1. Integers

Solutions to Ex 1.1:

Q1. Write down a pair of integers whose:

Ans:

(a) Pairs whose sum is -7 are:

-5 + (-2) = -7 or

-1 + (-6) = -7 or

-4 + (-3) = -7

(b) Pairs whose difference is -10 are:

-2 - 8 = -10 or

-9 - 1 = -10 or

-5 - 5 = -10 or

-6 - 4 = -10 

(c) Pairs whose sum is 0:  

-3 + 3 = 0 or

4 + (-4) = 0

Q2. Ans:

(a) Pairs of negative integers whose difference gives 8 are:

-2 - (-10) = 8 or

-4 - (-12) = 8

(b) A negative integer and a positive integer whose sum is -5:

-7 + 2 = -5 or

-10 + 5 = -5 or

-8 + 3 = -5

(c) a negative integer and a positive integer whose difference is -3:

-2 - 1 = -3

Q3. In a quiz, team A scored -40, 10, 0 and team B scored 10, 0, -40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?

Ans:

Team A scored = -40 + 10 + 0

      = -30

Team B scored = 10 + 0 + -40

      = -30

Thus, both the teams scored equal.

Yes, we can add integers in any order as addition is commutative for integers. 

Q4. Fill in the blanks to make the following statements true:

Ans:

(i) (-5) + (-8) = (-8) + (-5) (commutative property)

(ii) 53 + 0 = -53 (additive identity)

(ii) 17 + (-17) = 0 (additive inverse)

(iv) [13 + (-12)] + (-7) = 13 + [(-12) + (-7)] (associative property)

(v) (-4) + [15 + (-3)] = [-4 + 15] + (-3) (associative property)

Solutions to Ex 1.2 :

Q1. Find each of the following products:

Ans:

(a) 3 x (-1) = -3

(b) (-1) x 225 = -225

(c) (-21) x (-30) = 630

(d) (-316) x (-1) = 316

(e) (-15) x 0 x (-18) = 0

(f) (-12) x (-11) x 10 = 1320

(g) 9 x (-3) x (-6) = 162

(h) (-18) x (-5) x (-4) = -360

(i) (-1) x (-2) x (-3) x 4 = -24

(j) (-3) x (-6) x (-2) x (-1) = 36

Q2. Verify the following:

Ans:

(a) LHS = 18 x [ 7 + (-3)]

   = 18 x 4 = 72

    RHS = [18 x 7] + [18 x (-3)]

  = 126 - 54 = 72

Thus, LHS = RHS is verified

(b) LHS = (-21) x [(-4) + (-6)]

   = -21 x -10 = 210

    RHS = [(-21) x (-4)] + [(-21) x (-6)]

  = 84 + 126 = 210

Thus, LHS = RHS is verified

Q3. Ans:

(i) For any integer a, (-1) x a = -a

(ii) (a) 22 x (-1) = -22; the integer is 22.

     (b) -37 x (-1) = 37; the integer is -37.

     (c) 0 x (-1) = 0; the integer is 0.

Q4. Products showing the pattern are:

Ans: (-1) x 5 = -5

(-1) x 4 = -4

(-1) x 3= -3

(-1) x 2 = -2

(-1) x 1 = -1

(-1) x (-1) = 1

Solutions to Ex 1.3:

Q1. Evaluate each of the following:

Ans:

(a) (−30) ÷10= -3 

(b) 50 ÷ (−5) = -10

(c) (−36) ÷ (−9) = 4

(d) (−49) ÷ 49= -1

(e) 13 ÷ [(−2) + 1]= 13 ÷ (-1) = -13

(f) 0 ÷ (−12) = 0

(g) (−31) ÷ [(−30) + (−1)] = (−31) ÷ (−31) = 1

(h) [(−36) ÷ 12] ÷ 3 = (-3) ÷ 3 = -1

(i) [(−6) + 5] ÷  [(−2)+1] = (-1) ÷ (-1) = 1

Q2. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c.

Ans:

(a) a = 12, b = – 4, c = 2

Subsituting the given values:

LHS = 12 ÷ (-4 + 2)

       = 12 ÷ (-2)

LHS = -6

RHS = [12 ÷ (-4)] + (12 ÷ 2)3

        = (-3) + 6

RHS = 3

Hence, LHS ≠ RHS is verified.

(b) a = (–10), b = 1, c = 1

LHS = (-10) ÷ (1 + 1)

       = (-10) ÷ 2

LHS = -5

RHS = [(-10) ÷ 1} + [(-10) ÷ 1]

       = (-10) + (-10)

RHS = -20

Hence, LHS ≠ RHS is verified.

Q3. Fill in the blanks:

Ans:

(a) 369 ÷ 1 = 369 

(b) (–75) ÷ 75 = –1

(c) (–206) ÷ (-206) = 1 

(d) – 87 ÷ (-1) = 87

(e) (-87) ÷ 1 = – 87

(f) (-48) ÷ 48 = –1

(g) 20 ÷ (-10) = –2 

(h) (-12) ÷ (4) = –3

Q4. Write five pairs of integers (a, b) such that a ÷ b = –3. One such pair is (6, –2) because 6 ÷ (–2) = (–3)

Ans:

1. $-3$
2. 9 ÷ (−3) = −3
3. (−9) ÷ 3 = −3
4. (-12) ÷ 4 = -3
5. 15 ÷ (-5) = -3

Q5. The temperature at 12 noon was 10°C above zero. If it decreases at the rate of 2°C per hour until midnight, at what time would the temperature be 8°C below zero? What would be the temperature at mid-night?

Ans: 

Given: The temperature at 12 pm = 10^oC

   Temperature decreases 2^oC per hour,

1. For 8^oC below zero, total temperature decrease = 10 - (-8) = 18^oC

   Number of hours taken for temperature decrease = 18 ÷ 2 = 9 hours

   Time at which temperature will be 8^oC below 0 = 12 + 9 = 9pm

2. Number of hours till 12 midnight = 12

Till 12am, temperature will decrease by = 12 x 2 = 24^oC

Hence, temperature at 12 midnight = 10 - 24 = -14^oC

Q6. 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?

Ans:

Given: Marks for correct answer = 3

   Marks for incorrect answer = (-2) 

(i) Radhika's score = 20

    Number of answers correct = 12

    Score for correct answers = 12 x 3 = 36

    Marks deducted for incorrect answer = 36 - 20 = 16

    Number of questions attempted incorrectly = 16 ÷ 2 = 8 questions

(ii) Mohini's score = (-5)

     Number of answers correct = 7

     Score for correct answers = 7 x 3 = 21

     Marks deducted for incorrect answer = 21 - (-5) = 26

     Number of questions attempted incorrectly = 26 ÷ 2 = 13 questions

Thus, Radhika attempted 8 questions & Mohini attempted 13 questions incorrectly.

Q7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach – 350 m.

Ans:

Given: Elevator descends from = 10m

            Rate of descend = 6m/min

            Distance travelled = -350m

Total distance travelled = 10 - (-350) = 360m

Time taken to travel 360m = 360 ÷ 6 = 60min

Thus, the time taken to reach -350m is 60 minutes or 1 hour.