NCERT Solution : Integers Ex 1.3 Mathematics Chapter 1 Class 7
1. Find each of the following products :
(a) 3 ✖ (-1)
(b) (-1) ✖ 225
(c) (-21) ✖ (-30)
(d) (-316) ✖ (-1)
(e) (-15) ✖ 0 ✖ (-18)
(f) (-12) ✖ (-11) ✖ (10)
(g) 9 ✖ (-3) ✖ (-6)
(h) (-18) ✖ (-5) ✖ (-4)
(i) (-1) ✖ (-2) ✖ (-3) ✖ 4
(j) (-3) ✖ (-6) ✖ (-2) ✖ (-1)
Solution :
(a) 3 ✖ (-1) = -3
(b) (-1) ✖ 225 = -225
(c) (-21) ✖ (-30) = 630
(d) (-316) ✖ (-1) = 316
(e) (-15) ✖ 0 ✖ (-18) = 0
(f) (-12) ✖ (-11) ✖ (10) = 1320
(g) 9 ✖ (-3) ✖ (-6) = 162
(h) (-18) ✖ (-5) ✖ (-4) = -360
(i) (-1) ✖ (-2) ✖ (-3) ✖ (4) = - 24
(j) (-3) ✖ (-6) ✖ (-2) ✖ (-1) = 36
2. Verify the following :
(a) 18 ✖ [7 + (-3)] = [18 ✖7] + [18✖ (-3)]
(b) (-21) ✖ [(-4) + (-6)] = [(-21) ✖ (-4)] + [(-21) ✖ (-6)]
Solution :
(a) 18 ✖ [7 + (-3)] = [18 ✖7] + [18✖ (-3)]
LHS :
= 18 ✖ [7 + (-3)]
= 18 ✖ [7 - 3]
= 18 ✖ 4
= 72
RHS :
= [18 ✖ 7] + [18 ✖ (-3)]
= 126 + [-54]
= 126 - 54
= 72
RHS = LHS. Hence, verified
(b) (-21) ✖ [(-4) + (-6)] = [(-21) ✖ (-4)] + [(-21) ✖ (-6)]
LHS :
= (-21) ✖ [(-4) + (-6)]
= -21 ✖ [ -4 -6]
= -21 ✖ [-10]
= 210
RHS :
= [(-21) ✖ (-4)] + [(-21) ✖ (-6)]
=84 + 126
= 210
RHS = LHS. Hence, Verified
3. (i) For any integer a, what is (-1) ✖ a equal to ?
(ii) Determine the integer whose product with (-1) is
(a) -22 (b) 37 (c) 0
Solution :
(i) - a
(ii)
(a) 22 ✖ (-1)
= -22
So, the integer is 22
(b) -37 ✖ (-1)
= 37
So, the integer is -37
(c) 0 ✖ (-1)
= 0
So, the integer is 0
4. Starting from (-1) ✖ 5, write various products showing some pattern to show (-1) ✖ (-1) = 1.
Solution :
(-1) ✖ 5 = -5
(-1) ✖ 4 = - 4 = (-5) + 1
(-1) ✖ 3 = - 3 = (-4) + 1
(-1) ✖ 2 = - 2 = (-3) + 1
(-1) ✖ 1 = - 1 = (-2) + 1
(-1) ✖ 0 = 0 = (-1) + 1
(-1) ✖ (-1) = 1 = 0 + 1
5. Find the product, using suitable properties :
(a) 26 ✖ (-48) + (-48) ✖ (-36)
(b) 8 ✖ 53 ✖ (-125)
(c) 15 ✖ (-25) ✖ (-4) ✖ (-10)
(d) (-41) ✖ 102
(e) 625 ✖ (-35) + (-625) ✖ 65
(f) 7 ✖ (50 - 2)
(g) (-17) ✖ (-29)
(h) (-57) ✖ (-19) + 57
Solution :
(a) 26 ✖ (-48) + (-48) ✖ (-36)
= -48 [26 + (-36)]
= -48 [26 - 36]
= - 48 [-10]
= 480
{ Distributive property of multiplication over addition}
(b) 8 ✖ 53 ✖ (-125)
= 53 ✖ [ 8 ✖ (-125) ]
= 53 ✖ [-1000]
= - 53000
{Associative property of multiplication}
(c) 15 ✖ (-25) ✖ (-4) ✖ (-10)
= [ (-25) ✖ (-4)] ✖ [ 15 ✖ (-10)]
= [100] ✖ [-150]
= -15000
(d) (-41) ✖ 102
= (-41) ✖ [100 + 2]
= (-41) ✖ 100 + (-41) ✖ 2
{ Distributive property of multiplication over addition }
= -4100 + (-82)
= -4100 - 82
= -4182
(e) 625 ✖ (-35) + (-625) ✖ 65
= 625 [ (-35) + (-65)]
{ Distributive property of multiplication over addition }
= 625 [ -35 -65]
= 625 [-100]
= -62500
(f) 7 ✖ (50 - 2)
= 7 ✖ 48
= 336
or,
= 7 ✖ 50 - 7 ✖ 2
{ Distributive property of multiplication over addition }
= 350 - 14
= 336
(g) (-17) ✖ (-29)
Using distributive property, we get
= (-17) ✖ [(-30) + 1)]
{ a ✖ (b + c) = a ✖ b + a ✖ c }
= (-17) ✖ (-30) + (-17) ✖ 1
= 510 - 17
= 493
(h) (-57) ✖ (-19) + 57
Using distributive property, we get
= (-57) ✖ (-19) + 57 ✖ 1
{a ✖ b + a ✖ c = a ✖ (b + c)}
= 57 ✖ 19 + 57 ✖ 1
= 57 ✖ (19 + 1)
= 57 ✖ 20
= 1140
6. A certain freezing process requires that room temperature be lowered from 40℃ at the rate of 5℃ every hour. What will be the room temperature 10 hours after the process begins ?
Solution :
Present temperature of room = 40℃
Decrease in temperature after every hour = 5℃
Temperature of room after 10 hours :
= 40℃ + 10 ✖ (-5℃)
= 40℃ - 50℃
= - 10℃
Thus, the room temperature after 10 hours is -10℃, after the process begins.
7. In a class test containing 10 questions, 5 marks are awarded for every correct answer and (-2) marks are awarded for every incorrect answer and 0 for questions not attempted.
(i) Mohan gets four correct and six incorrect answers. What is his score ?
(ii) Reshma gets five correct answers and five incorrect answers, what is her score ?
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score ?
Solution :
Given :-
Total number of questions = 10
Marks awarded for every correct answer = 5
Marks awarded for every incorrect answer = (-2)
Marks for not attempted question = 0
(i) Marks obtained by mohan by 4 correct answers
= 4 ✖ 5
= 20 marks
Marks for 6 incorrect answers
= 6 ✖ (-2)
= -12
∴ Total score of mohan
= marks for correct answers + marks for incorrect answers
= 20 + (-12)
= 20 - 12
= 8 marks
(ii) Makrs obtained by Reshma for 5 correct answers
= 5 ✖ 5
= 25 marks
marks for 5 incorrect answers
= 5 ✖ (-2)
= -10 marks
∴ Total score of Reshma
= marks for correct answers + marks for incorrect answers
= 25 + (-10)
= 25 - 10
= 15 marks
(iii) Marks obtained by Heena for 2 correct answers
= 2 ✖ 5
= 10 marks
marks for 5 incorrect answers
= 5 ✖ (-2)
= -10 marks
marks for not attempted questions
= 3 ✖ 0
= 0 marks
∴ Total score of heena
= marks for correct answers + marks for incorrect answers + marks for not attempted questions
= 10 + (-10) + 0
= 10 - 10 + 0
= 0 marks
8. A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bag of grey cement sold.
(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss ?
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number pf grey bags sold is 6,400 bags.
Solution :
Given :-
Profit of 1 bag of white cement = ₹8
Loss od 1 bag of grey cement = ₹5
(a) Profit on selling 3000 bags of white cement
= 3000 ✖ 8
= 24000
Loss of selling 5000 bags of grey cement
= 5000 ✖ 5
= 25,000
Since, Profit < Loss
∴ his total loss on selling the grey cement bags
= Loss - Profit
= 25,000 - 24,000
= 1,000
Thus, he has lost of ₹1000 on selling the grey cement bags .
(b) Let the number of bags of white cement be x
According to the questions,
Loss = Profit
∴ 5 ✖ 6400 = x ✖ 8
x = 5 ✖ 6,400/8
x = 5,000 bags
Thus, he must sell 4000 white cement bags to have neither profit or loss.
9. Replace the blank with an integer to make it true statement .
(a) (-3) ✖ _____ = 27
(b) 5 ✖ ______ = -35
(c) ____ ✖ (-8) = -56
(d) ____ ✖ (-12) = 132
Solution :
(a) (-9)
(b) (-7)
(c) 7
(d) (-11)