x – 11 = -5
Therefore, x = 6
(9) Find the number in standard form for the following expansions:
(i) 8 × 10^4 + 7×10^3 + 6×10^2 + 5 × 10^1 + 2×1 + 4 × 10^-2+7×10^-4
(ii) 5 × 10^3+5 × 10^1 + 5 × 10^-1+5 × 10^-3
(iii) The radius of a hydrogen atom is 2.5 × 10^–11 m
Solution :
(i) 8 × 10^4+7 × 10^3+6 × 10^2+5 × 10^1+2 × 1+4 × 10^-2+7 × 10^-4
= 10^3(80+7) + 10 (60+5) + 2 + 10^-2 (4+0.07)
= 87 × 10^3 + 65 × 10 + 2 + 4.07 × 10^-2
= 10(8700+65) + 2 + 407/100 × 1/100
= 87650 + 2 +0.0407
= 87652.0407
(ii) 5 × 10^3+5 × 10^1+5 × 10^-1+5 × 10^-3
= 5 × 10[100+1] +5 × 10^-1 [1+1/100]
= 50101 + 5/10 × 101/100
= 5 × 101 [1+1/1000]
= 5 × 101 × 1001/1000
= 505.505
(iii) The radius of a hydrogen atom is 2.5 × 10^–11 m
= 2.5 × 10^–11 m
= 0.025 × 10^-13
(10) Write the following numbers in scientific notation:
(i) 467800000000
(ii) 0.000001972
(iii) 1642.398
(iv) Earth’s volume is about 1,083,000,000,000 cubic kilometres
(v) If you fill a bucket with dirt, the portion of the whole Earth that is in the bucket will be 0.0000000000000000000000016 kg
Solution :
Numbers in scientific notation :
(i) = 4.678×10^11
(ii) = 1.972×10^-6
(iii) = 1.642398×10^3
(iv) = 1.083 × 10^12
(v) = 1.6 × 10^-24
Objective Type Questions :
(11) By what number should (-4)^-1 be multiplied so that the product becomes 10^-1
(A) 2/3
(B) -2/5
(C) 5/2
(D) -5/2
Solution :
(-4)^-1 × -2/5
= -1/4 × -2/5 = 1/10 = (10)^-1
Hence, option (B) -2/3 is the correct option.
(13) (-2)^-3 × (-2)^-2 = ____
(A) -1/32
(B) 1/32
(C) 32
(D) -32
Solution :
(-2)^-3 × (-2)^-2
= (-2)^-3-2
= (-2)^-5
= -1/2^5
= -1/32
Hence, the correct answer is option (A) -1/32.
(13) Which is not correct?
(A) (-1/4)^2 = 4^-2
(B) (-1/4)^2 = (-1/2)^4
(C) (-1/4)^2 = 16^-1
(D) – (1/4)^2 = 16^-1
Solution :
Option (D) – (1/4)^2 = 16^-1 is not correct.
(14) If 10^x/10^-3 = 10^9, then x is
(A) 4
(B) 5
(C) 6
(D) 7
Solution :
10^x/10^-3 = 10^9
= 10^x + 3 = 10^9
∴ x + 3 = 9
x = 6
Hence, the correct answer is option (C) 6.
(15) 0.0000000002020 in scientific form is ___
(A) 2.02×10^9
(B) 2.02×10^-9
(C) 2.02×10^-8
(D) 2.02×10^-10
Solution :
Correct option – (B)
0.0000000002020 in scientific form is 2.02×10^-9.
Numbers Exercise 1.7 Solutions :
Question no – (1)
If 3/4 of a box of apples weighs 3 kg and 225 gm, how much does a full box of apples weigh?
Solution :
3 kg and 225 gm = 3225 gm
3/4 of a box weights 3225 gm
1 of a box weights 3225 × 4/3
∴ (1-3/4) = 1/4 of box weights 3225 × 4/3 × 1/4 = 175 gm
∴ (3/4 + 1/4) = total box of apple weight (3225 + 175) gm
= 3400 gm
= 3 kg 400 gm
Hence, a full box of apples weigh 3 kg 400 gm.
Question no – (2)
Mangalam buys a water jug of capacity 3 4/5 litre. If she buys another jug which is 2 2/3 times as large as the smaller jug, how many litre can the larger one hold.
Solution :
The large one hold,
3 4/5 × 2 2/3
= 19/5 × 8/3
= 152/15
= 10 2/15 liter
∴ The large one hold 10 2/15 liter.
Question no – (3)
Ravi multiplied 25/8 and 16/15 and he says that the simplest form of this product is 10/3 and Chandru says the answer in the simplest form is 3 1/3 who is correct. are they both correct. explain.
Solution :
= 25/8 × 16/15
= 10/3
= 3 1/3
Therefore, they both are correct.
Question no – (4)
Find the length of a room whose area is 153/10 sq.m and whose breadth is 2 11/20 m.
Solution :
Let, the length of room is L
We know, Area = Length × breadth
153/10 = L × 2 11/20
= 51/20 × L = 153/10
= L = 153/10 × 20/51
= L = 6 m
Therefore, the length of a room is 6 m.
Question no – (6)
A greeting card has an area 90 cm². between what two whole numbers is the length of its side
Solution :
Given area 90 cm²
81 < 90 < 100
9² < 90 < 10²
Hence, between 9 and 10 is the length of its side.
Question no – (9)
If 2^m-1 + 2^m+1 = 640, then find m
Solution :
2^m-1 + 2^m+1 = 640
= 2^m/2 + 2^m × 2 = 640
= 2^m(1/2+2) = 640
= 2^m × 5/2 = 640
= 2^m = 640 × 2/5
= 2^m = 256
= 2^m = 2^8
Therefore, m = 8
Question no – (10)
Give the answer in scientific notation :
A human heart beats at an average of 80 beats per minute. how many times does it beat in
(i) an hour?
(ii) a day?
(iii) a year?
(iv) 100 years
Solution :
Average of 80 beets per minute
(i) 1 hour it beets,
= 80 × 60
= 4800 beats
(ii) A day = 24 hours in beets,
= 80 × 60 × 24
= 115200 beats
(iii) A year = 365 days in beats,
= 80 × 60 × 24 × 365
= 42048000 beats
(iv) 100 years = 36500 days in beats,
= 80 × 60 × 24 × 365 × 36500
= 1.534752×10^12 beats
Challenging Problem Solutions :
Question no – (11)
In a map, if 1 inch refers to 120 km, then find the distance between two cities B and C which are 4 1/6 inches and 3 1/3 inches from the city A which lies between the cities B and C
Solution :
1 inch = 120 cm,
4 1/6 = 25/6 inch
= 25/6 × 120 km
= 500 km
3 1/3 = 10/3 inch
= 10/3 × 120
= 400 km
∴ A lies between B an C
∴ 500 + 400/2 = 450
Distance between B and A,
= (500 – 450)
= 50 km
Distance between A and C,
= (450 – 400)
= 50 km
Question no – (12)
Give an example and verify each of the following statements
(i) Subtraction is not commutative for rational numbers.
(ii) Division is not associative for rational numbers.
(iii) Distributive property of multiplication over subtraction is true for rational numbers. That is,a(b-c) = ab – ac.
(iv) The mean of two rational numbers is rational and lies between them.
Solution :
(i) Let two rational number 1/5, 2/5
∴ If we subtract
1/5 – 2/5 = 1-2/5 = -1/5
And 2/5 – 1/5 = 2-1/5 = 1/5
∴ Subtraction is not commutative for rational numbers
(ii) Let, a = 1/5, b = 2/5, c = 3/5 be three rational numbers
(a ÷ b) ÷ c = (1/5 ÷ 2/5) ÷ 3/5
= (1/5 × 5/2) × 5/3
= 1/2 × 5/3
= 5/6
b ÷ (b ÷ c) = 1/5 ÷ (2/5 ÷ 3/5)
= 1/5 ÷ (2/5 × 5/3)
= 1/5 × 2/3
= 3/10
∴ Division is not associative for rational numbers
(iii) Let, any three rational numbers
a = 1/5, b = 2/5, c = 3/5
∴ a(b-c) = 1/5 (2/5 – 3/5)
= 1/5 × (-1/5)
= – 1/25
ab – ac = 1/5 × 2/5 – 1/5 × 3/5
= 2-3/25
= -1/25
∴ Multiplication over subtraction is true
(iv) Let, a = 1/5 and b = 2/5 and two rational mean of a and b
= (1/5 + 2/5)/2
= (3/5)/2
= 3/10
∴ 1/5 < 3/10 < 2/5
∴ Mean of any two rational numbers is rational and lies between them.
Question no – (13)
If 1/4 of a ragi adai weighs 120 grams, what will be the weight of 2/3 of the same ragi adai?
Solution :
1/4 of ragi adai wight 120 gm
1 of adai weigh 120 × 1/4 gm
2/3 of ragi adai weight 120×1/4×2/3 gm
= 120 × 1/4 × 2/3
= 20 gm
So, the weight of 2/3 of the same ragi adai is 20 gm.
Question no – (14)
If p +2q = 18 and pq = 40, find 2/p + 1/q
Solution :
p + 2q = 18, pq = 40
2/p + 1/q
= 2q + p/pq
= 18/40
= 9/20
Question no – (15)
Find x if 5 x/5 3 3/4 = 21
Solution :
5 x/5 × 3 3/4 = 21
= (25+x)/5 15 = 21
= 25+x/5 = 21-15
= 25+x = 6×5
= x = 30 – 25
= 5
Hence, x = 5
Question no – (16)
By how much does 1/(10/11) exceed (1/10)/11
Solution :
(1/10)/11 = 11/10
Multiplying this by 1/121 we get
11/10 × 1/121
= 1/10 × 11
= (1/10)/11
Question no – (17)
A group of 1536 cadets wanted to have a parade forming a square design. is it possible if it is not possible, how many more cadets would be required.
Solution :
1536 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 23
= (2 × 2 × 2 × 2)^2 × 2 × 3
It is not possible.
It we multiplies 6 cadets then it is possible
1536 × 6 = (2 × 2 × 2 × 2)^2 × 2 × 3 × 2 × 3
= (2 × 2 × 2 × 2 × 2 × 3)^2
= 96^2
Question no – (19)
Simplify (3.769 × 10⁵) + (4.21 × 10⁵)
Solution :
(3.769 × 10⁵) + (4.21 × 10⁵)
= 10⁵ (3.769 + 4.21)
= 10⁵ × 7.979
= 7.979/1000 × 10⁵
= 79790 …(Simplified)
Question no – (20)
Order the following from the least to the greatest : 16^25, 8^100, 3^500, 4^400, 2^600
Solution :
16^25, 8^100, 3^500, 4^400, 2^600
= (2^4)^25, (2^3)6100, 3^500, (2^2)^400, 2^600
= 2^100, 2^300, 3^500, 2^600, 2^800
Next Chapter Solution :
👉 Measurements
