- Find each of the following products:
(a) 3 × (–1) = -(3 x 1) = -3
(b) (–1) × 225 = -(1 x 225) = -225
(c) (–21) × (–30) = 21 x 30 = 630
(d) (–316) × (–1) = 316 x 1 = 316
(e) (–15) × 0 × (–18) = -( 15 x 0) x (-18) = 0 x (-18) = 0
(f) (–12) × (–11) × (10) = (-12) x 10 x (-11) = -(12 x 10) x (-11) = -(120) x -11 = 120 x 11 = 1320
(g) 9 × (–3) × (– 6) = 9 x (3 x 6) = 9 x 18 = 162
(h) (–18) × (–5) × (– 4) = (-18) x (5 x 4) = -18 x 20 = -(18 x 20) = -360
(i) (–1) × (–2) × (–3) × 4 = (1 x 2) x -(3 x 4) = 2 x -12 = -(12 x 2) = -24
(j) (–3) × (–6) × (–2) × (–1) = (3 x 6) x (2 x 1) = 18 x 2 = 36
2. Verify the following:
(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]
L.H.S = 18 × [7 + (–3)] = 18 × [4] = 72
R.H.S = [18 × 7] + [18 × (–3)] = 126 + [-54] = 126 – 54 = 72
Therefore, L.H.S = R.H.S
(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]
L.H.S = (–21) × [(– 4) + (– 6)] = (-21) × [-4-6] = (-21) × [-10] = 210
R.H.S = [(–21) × (– 4)] + [(–21) × (– 6)] = [21 × 4] + [21 × 6] = 84 + 126 = 210
Therefore, L.H.S = R.H.S
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
Answer (i) For any integer a, (-1) × a = -a. In other words, we can say that (-1) × a is equal to additive inverse of a.
Answer (ii)
a. Integer whose product with (-1) is equal to -22 is 22. -1 × (22) = -(1 × 22) = -22
b. Integer whose product with (-1) is equal to 37 is -37. -1 × (-37) = (1 × 37) = 37
(c) Integer whose product with (-1) is equal to 0 is 0. -1 × 0 = 0
4. Starting from (–1) × 5, write various products showing some pattern to show
(–1) × (–1) = 1.
(-1) x 4 = -4 = -5+1
(-1)x3 = -3 = -4+1
(-1)x2= -2 = -3+1
(-1)x1=-1=-2+1
(-1)x0=0=-1+1
According to pattern, we will have, (-1)x(-1)= 0+1 = 1
5. Find the product, using suitable properties:
(a) 26 × (– 48) + (– 48) × (–36)
Using distributive property, we have a × b + a × c = a × (b+c)
Therefore, 26 × (– 48) + (– 48) × (–36) = -48(26-36) = (-48) × (-10) = 48 × 10 = 480
(b) 8 × 53 × (–125) = 53 × (8 × (-125)) = 53 × (-1000) = -53000
(c) 15 × (–25) × (– 4) × (–10) = 15 × (-10) × (-25) × (-4) = -150 × 100 = -15000
(d) (– 41) × 102 = -41 × (100+2) = -41 × 100 + (-41) × 2 = -4100-82 = -4182
(e) 625 × (–35) + (– 625) × 65 = 625 × (-35) + (625) × (-65) = 625 × (-35-65) = 625 × -100 = -62500
(f) 7 × (50 – 2) = (7 × 50) – (7 × 2) = 350 – 14 = 336
(g) (–17) × (–29) = (-17) × (-30+1) = (-17) × (-30) + (-17) × 1 = 510 -17 = 493
(h) (–57) × (–19) + 57 = 57 × 19 + 57 × 1 = 57 × (19 + 1) = 57 × 20 = 1140
6. A certain freezing process requires that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?
Answer: Current room temperature = 40°C
At the rate of 5°C every hour, total temperature lowered in 10 hours = 10 × 5 = 50°C
Room temperature after 10 hours = 40 – 50 = -10°C
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?
Mohan’s score = (5 × 4) + (-2 ×6) = 20 -12 = 8
(ii) Reshma gets five correct answers and five incorrect answers, what is her score?
Reshma’s score = (5 × 5) + (-2 × 5) = 25 -10 = 15
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?
Heena’s score = (2 × 5) + (-2 × 5) = 10 -10 = 0
8. A cement company earns a profit of Rs. 8 per bag of white cement sold and a loss of
Rs. 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?
Profit or loss = (3000 × 8) + (5000 × (-5)) = 24000 – 25000 = -1000. Therefore, it is a loss of Rs. 1000
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags.
Loss from 6400 grey bags sold = 6400 × 5 = Rs 32000
Total number of white cement bags needed to be sold to break even
9. Replace the blank with an integer to make it a true statement.
(a) (–3) × -9 = 27 (b) 5 × -7 = –35
(c) 7 × (– 8) = –56 (d) -11× (–12) = 132