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6. In a class test (+ 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. (i) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly? (ii) Mohini scores –5 marks in this test, though she has got 7 correct answers. How many questions has she attempted incorrectly?

Given, Marks for every correct answer = Marks for every wrong answer = (i) According to question, Marks obtained by Radhika = Number of correct answers =  Marks obtained for correct answers =  Marks obtained for incorrect answers = Total marks – Marks obtained for correct answers  Number of incorrect answers =  Therefore, Radhika attempted 8 questions incorrectly. (ii)  According to...

Q. Is   (i) 1 ÷ a = 1?
     (ii) a ÷ (–1) = – a? for any integer a.
Take different values of a and check.

(i) is true only for . (ii) Taking,  Hence, this is true for every integer .

Find:    (a) (–36) ÷ (– 4)        (b) (–201) ÷ (–3)        (c) (–325) ÷ (–13)

When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+). (a)           (b)          (c)

Find:     (a) 125 ÷ (–25)        (b) 80 ÷ (–5)        (c) 64 ÷ (–16)

When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient. (a)         (b)          (c)

Find:
                (a) (–100) ÷ 5            (b) (–81) ÷ 9

                (c) (–75) ÷ 5               (d) (–32) ÷ 2

When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient (a)             (b)               (c)              (d)

9. Replace the blank with an integer to make it a true statement.
                    (a) (–3) × _____ = 27            (b) 5 × _____ = –35
                    (c) _____ × (– 8) = –56         (d) _____ × (–12) = 132

(a)Given,   The product is positive, therefore there should be even number of negative integers.              (b) Given, The product is negative, therefore there should be odd number of negative integers. (c)          The product is negative, therefore there should be odd number of negative integers. (d) The product is positive, therefore there should be even number of negative integers.

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.

(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.

Given, Profit earned by selling 1 bag of white cement =  Loss on 1 bag of grey cement =  Loss on 6400 bags of grey cement =  To be neither in profit nor loss, the profit from white cement must be  Number of white cement bags to be sold =  Therefore the company has to sell 4000 white cement bags.

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?

Given, Profit earned by selling 1 bag of white cement =  Profit earned by selling 3000 bags of white cement=  Loss on 1 bag of grey cement = Loss of 5000 bags of grey cement =  Total earnings =  This means that the company incurred a loss of  

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.

 (iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?

Given, Total questions =  Marks for correct answer =  Marks for incorrect answers =  Marks for unattempted question =  According to question, Heena gets two correct and five incorrect answers. Therefore, number of unattempted questions =  Total marks =  Therefore, Heena scored 0 marks.

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.

 (ii) Reshma gets five correct answers and five incorrect answers, what is her score?

Given, Total questions =  Marks for correct answer =  Marks for incorrect answers =  Marks for unattempted question =  According to question, Reshma gets five correct answers and five incorrect answers Total marks =  Therefore, Reshma scored 15 marks.

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?

Given, Total questions =  Marks for correct answer =  Marks for incorrect answers =  Marks for unattempted question =  According to question, Mohan gets four correct and six incorrect answers Total marks =  Therefore, Mohan scored 8 marks.

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?

Given, Initial temperature = Also, Change in temperature per hour = Therefore,Change in temperature after 10 hours = Hence, Final temperature =

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

Using suitable properties of multiplication: (a)              (b)                  (c)                 (d)                  (e) (f) (g)                                      (h)

4. Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1.

Given, Therefore,

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

(i) For any integer a,  , i.e, the additive inverse of the given integer. (ii) The integer whose product with  gives the following are:

3. Verify the following:

 (b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]

L.H.S =  R.H.S  Therefore, L.H.S = R.H.S Hence verified.

2. Verify the following:

 (a)    18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]

L.H.S =  R.H.S =  L.H.S = R.H.S Hence, verified.

1.  Find each of the following products:

                (f) (–12) × (–11) × (10)        (g) 9 × (–3) × (– 6)

                (h) (–18) × (–5) × (– 4)         (i) (–1) × (–2) × (–3) × 4

                (j) (–3) × (–6) × (–2) × (–1)

Product of even number of negative integers is an even integer. (f)          (g)               (h)          (i)                 (j)

1. Find each of the following products:

            (a) 3 × (–1)                (b) (–1) × 225
            (c) (–21) × (–30)        (d) (–316) × (–1)
            (e) (–15) × 0 × (–18)

(a)                (b)              (c) Product of two negative integers is a positive integer         (d)              (e)
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