ICSE Class 7 Mathematics · Solutions for Class 7 RS Aggarwal ICSE Foundation Mathematics · 97 solved questions
Write all integers between:
(i) -9 and -3
(ii) -33 and -26
(iii) -5 and 2
(iv) -1 and 6
(i) All integers between -9 and -3 are:
-8, -7, -6, -5, -4
(ii) All integers between -33 and -26 are:
-32, -31, -30, -29, -28, -27
(iii) All integers between -5 and 2 are:
-4, -3, -2, -1, 0, 1
(iv) All integers between -1 and 6 are:
0, 1, 2, 3, 4, 5
Compare the integers :
(i) -16 and -61
(ii) -39 and 13
(iii) -236 and -362
(iv) -2 and 0
(i) -16 and -61
(16 < 61) ⇒ (–16 > –61) $\quad$ [∵ a < b ⇒ -a > -b]
(ii) -39 and 13
We know that every negative integer is less than every positive integer.
∴ –39 < 13
(iii) -236 and -362
(236 < 362) ⇒ (–236 > –362) $\quad$ [∵ a < b ⇒ -a > -b]
(iv) -2 and 0
We know that every negative integer is less than 0.
∴ –2 < 0
Evaluate :
(i) 20 - | -11 |
(ii) | -23 | - | -16 |
(iii) | -137 | + | 13 |
(iv) 6 - | -4 |
(i)
We have
20 - | -11 |
= 20 - 11
= 9 $\quad$ [∵ |-11| = 11]
(ii)
We have
| -23 | - | -16 |
= 23 - 16
= 7 $\quad$ [∵ |-23| = 23 and |-16| = 16]
(iii)
We have
| -137 | + | 13 |
= 137 + 13
= 150 $\quad$ [∵ |-137| = 137 and |13| = 13]
(iv)
We have
6 - |-4|
= 6 - 4
= 2 $\quad$ [∵ |-4| = 4 ]
Arrange the following integers in ascending order :
-9, 11, -23, 41, -37, 0, -5
Given negative integers are −9, −23, −37, −5.
In ascending order they are −37 < −23 < −9 < −5.
Given positive integers are 11, 41, 0.
In ascending order they are 0 < 11 < 41.
Hence, all the given integers in ascending order are:
−37 < −23 < −9 < −5 < 0 < 11 < 41.
i.e., −37, −23, −9, −5, 0, 11, 41.
Arrange the following integers in ascending order :
-31, 19, -50, -8, -23, 3, 8
Given negative integers are −31, −50, −8, −23.
In ascending order they are −50 < −31 < −23 < −8.
Given positive integers are 19, 3, 8.
In ascending order they are 3 < 8 < 19.
Hence, all the given integers in ascending order are:
−50 < −31 < −23 < −8 < 3 < 8 < 19.
i.e., −50, −31, −23, −8, 3, 8, 19.
Arrange the following integers in ascending order :
-2, 12, -43, 31, 7, -35, -10
Given negative integers are −2, −43, −35, −10.
In ascending order they are −43 < −35 < −10 < −2.
Given positive integers are 12, 31, 7.
In ascending order they are 7 < 12 < 31.
Hence, all the given integers in ascending order are:
−43 < −35 < −10 < −2 < 7 < 12 < 31.
i.e., −43, −35, −10, −2, 7, 12, 31.
Arrange the following integers in descending order:
-24, 16, -40, -5, -13, 6, -1
Given positive integers are 16, 6.
In descending order they are 16 > 6.
Given negative integers are −24, −40, −5, −13, −1.
In descending order they are −1 > −5 > −13 > −24 > −40.
Hence, all the given integers in descending order are:
16 > 6 > −1 > −5 > −13 > −24 > −40.
i.e., 16, 6, −1, −5, −13, −24, −40.
Arrange the following integers in descending order:
0, -7, 19, -23, -3, 8, 46
Given positive integers are 19, 8, 46.
In descending order they are 46 > 19 > 8.
Given negative integers are −7, −23, −3.
In descending order they are −3 > −7 > −23.
Also, 0 is less than positive integers and greater than negative integers.
Hence, all the given integers in descending order are:
46 > 19 > 8 > 0 > −3 > −7 > −23.
i.e., 46, 19, 8, 0, −3, −7, −23.
Arrange the following integers in descending order:
-31, -13, -49, 4, 37, -9, -52
Given positive integers are 4, 37.
In descending order they are 37 > 4.
Given negative integers are −31, −13, −49, −9, −52.
In descending order they are −9 > −13 > −31 > −49 > −52.
Hence, all the given integers in descending order are:
37 > 4 > −9 > −13 > −31 > −49 > −52.
i.e., 37, 4, −9, −13, −31, −49, −52.
Fill in the blanks:
(i) 0 is greater than every ............... integer.
(ii) Every negative integer is less than every ............... integer .
(iii) For any two positive integers a and b, if a < b, then -a ............... -b .
(iv) Modulus of a negative integer is always ............... .
(v) The largest negative integer is ............... .
(vi) The smallest positive integer is ............... .
(i) 0 is greater than every negative integer.
(ii) Every negative integer is less than every positive integer.
(iii) For any two positive integers a and b, if a < b, then −a > −b.
(iv) Modulus of a negative integer is always positive.
(v) The largest negative integer is −1.
(vi) The smallest positive integer is 1.
Add the following integers :
(i) 64 and 36
(ii) 73 and -37
(iii) -26 and -45
(iv) -51 and 25
(v) 100 and -32
(vi) 0 and -21
(i)
We have
64 + 36
= 100
(ii)
We have
73 + (-37)
Since one integer is positive and the other is negative, we subtract their values and keep the sign of the greater number (73).
73 + (-37)
= 73 - 37
= 36
(iii)
We have
-26 + (-45)
Since both integers are negative, we add their values and keep the negative sign.
-26 + (-45)
= -26 - 45
= -71
(iv)
We have
−51 + 25
Since one integer is negative and the other is positive, we subtract their values and keep the sign of the greater number (−51).
−51 + 25
= −(51 − 25)
= −26
(v)
We have
100 + (−32)
Since one integer is positive and the other is negative, we subtract their values and keep the sign of the greater number (100).
100 + (−32)
= 100 − 32
= 68
(vi)
We have
0 + (-21)
Zero is the additive identity. Adding 0 to any integer does not change its value.
∴ 0 + (-21)
= -21
Find the additive inverse of :
(i) 23
(ii) -33
(iii) -1
(iv) -476
(i) 23
Since 23 + (−23) = 0
∴ the additive inverse of 23 is -23
(ii) -33
Since (−33) + 33 = 0
∴ the additive inverse of −33 is 33
(iii) -1
Since (−1) + 1 = 0
∴ the additive inverse of −1 is 1
(iv) -476
Since (−476) + 476 = 0
∴ the additive inverse of −476 is 476
Evaluate :
(i) 6 - 24
(ii) 18 - (-8)
(iii) (-16) - (-5)
(iv) (-20) - 6
(v) (-1) - (-19)
(vi) 8 - (-23)
(i)
We have
6 - 24
= 6 + (-24)
= -18
(ii)
We have
18 - (-8)
= 18 + 8
= 26 $\quad$ [∵ a - (-b) = a + b]
(iii)
We have
(-16) - (-5)
= -16 + 5
= -11
(iv)
We have
(-20) - 6
= (-20) + (-6)
= -26
(v)
We have
(-1) - (-19)
= (-1) + 19
= 18
(vi)
We have
8 - (-23)
= 8 + 23
= 31 $\quad$ [∵ a - (-b) = a + b]
Verify the following :
(-14) + 9 = 9 + (-14)
We have
(-14) + 9 = 9 + (-14)
LHS = (−14) + 9
= −5
RHS = 9 + (−14)
= −5
Since LHS = RHS,
∴ (-14) + 9 = 9 + (-14)
Verify the following :
(-8) + (-12) = (-12) + (-8)
We have
(-8) + (-12) = (-12) + (-8)
LHS = (−8) + (−12)
= −20
RHS = (−12) + (−8)
= −20
Since LHS = RHS,
∴ (-8) + (-12) = (-12) + (-8)
Verify the following :
(-6) + [(-8) + 12] = [(-6) + (-8)] + 12
We have
(-6) + [(-8) + 12] = [(-6) + (-8)] + 12
LHS = (−6) + [−8 + 12]
= (−6) + 4
= −2
RHS = [(−6) + (−8)] + 12
= (−14) + 12
= −2
Since LHS = RHS,
∴ (-6) + [(-8) + 12] = [(-6) + (-8)] + 12
Verify the following :
[(-9) + (-7)] + (-14) = (-9) + [(-7) + (-14)]
We have
[(-9) + (-7)] + (-14) = (-9) + [(-7) + (-14)]
LHS = [−9 + (−7)] + (−14)
= (−16) + (−14)
= −30
RHS = (−9) + [−7 + (−14)]
= (−9) + (−21)
= −30
Since LHS = RHS,
∴ [(-9) + (-7)] + (-14) = (-9) + [(-7) + (-14)]
Fill in the blanks:
(i) 8 + ............... = 0
(ii) (-10) + ............... = -10
(iii) (-6) + (-8) = (-8) + ...............
(iv) (-6) + ............... = -14
(v) (-11) + ............... = (-7)
(vi) (-9) + ............... = -1
(i) 8 + (-8) = 0
(ii) (-10) + 0 = -10
(iii) (-6) + (-8) = (-8) + (-6)
(iv) (-6) + (-8) = -14
(v) (-11) + 4 = (-7)
(vi) (-9) + 8 = -1
Subtract the sum of -137 and -43 from the sum of -103 and 27.
Calculate the sum of -137 and -43:
(-137) + (-43) = -180
Calculate the sum of -103 and 27:
(-103) + 27 = -76
Subtract the first sum from the second sum:
(-180) - (-76)
= -180 + 76
= -104
∴ The answer is -104
Subtract -29 from -53 and add -16 to the result.
Subtract -29 from -53:
-53 - (-29) = -53 + 29 = -24
Add -16 to the result: -24 + (-16) = -40
Final Answer is -40
The sum of two integers is 43. If one of them is -27, find the other.
Let the two integers be x and y.
Given that x + y = 43 and x = -27.
Substitute x in the below equation
x + y = 43
-27 + y = 23 $\hspace{2cm}$(Substituting x)
y = 23 + 27 $\hspace{2.2cm}$(Solve for y)
y = 23 + 27
y = 70
∴ The other integer is 70
Find the successor of :
(i) 30
(ii) -70
(iii) -206
(iv) -1
(i)
The successor of 30 is:
30 + 1 = 31
(ii)
The successor of -70 is:
-70 + 1 = -69
(iii)
The successor of -206 is:
-206 + 1 = -205
(iv)
The successor of -1 is:
-1 + 1 = 0
Find the predecessor of:
(i) 60
(ii) -351
(iii) 0
(iv) -99
(i)
The predecessor of 60 is:
60 - 1 = 59
(ii)
The predecessor of -351 is:
-351 - 1 = -352
(iii)
The predecessor of 0 is:
0 - 1 = -1
(iv)
The predecessor of -99 is:
-99 - 1 = -100
Find the product :
18 x 4
We have
18 x 4 = 72
Find the product :
(-25) x 6
We have
(-25) x 6 = -150
[∵ Negative x Positive = Negative]
Find the product :
(-30) x 7
We have
(-30) x 7 = -(30 x 7) = -210
[∵ Negative x Positive = Negative]
Find the product :
8 x (-15)
We have
8 x (-15) = -(8 x 15) = -120
[∵ Positive x Negative = Negative]
Find the product :
20 x (-10)
We have
20 x (-10) = -(20 x 10) = -200
[∵ Positive x Negative = Negative]
Find the product :
(-12) x (-15)
We have
(-12) x (-15) = +(12 x 15) = 180
[∵ Negative x Negative = Positive]
Find the product :
(-8) x (-13)
We have
(-8) x (-13) = +(8 x 13) = 104
[∵ Negative x Negative = Positive]
Find the product :
(-20) x (-1)
We have
(-20) x (-1) = +(20 x 1) = 20
[∵ Negative x Negative = Positive]
Find the product :
(-9) x 0
We have
(-9) x 0 = -(9 x 0) = 0
[∵ Anything multiplied with 0 becomes 0]
Find the product :
0 x (-11)
We have
0 x (-11) = -(0 x 11) = 0
[∵ Anything multiplied with 0 becomes 0]
Find the product :
{(-9) x 8} x (-5)
We have
{(-9) x 8} x (-5)
First multiply inside the brace:
(-9) x 8 = -72
Now multiply the result with last number:
(-72) x (-5) = +(72 x 5) = 360
∴ {(-9) x 8} x (-5) = 360
Find the product :
{(-10) x (-5)} x 6
We have
{(-10) x (-5)} x 6
First multiply inside the brace:
(-10) x (-5) = 50
Now multiply the result with last number:
50 x 6 = 300
∴ {(-10) x (-5) x 6} = 300
Find the product :
{(-12) x (-15)} x (-2)
We have
{(-12) x (-15)} x (-2)
First multiply inside the brace:
(-12) x (-15) = 180
Now multiply:
180 x (-2) = -(180 x 2) = -360
∴ {(-12) x (-15)} x (-2) = -360
Find the product :
(-8) x {(-5) x (-3)}
We have
(-8) x {(-5) x (-3)}
First multiply inside the brace:
(-5) x (-3) = 15
Now multiply the result with the first number:
(-8) x 15 = -(8 x 15) = -120
∴ (-8) x {(-5) x (-3)} = -120
Find the product :
(-11) x {(-8) x 5}
We have
(-11) x {(-8) x 5}
First multiply inside the brace:
(-8) x 5 = -40
Now multiply the result with the first number:
(-11) x (-40) = +(11 x 40) = 440
∴ (-11) x {(-8) x 5} = 440
Verify the following :
(-14) x (-8) = (-8) x (-14)
We have
(-14) x (-8) = (-8) x (-14)
LHS = (-14) x (-8) = 112
RHS = (-8) x (-14) = 112
Since LHS = RHS,
∴ (-14) x (-8) = (-8) x (-14) $\hspace{1.5cm}$[a x b = b x a ⇒ Commutative Property of multiplication.]
Verify the following :
{(-7) x 5} x (-6) = (-7) x {5 x (-6)}
We have
{(-7) x 5} x (-6) = (-7) x {5 x (-6)}
LHS = {(-7) x 5} x (-6)
= (-35) x (-6)
= 210
RHS:(-7) x {5 x (-6)}
= (-7) x (-30)
= 210
Since LHS = RHS,
∴ {(-7) x 5} x (-6) = (-7) x {5 x (-6)} $\hspace{1.5cm}$[{a x b} x c = a x {b x c} ⇒ Associative Property of multiplication.]
Verify the following :
(-10) x {(-7) + (-9)} = {(-10) x (-7)} + {(-10) x (-9)}
We have
(-10) x {(-7) + (-9)} = {(-10) x (-7)} + {(-10) x (-9)}
LHS = (-10) x {(-7) + (-9)}
=(-10) x {-16}
= 160
RHS = {(-10) x (-7)} + {(-10) x (-9)}
= 70 + 90
= 160
Since LHS = RHS,
∴ (-10) x {(-7) + (-9)} = {(-10) x (-7)} + {(-10) x (-9)} $\hspace{1.5cm}$[a x {b + c} = {a x b} + {a x c} ⇒ Distributive Property of multiplication over addition.]
Find the quotient :
28 ÷ (-7)
We have
28 ÷ (-7)
= $\frac {+28}{-7}$
= -4
Find the quotient :
(-65) ÷ 13
We have
(-65) ÷ 13
= $\frac {-65}{+13}$
= -5
Find the quotient :
(-66) ÷ (-6)
We have
(-66) ÷ (-6)
= $\frac {-66}{-6}$
= 11
Find the quotient :
(-9) ÷ (-1)
We have
(-9) ÷ (-1)
= $\frac {-9}{-1}$
= 9
Find the quotient :
0 ÷ (-2)
We have
0 ÷ (-2)
= $\frac {0}{-2}$
= 0
Find the quotient :
(-12) ÷ (-12)
We have
(-12) ÷ (-12)
= $\frac {-12}{-12}$
= 1
Write all even integers between
(i) (-4) and 11
(ii) (-13) and (-7)
We have
(i) All even integers between -4 and 11 are:
-2, 0, 2, 4, 6, 8 and 10
(ii) All even integers between -13 and -7 are:
-12, -10 and -8
Write all odd integers between
(i) (-1) and 7
(ii) (-20) and (-14)
(i) All odd integers between -1 and 7 are:
1, 3 and 5
(ii) All odd integers between -20 and -14 are:
-19, -17 and -15
Write five consecutive even integers succeeding -21.
The required consecutive even integers succeeding -21 are:
-20, -18, -16, -14 and -12
Write five consecutive odd integers preceding -36.
The required consecutive odd integers preceding -36 are:
-37, -39, -41, -43 and -45
The product of two integers is -120. If one number is 15, find the other.
Let the two integers be p and q.
Given that p x q = -120 and p = 15
Substitute p in the below equation:
p x q = -120
15 x q = −120 $\hspace{2cm}$(Substitute p)
q = $\frac{-120}{15}$ $\hspace{2.7cm}$(Solve for q)
q = -8
∴ The other number is -8.
Simplify:
5 {(-6) + (12 ÷ 4)}
Given expression:
5 {(-6) + (12 ÷ 4)}
= 5 {(-6) + 3} $\quad$ [Simplifying ( )]
= 5 x -3 $\quad$ [Simplifying { }]
= -15
The answer is -15
Simplify:
10 - {56 ÷ (-16 + 9)}
Given expression:
10 - {56 ÷ (-16 + 9)}
= 10 - {56 ÷ (-7)} $\quad$ [Simplifying ( )]
= 10 - (-8) $\quad$ [Simplifying { }]
= 10 + 8 $\quad$ [Removing ( )]
= 18
The answer is 18
Simplify:
(-6) - {(-28) ÷ (-7)}
Given expression:
(-6) - {(-28) ÷ (-7)}
= (-6) - {4} $\quad$ [Simplifying ÷]
= (-6) - 4 $\quad$ [Removing { }]
= -6 - 4 $\quad$ [Removing ( )]
= -10
The answer is -10
Simplify:
$8 - \text{\textbraceleft}12 - (9 - \overline{7 - 4})\text{\textbraceright}$
Given expression:
$8 - \text{\textbraceleft}12 - (9 - \overline{7 - 4})\text{\textbraceright}$
$$\begin{aligned}= 8 - \text{\textbraceleft}12 - (9 - 3)\text{\textbraceright} \quad \text{[Removing bar]} \\[1em] = 8 - \text{\textbraceleft}12 - 6\text{\textbraceright} \quad \text{[Removing ( )]} \\[1em] = 8 - 6 = 2 \quad \text{[Removing \text{\textbraceleft} \text{\textbraceright}]}\end{aligned}$$
The answer is 2
Simplify:
$11 - [(-8) - \text{\textbraceleft}10 - (9 - \overline{7 - 4})\text{\textbraceright}]$
Given expression:
$11 - [(-8) - \text{\textbraceleft}10 - (9 - \overline{7 - 4})\text{\textbraceright}]$
$$\begin{aligned}= 11 - [(-8) - \text{\textbraceleft}10 - (9 - 3)\text{\textbraceright}] \quad \text{[Removing bar]} \\[1em] = 11 - [(-8) - \text{\textbraceleft}10 - 6\text{\textbraceright}] \quad \text{[Simplifying ( )]} \\[1em] = 11 - [(-8) - 4] \quad \text{[Simplifying \text{\textbraceleft} \text{\textbraceright}]} \\[1em] = 11 - [-12] \quad\text{[Removing ( )]} \\[1em] = 11 + 12 = 23 \quad \text{[Removing [ ]]}\end{aligned}$$
The answer is 23
Simplify:
$(-9) - [(-5) - \text{\textbraceleft}11 - (8 - \overline{5 - 2})\text{\textbraceright}]$
Given expression:
$(-9) - [(-5) - \text{\textbraceleft}11 - (8 - \overline{5 - 2})\text{\textbraceright}]$
$$\begin{aligned}= (-9) - [(-5) - \text{\textbraceleft}11 - (8 - 3)\text{\textbraceright}] \quad\text{[Removing bar]} \\[1em] = (-9) - [(-5) - \text{\textbraceleft}11 - 5\text{\textbraceright}] \quad\text{[Simplifying ( )]} \\[1em] = (-9) - [(-5) - 6] \quad\text{[Simplifying \text{\textbraceleft} \text{\textbraceright}]} \\[1em] = (-9) - [-11] \quad\text{[Removing ( )]} \\[1em] = -9 + 11 = 2 \quad\text{[Removing [ ]]}\end{aligned}$$
The answer is 2
Simplify:
$10 - [8 - \text{\textbraceleft}11 + 30 ÷ (4 - \overline{5 - 7})\text{\textbraceright}]$
Given expression:
$10 - [8 - \text{\textbraceleft}11 + 30 ÷ (4 - \overline{5 - 7})\text{\textbraceright}]$
$$\begin{aligned}= 10 - [8 - \text{\textbraceleft}11 + 30 ÷ (4 - (-2))\text{\textbraceright}] \quad\text{[Removing bar]} \\[1em] = 10 - [8 - \text{\textbraceleft}11 + 30 ÷ (6)\text{\textbraceright}] \quad\text{[Simplifying ( )]} \\[1em] = 10 - [8 - \text{\textbraceleft}11 + 5\text{\textbraceright}] \quad\text{[Simplifying ÷]} \\[1em] = 10 - [8 - 16] \quad\text{[Simplifying \text{\textbraceleft} \text{\textbraceright}]} \\[1em] = 10 - [-8] \quad\text{[Simplifying [ ]]} \\[1em] = 10 + 8 = 18 \quad\text{[Removing [ ]]}\end{aligned}$$
The answer is 18
By how much does 5 exceed -5?
To find how much one number exceeds another, we subtract the smaller number from the larger one:
5 - (-5)
= 5 + 5
= 10
The answer is 10
How much -7 less than -1?
We need to find the difference between −1 and −7.
So we calculate:
(-1) - (-7)
= -1 + 7
= 6
The answer is 6
What must be subtracted from 7 to get -6?
Let the required number be x.
7 - x = -6
To solve for x, we rearrange:
x = 7 - (-6)
x = 7 + 6
x = 13
The answer is 13
What must be subtracted from -1 to get -19?
Let the required number be x
(-1) - x = -19
Rearrange to solve for x:
x = -1 - (-19)
x = -1 + 19
x = 18
The answer is 18
The sum of two integers is -11. If one of them is 5, then find the other.
Let the two integers be p and q.
Given,
p = 5
p + q = -11
Substitute p in the below equation:
p + q = -11
5 + q = -11 $\hspace{2cm}$[Substituting p]
q = -11 - 5 $\hspace{2.1cm}$[Solve for q]
q = -(11 + 5)
q = -16
The answer is -16
Add the product of (-13) and (-17) to the quotient of (-187) and 11.
Calculate the product of (-13) and (-17):
(-13) x (-17) = 221 $\hspace{5cm}$(Negative x Negative = Positive)
Calculate the quotient of (-187) and 11:
(-187) ÷ 11 = -17 $\hspace{5cm}$(Negative ÷ Positive = Negative)
Now add them together:
221 + (-17)
= 221 - 17 = 204
The answer is 204
A shopkeeper bought a pen for ₹75, a book for ₹240 and a pencil box for ₹46. He sold the pen for ₹81, book for ₹255 and the pencil box for ₹40. What was his gain or loss?
Given:
Cost Price (CP) of pen = ₹75
Cost Price (CP) of book = ₹240
Cost Price (CP) of pencil box = ₹46
∴ The total Cost Price(CP) = 75 + 240 + 46 = ₹361
Selling Price of pen = ₹81
Selling Price of book = ₹255
Selling Price of pencil box = ₹40
∴ The total Selling Price(SP) = 81 + 255 + 40 = ₹376
Since SP is greater than CP, it is a gain.
Gain = SP - CP
Gain = 376 - 361 = 15 $\hspace{1cm}$[Substituting the values of total SP and total CP]
∴ The shopkeeper gains ₹15
A man starts from his home and drives 169 km to the East and then 192 km to the West. How far is he from his home finally and in which direction?
Let East be positive direction so West will be negative direction.
Given:
Distance travelled towards East = 169 km
Distance travelled towards West = -192 km
Total displacement = Distance travelled towards East + Distance travelled towards West
By substituting the values, we get
Total displacement = 169 + (-192)
= 169 - 192 = -23
The negative sign indicates that he is in West direction.
∴ He is 23 km to the West of his home.
During a month, the average day temperature of a desert region was 43°C and the average night temperature was -6°C. Find the difference between the average day temperature and the average night temperature.
Given:
Average day temperature = 43°C
Average night temperature = -6°C
Difference = Average day temperature - Average night temperature
By substituting the values, we get
Difference = 43 - (-6)
= 43 + 6 = 49
∴ The difference between the average day and the average night temperature = 49°C
Radhey bought 4 pairs of jeans at ₹1256 each . How much did he pay for the jeans in all?
Given:
Number of pairs of jeans = 4
Cost of 1 pair of jeans = ₹1256
Total cost = Number of pairs x Cost of 1 pair
By substituting the values, we get
Total cost = 4 x 1256 = ₹5024
∴ The total amount he paid is ₹5024
Mathew had no money to pay for his house rent of ₹14240. His 5 friends decided to contribute equally to pay for the rent. How much will each friend pay?
Given:
Total house rent = ₹14240
Number of friends = 5
Amount paid by each friend = Total house rent ÷ Number of friends
By substituting the values, we get:
Amount paid by each friend = 14240 ÷ 5 = 2848
∴ Each friend contributes ₹2848
A class of 36 students contributed ₹540 each for the picnic. Out of these, 7 students decided not to go. They were returned ₹490 each. What amount was spent on the picnic?
Given:
Total number of students = 36
Contribution per student = ₹540
Total amount collected = 36 x 540 = 19440
Number of students who did not go = 7
Amount returned per student = ₹490
Total amount returned = 7 x 490 = 3430
Amount spent on picnic = Total amount collected - Total amount returned
By substituting the values, we get:
Amount spent on picnic = 19440 - 3430 = 16010
∴ The amount spent on the picnic = ₹16010
The absolute value of 5 is
Absolute value of 5 = |5| = 5
Hence, option 4 is the correct option.
The additive inverse of -7 is
Since -7 + 7 = 0
∴ The additive inverse of -7 is 7
Hence, option 1 is the correct option.
The sum of two integers is -8. If one of them is 5, then the other is
Given:
Sum of two integers = -8
One integer = 5
Let the other integer be x
According to the question, the equation can be written as:
5 + x = -8
x = -8 - 5 = -(8 + 5) $\hspace{2cm}$[Solve for x]
x = -13
∴ The other integer is -13
Hence, option 4 is the correct option.
The successor of -41 is
To find the successor, add 1 to the given integer
-41 = -41 + 1 = -40
∴ The successor of -41 is -40
Hence, option 4 is the correct option.
The value of (-7) x 6 + (-7) x 14 is
Given expression:
(-7) x 6 + (-7) x 14
= -7 x (6 + 14) $\hspace{2cm}$[a x (b + c) = (a x b) + (a x c) ⇒ Distributive property]
=-7 x 20 $\hspace{3cm}$[Simplifying ( )]
= -140
∴ The value of (-7) x 6 + (-7) x 14 is -140
Hence, option 3 is the correct option.
Fill in the blanks :
(i) The absolute value of any integer can never be ............... .
(ii) The sum of any integer and its additive inverse is always ............... .
(iii) a x (b + c) = (a x b) + (a x c) is ............... property of multiplication over addition.
(iv) The product of two integers having unlike signs is always a ............... integer.
(v) The additive inverse of an integer a has the ............... sign as a.
(i) The absolute value of any integer can never be negative.
(ii) The sum of any integer and its additive inverse is always 0.
(iii) a x (b + c) = (a x b) + (a x c) is distributive property of multiplication over addition.
(iv) The product of two integers having unlike signs is always a negative integer.
(v) The additive inverse of an integer a has the opposite sign as a.
State True or False :
For any integer a, the multiplicative inverse is 1.
False
Reason
The multiplicative inverse of an integer a is $\frac{1}{a}$ not a (except when a = 1)
State True or False :
The sum of two integers having unlike signs is always a positive integer.
False
Reason
The sign of the sum depends on the absolute value of the integers. If the negative integer has a greater absolute value, the sum will be negative. If the positive integer has a greater absolute value, the sum will be positive.
State True or False :
The multiplicative inverse of an integer is never an integer.
False
Reason
The multiplicative inverse of 1 is 1 and of −1 is −1, which are integers. However, for all other integers, the inverse is a fraction. Hence, it is not true for all integers.
State True or False :
Any integer multiplied to its multiplicative inverse always gives the multiplicative identity.
True
Reason
By definition, $a \times \frac{1}{a} = 1 \text{ for } (a \neq 0).$ Since 1 is the multiplicative identity, this statement is correct.
State True or False :
Every integer has a multiplicative inverse.
False
Reason
0 does not have a multiplicative inverse because division by zero is not defined.
State True or False :
The quotient of two integers with unlike signs is always negative.
True
Reason
The rules for division are the same as multiplication. When you divide a positive integer by a negative one (or vice versa), the result is always negative.
Aditya went to Australia during the holidays of his children. They visited the coral reef and they were all excited to go for scuba diving. Aditya dived 36 feet to reach the brain coral as his instructor had directed him. He then rose by 19 feet to travel over a ridge.
(1) What is the depth of the ridge ?
(2) Aditya again dived 58 feet to reach the base of the reef. What is the depth of the reef ?
(3) In order to see an underwater cave, Aditya rose 26 feet. What is his location with respect to sea-level ?
(4) If a whale moving at a depth of 21 feet from the sea level passed over Aditya, what was its distance from Aditya ?
(1) Given:
Initial depth = -36 feet $\hspace{2cm}$(36 feet below sea level)
He rises 19 feet = +19
Depth = -36 + 19 = -17 $\hspace{2cm}$(17 feet below sea level)
Depth = 17 feet
Hence, option 3 is the correct option.
(2) Starting position = -17 feet $\hspace{2cm}$(from the previous step)
Again he dives 58 feet = -58 feet $\hspace{2cm}$(Given)
Depth = -17 + (-58)
= -17 - 58
= -(17 + 58) = -75 feet
Depth = 75 feet
Hence, option 4 is the correct option.
(3) Starting position = -75 feet $\hspace{2cm}$(from the previous step)
He rises by 26 feet = +26 $\hspace{3cm}$(Given)
Location = -75 + 26 = -49
Location = 49 feet below the sea level
Hence, option 1 is the correct option.
(4) Given:
Whale's position: -21 feet
Aditya's position: -49 feet (from the previous step)
Distance of Whale from Aditya = -21 - (-49)
= -21 + 49 = 28
Distance of Whale from Aditya = 28 feet
Hence, option 4 is the correct option.
Tanvi had a keen interest to invest in share market. So, she took lessons from an investment coach. The coach told her to watch the fluctuations in the value of share over a period of time. She found that on April 1, the price of a share of XYZ company was ₹2552.
(1) On April 2, the price of this share changed by gaining ₹37. What was the price of each share of XYZ on April 2 ?
(2) On April 3, the price changed by losing ₹16 by 12 pm and then again losing ₹8 by the end of the day. What was the price of each share of XYZ by the end of the day on April 3 ?
(3) On April 4, the price of XYZ company's share changed by gaining ₹11 by 12 pm and then again gaining ₹14 by the end of the day. What was the price of each share of XYZ by the end of the day on April 4?
(4) On April 5, the price of the share changed by losing ₹13 by 12 pm and then gaining ₹4 by the end of the day. What was the price of each share of XYZ by the end of the day on April 5 ?
(1) Price on April 2
Given:
Opening price (April 1) = ₹2552
Change = gain of ₹37
Price on April 2 = ₹2552 + ₹37 = ₹2589
Hence, option 2 is the correct option.
(2) Price by the end of April 3
Given:
Change 1 = Loss of ₹16
Change 2 = Loss of ₹8
Total loss = ₹16 + ₹8 = ₹24
Price on April 2 = ₹2589 $\hspace{2cm}$(from the previous step)
Price by the end of April 3 = Price on April 2 - Total loss
Substituting the values, we get:
Price by the end of April 3 = ₹2589 - ₹24 = ₹2565
Hence, option 2 is the correct option.
(3) Price by the end of April 4
Given:
Change 1 = Gain of ₹11
Change 2 = Gain of ₹14
Total Gain = ₹11 + ₹14 = ₹25
Price on April 3 = ₹2565 $\hspace{2cm}$(from the previous step)
Price by the end of April 4 = Price on April 3 + Total Gain
Substituting the values, we get:
Price by the end of April 4 = ₹2565 + ₹25 = ₹2590
Hence, option 3 is the correct option.
(4) Price by the end of April 5
Given:
Change 1 = Loss of ₹13
Change 2 = Gain of ₹4
Price on April 4 = ₹2590 $\hspace{2cm}$(from the previous step)
Price by the end of April 5 = ₹2590 - ₹13 + ₹4 = ₹2581
Hence, option 4 is the correct option.
Assertion: Difference of two negative integers cannot be a positive integer.
Reason: For any two integers a and b, a - b = a + (additive inverse of b)
Assertion (A) is false but Reason (R) is true.
Explanation
The assertion is false because the difference of two negative integers can be positive.
Example:
(-3) - (-7) = -3 + 7 = 4
which is positive.
The reason is true because subtraction of integers is defined as adding the additive inverse:
a - b = a + (-b)
Hence, option 4 is the correct option.
Assertion: Product of three negative integers and a positive integer is negative.
Reason: (-1) x (-1) x (-1) x ............... n times = -1, if n is odd.
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Explanation
Product of three negative integers:
(negative x negative x negative) = (negative)
So, product of three negative integers is negative.
Multiplying this by a positive integer keeps it negative:
(negative result x positive) = (negative)
The reason correctly explains that an odd number of negative integers gives a negative result.
Hence, option 1 is the correct option.
In a competition 3 marks are given for every correct answer and (−2) marks are given for every incorrect answer and no marks for not attempting any question.
Sachin scored 24 marks. If he got 14 correct answers, how many questions has he attempted incorrectly?
Given:
Marks for each correct answer = 3
Marks for each incorrect answer = −2
Number of correct answers = 14
Total score = 24
Marks scored from correct answers = Number of correct answers × Marks for each correct answer
= 14 × 3
= 42
Since the total score is 24, the marks lost due to incorrect answers:
Marks lost = 42 − 24 = 18
Let the number of incorrect answers be x.
Each incorrect answer deducts 2 marks. Therefore,
x × 2 = 18
x = $\frac{18}{2}$
x = 9
∴ Sachin attempted 9 questions incorrectly.
Hence, option 2 is the correct option.
A pair of integers whose product is −12 and there lies seven integers between them, is:
We need a pair of integers whose product is −12 and which has exactly 7 integers lying between them.
Checking option 1: −2 and 6
Product = (−2) × 6 = −12
Integers between −2 and 6 are: −1, 0, 1, 2, 3, 4, 5
Number of integers between them = 7
Checking option 2: −12 and 1
Product = (−12) × 1 = −12
Integers between −12 and 1 are: -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0
Number of integers between them = 12
Checking option 3: −3 and 4
Product = (−3) × 4 = −12
Integers between −3 and 4 are: -2, -1, 0, 1, 2, 3
Number of integers between them = 6
Checking option 4: 3 and -4
Product = 3 × (−4) = −12
Integers between −4 and 3 are: −3, −2, −1, 0, 1, 2
Number of integers between them = 6
Only option 1 satisfies both conditions.
∴ The required pair is −2 and 6.
Hence, option 1 is the correct option.
Which of the following number sentences best describes the problem shown on the number line?
From the number line:
The first arrow starts from 0 and moves 3 units to the left, reaching −3. $\hspace{1cm}$[represents −3]
The second arrow starts from −3 and moves 2 units further to the left, reaching −5. $\hspace{1cm}$[represents + (−2)]
The third arrow starts from −5 and moves 8 units to the right, reaching 3. $\hspace{1cm}$[represents + 8]
∴ The number sentence is −3 + (−2) + 8.
Hence, option 4 is the correct option.
Compare and fill in the box.
Successor of ((−8) × ((−3) − (−5))) ☐ predecessor of ((−12) × ((36) ÷ (−6))).
Step 1: Evaluate the first expression
(−8) × ((−3) − (−5))
= (−8) × (−3 + 5) $\hspace{2cm}$[∵ a − (−b) = a + b]
= (−8) × 2
= −16
Successor of −16 = −16 + 1 = −15 $\hspace{2cm}$[Successor of a is (a + 1)]
Step 2: Evaluate the second expression
(−12) × (36 ÷ (−6))
= (−12) × (−6) $\hspace{2cm}$[∵ 36 ÷ (−6) = −6]
= 72
Predecessor of 72 = 72 − 1 = 71 $\hspace{2cm}$[Predecessor of a is (a − 1)]
Step 3: Compare the two values
−15 < 71
∴ Successor of ((−8) × ((−3) − (−5))) < predecessor of ((−12) × ((36) ÷ (−6))).
Hence, option 2 is the correct option.
If we together add 199 negative integers, the resulting number will be:
The sum of two negative integers is always a negative integer.
When more negative integers are added, the sum remains negative.
So, the sum of 199 negative integers will also be a negative integer.
∴ The resulting number will be negative.
Hence, option 1 is the correct option.
Find x and y.
(i) The sum of two integers is 71. If one of them is −101, then other integer is x.
(ii) The product of an integer and y is zero.
| x | y | |
|---|---|---|
| 1. | 172 | 0 |
| 2. | 184 | 1 |
| 3. | 172 | 1 |
| 4. | 172 | 2 |
Finding x:
Given:
Sum of two integers = 71
One integer = −101
According to the question, the equation can be written as:
x + (−101) = 71
x = 71 − (−101)
x = 71 + 101 $\hspace{2cm}$[∵ a − (−b) = a + b]
x = 172
Finding y:
Given:
Product of an integer and y is zero.
By the multiplicative property of 0, for every integer a, we have a × 0 = 0 × a = 0.
∴ y = 0
So, x = 172 and y = 0.
Hence, option 1 is the correct option.
We have, °C = (°F − 32) × $\frac{5}{9}$. If °C = −35, then, °F is equal to:
Given:
°C = (°F − 32) × $\frac{5}{9}$
°C = −35
Substituting the value of °C, we get:
−35 = (°F − 32) × $\frac{5}{9}$
(−35) × $\frac{9}{5}$ = °F − 32 $\hspace{2cm}$[Multiplying both sides by $\frac{9}{5}$]
$\frac{-35 \times 9}{5}$ = °F − 32
$\frac{-315}{5}$ = °F − 32
−63 = °F − 32
°F = −63 + 32
°F = −31
∴ The value of °F is −31.
Hence, option 2 is the correct option.
A shopkeeper earns a profit of ₹2 by selling a pen and incurs a loss of 50 paise per pencil and loss of 15 paise per eraser while selling pencils and erasers of old stock. On a particular day, he earns a profit of ₹10. If he sold 10 pens and the number of pencils and erasers he sold are in the ratio 7 : 10, then the number of pencils and erasers he sold on the day is:
Given:
Profit per pen = ₹2
Loss per pencil = 50 paise = ₹$\frac{1}{2}$ = ₹0.50
Loss per eraser = 15 paise = ₹$\frac{15}{100}$ = ₹0.15
Number of pens sold = 10
Total profit on the day = ₹10
Ratio of pencils to erasers = 7 : 10
Step 1: Find the profit from selling pens
Profit from pens = 10 × 2 = ₹20
Step 2: Set up the equation using the ratio
Let the number of pencils = 7k and the number of erasers = 10k.
Loss from pencils = 7k × 0.50 = ₹3.5k
Loss from erasers = 10k × 0.15 = ₹1.5k
Total loss = 3.5k + 1.5k = ₹5k
Step 3: Apply the net profit condition
Net profit = Profit from pens − Total loss
According to the question, the equation can be written as:
20 − 5k = 10
5k = 20 − 10
5k = 10
k = 2
Step 4: Find the number of pencils and erasers
Number of pencils = 7k = 7 × 2 = 14
Number of erasers = 10k = 10 × 2 = 20
∴ The shopkeeper sold 14 pencils and 20 erasers.
Hence, option 1 is the correct option.
In a magic square, if each row, column and diagonal have the same sum, then the values of x and y respectively are
| 1 | −10 | 0 |
|---|---|---|
| x | −3 | -2 |
| −6 | 4 | y |
In a magic square, the sum of the integers in every row, every column and both diagonals is the same.
The given magic square is:
| 1 | −10 | 0 |
|---|---|---|
| x | −3 | -2 |
| −6 | 4 | y |
Step 1: Find the magic sum using a complete row
Using the first row:
1 + (−10) + 0 = −9
So, the sum of each row, column and diagonal is −9.
Step 2: Find x using the middle row
x + (−3) + (−2) = −9
x - 3 - 2 = -9
x − 5 = −9
x = -9 + 5
x = −4
Step 3: Find y using the second row
−6 + 4 + y = −9
−2 + y = -9
y = −9 + 2
y = −7
Verification (using diagonals):
Diagonal 1:
1 + (−3) + (−7) = −9
1 - 3 - 7 = -9
1 - 10 = -9
-9 = - 9
LHS = RHS
∴ x = −4 and y = −7.
Hence, option 2 is the correct option.
The additive inverse of −28 + 12 + 42 − 53 is:
Step 1: Simplify the given expression
−28 + 12 + 42 − 53
Grouping positive and negative integers separately:
= (12 + 42) + (−28 − 53)
= 54 + (−81)
= 54 − 81
= −27
Step 2: Find the additive inverse
The additive inverse of an integer a is −a, since a + (−a) = 0.
So, the additive inverse of −27 is 27.
Verification: −27 + 27 = 0
∴ The additive inverse of −28 + 12 + 42 − 53 is 27.
Hence, option 2 is the correct option.
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