Mathematics - Real Life | Maths 7th STd

Exercise 1.1 Solutions

Question 1. Write down a pair of integers whose: (a) sum is –7 (b) difference is –10 (c) sum is 0

Answer (a) One possible pair of integers whose sum is -7 is -5 and -2. This is because (-5) + (-2) = -7.

(b) One possible pair of integers whose difference is -10 is 10 and 20. This is because 10 - 20 = -10. (c) One possible pair of integers whose sum is 0 is -5 and 5. This is because (-5) + 5 = 0.

Question 2. (a) Write a pair of negative integers whose difference gives 8.

(b) Write a negative integer and a positive integer whose sum is –5.

(c) Write a negative integer and a positive integer whose difference is –3.

Answer (a) One possible pair of negative integers whose difference is 8 is -4 and -12. This is because (-4) - (-12) = 8.

(b) One possible pair of negative integer and positive integer whose sum is -5 is -7 and 2. This is because (-7) + 2 = -5.

(c) One possible pair of negative integer and positive integer whose difference is -3 is -1 and 2. This is because (-1) - 2 = -3.

Question 3. 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?

Answer Team A's total score is (-40) + 10 + 0 = -30. Team B's total score is 10 + 0 + (-40) = -30. Therefore, both teams scored the same.

Yes, integers can be added in any order. This is because addition is commutative for integers. For any two integers 'a' and 'b', a + b = b + a.

Question 4. Fill in the blanks to make the following statements true:

(i) (–5) + (– 8) = (– 8) + (............)

(ii) –53 + ............ = –53

(iii) 17 + ............ = 0

(iv) [13 + (– 12)] + (............) = 13 + [(–12) + (–7)]

(v) (– 4) + [15 + (–3)] = [– 4 + 15] + ............

Answer (i) (-5) + (-8) = (-8) + (-5)

(ii) -53 + 0 = -53

(iii) 17 + (-17) = 0

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

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

Exercise 1.2 Solutions

Question 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) (–18) × (–5) × (– 4) (i) (–1) × (–2) × (–3) × 4 (j) (–3) × (–6) × (–2) × (–1)

Answer (a) 3 x (-1) = -3 The product of a positive integer and a negative integer is always a negative integer. (b) (-1) x 225 = -225 The product of a negative integer and a positive integer is always a negative integer. (c) (-21) x (-30) = 630 The product of two negative integers is always a positive integer. (d) (-316) x (-1) = 316 The product of two negative integers is always a positive integer. (e) (-15) x 0 x (-18) = 0 Any integer multiplied by zero gives a product of zero. (f) (-12) x (-11) x (10) = 1320 We can solve this by multiplying in steps: (-12) x (-11) = 132, and then 132 x 10 = 1320. (g) 9 x (-3) x (-6) = 162 We can solve this by multiplying in steps: 9 x (-3) = -27, and then (-27) x (-6) = 162 (-18) x (-5) x (-4) = -360 We can solve this by multiplying in steps: (-18) x (-5) = 90, and then 90 x (-4) = -360 (i) (-1) x (-2) x (-3) x 4 = -24 We can solve this by multiplying in steps: (-1) x (-2) = 2, then 2 x (-3) = -6, and finally -6 x 4 = -24 (j) (-3) x (-6) x (-2) x (-1) = 36 We can solve this by multiplying in steps: (-3) x (-6) = 18, then 18 x (-2) = -36, and finally -36 x (-1) = 36

Question 2. Verify the following: (a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)] (b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]

Answer (a) We can verify this by solving each side of the equation separately:

• Left side: 18 x [7 + (-3)] = 18 x 4 = 72.
• Right side: [18 x 7] + [18 x (-3)] = 126 + (-54) = 72. Since the left side equals the right side, the equation is verified.

(b) We can verify this equation in the same way, by solving both sides:

• Left side: (-21) x [(-4) + (-6)] = (-21) x (-10) = 210.
• Right side: [(-21) x (-4)] + [(-21) x (-6)] = 84 + 126 = 210 Since the left side equals the right side, the equation is verified.

These examples demonstrate the distributive property of multiplication over addition for integers. This property states that for any integers a, b, and c: a x (b + c) = (a x b) + (a x c).

Question 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) x a = -a. This means that multiplying any integer by -1 will result in the same integer but with the opposite sign.

(ii) (a) The integer whose product with (-1) is -22 is 22. This is because (-1) x 22 = -22. (b) The integer whose product with (-1) is 37 is -37. This is because (-1) x (-37) = 37. (c) The integer whose product with (-1) is 0 is 0. This is because (-1) x 0 = 0.

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

Answer Here's how we can show this pattern:

(-1) x 5 = -5 (-1) x 4 = -4 (-1) x 3 = -3 (-1) x 2 = -2 (-1) x 1 = -1 (-1) x 0 = 0 (-1) x (-1) = 1

Notice that as we decrease the integer multiplied by -1, the product increases by 1. Following this pattern, when we get to (-1) x (-1), the product is 1.

Exercise 1.3 Solutions

Question 1. Evaluate each of the following: (a) (–30) ÷ 10 (b) 50 ÷ (–5) (c) (–36) ÷ (–9) (d) (– 49) ÷ (49) (e) 13 ÷ [(–2) + 1] (f ) 0 ÷ (–12) (g) (–31) ÷ [(–30) + (–1)] [(–36) ÷ 12] ÷ 3 (i) [(– 6) + 5)] ÷ [(–2) + 1]

Answer (a) (-30) ÷ 10 = -3 When dividing a negative integer by a positive integer, the quotient (answer) is negative. (b) 50 ÷ (-5) = -10 When dividing a positive integer by a negative integer, the quotient is negative. (c) (-36) ÷ (-9) = 4 When dividing a negative integer by a negative integer, the quotient is positive. (d) (-49) ÷ 49 = -1 When dividing a negative integer by its positive counterpart, the quotient is -1. (e) 13 ÷ [(-2) + 1] = 13 ÷ (-1) = -13 First, we simplify the expression inside the brackets. Then, we divide 13 by -1, which results in -13. (f) 0 ÷ (-12) = 0 Zero divided by any non-zero integer is always zero. (g) (-31) ÷ [(-30) + (-1)] = (-31) ÷ (-31) = 1 First, simplify the expression in the brackets. Then, we divide -31 by -31, which gives us 1. [(-36) ÷ 12] ÷ 3 = (-3) ÷ 3 = -1 We solve this step-by-step, starting with the operation inside the brackets. (-36) ÷ 12 = -3, and then -3 ÷ 3 = -1. (i) [(-6) + 5] ÷ [(-2) + 1] = (-1) ÷ (-1) = 1 First, we simplify inside the brackets: (-6) + 5 = -1 and (-2) + 1 = -1. Then, we divide -1 by -1, resulting in 1.

Question 2. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c. (a) a = 12, b = – 4, c = 2 (b) a = (–10), b = 1, c = 1

Answer (a) Let's substitute the values of a, b, and c into both sides of the equation and see if they are equal.

• Left side: a ÷ (b + c) = 12 ÷ (-4 + 2) = 12 ÷ (-2) = -6
• Right side: (a ÷ b) + (a ÷ c) = (12 ÷ -4) + (12 ÷ 2) = -3 + 6 = 3 Since -6 ≠ 3, the equation is not true for these values of a, b, and c.

(b) Now let's substitute the values a = -10, b = 1, and c = 1 into the equation.

• Left side: a ÷ (b + c) = -10 ÷ (1 + 1) = -10 ÷ 2 = -5
• Right side: (a ÷ b) + (a ÷ c) = (-10 ÷ 1) + (-10 ÷ 1) = -10 + (-10) = -20 Since -5 ≠ -20, the equation is not true for these values of a, b, and c either.

This exercise demonstrates that, in general, division is not distributive over addition or subtraction.

Question 3. Fill in the blanks: (a) 369 ÷ _____ = 369 (b) (–75) ÷ _____ = –1 (c) (–206) ÷ _____ = 1 (d) – 87 ÷ _____ = 87 (e) _____ ÷ 1 = – 87 (f) _____ ÷ 48 = –1 (g) 20 ÷ _____ = –2 _____ ÷ (4) = –3

Answer (a) 369 ÷ 1 = 369 Any number divided by 1 is equal to itself. (b) (-75) ÷ 75 = -1 Any integer divided by its positive counterpart equals -1. (c) (-206) ÷ -206 = 1 Any integer divided by itself equals 1. (d) -87 ÷ (-1) = 87 Dividing a negative integer by -1 results in the positive counterpart of that integer. (e) -87 ÷ 1 = -87 Any number divided by 1 equals itself. (f) -48 ÷ 48 = -1 Any integer divided by its positive counterpart equals -1. (g) 20 ÷ (-10) = -2 A positive integer divided by a negative integer results in a negative quotient. To get -2, we divide 20 by -10. -12 ÷ 4 = -3 To get -3, we need to divide a negative integer by a positive integer. We divide -12 by 4.

Question 4. Write five pairs of integers (a, b) such that a ÷ b = –3. One such pair is (6, –2) because 6 ÷ (–2) = (–3).

Answer Here are five pairs of integers (a, b) such that a ÷ b = -3:

• (6, -2)
• (9, -3)
• (12, -4)
• (-3, 1)
• (-15, 5)

Question 5. 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?

Answer

• Step 1: Determine the total temperature change. The temperature needs to decrease from 10°C above zero to 8°C below zero, a total change of 10 + 8 = 18°C.

• Step 2: Calculate the number of hours. The temperature decreases 2°C per hour. To decrease 18°C, it will take 18°C / 2°C/hour = 9 hours.

• Step 3: Determine the time. Since the temperature starts decreasing at 12 noon and it takes 9 hours to reach 8°C below zero, it will be 9:00 PM.

• Step 4: Calculate the temperature at midnight. From 9:00 PM to midnight, there are 3 more hours. The temperature will decrease another 3 hours * 2°C/hour = 6°C. Since the temperature is already 8°C below zero at 9:00 PM, at midnight, it will be -8°C - 6°C = -14°C.

Therefore, the temperature will be 8°C below zero at 9:00 PM, and the temperature at midnight will be -14°C.

Question 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?

Answer (i) Let's figure out how many questions Radhika answered incorrectly:

• Calculate marks from correct answers: Radhika got 12 correct answers * 3 marks/correct answer = 36 marks.
• Calculate marks from incorrect answers: Her total score is 20, and she earned 36 marks from correct answers. Therefore, she lost 20 marks - 36 marks = -16 marks from incorrect answers.
• Calculate the number of incorrect answers: She loses 2 marks per incorrect answer, and she lost 16 marks total. So, she answered -16 marks / -2 marks/incorrect answer = 8 questions incorrectly.

Therefore, Radhika attempted 8 questions incorrectly.

(ii) Now let's figure out how many questions Mohini answered incorrectly:

• Calculate marks from correct answers: Mohini answered 7 questions correctly * 3 marks/correct answer = 21 marks.
• Calculate marks from incorrect answers: Her total score is -5, and she earned 21 marks from correct answers. Therefore, she lost -5 marks - 21 marks = -26 marks from incorrect answers.
• Calculate the number of incorrect answers: She loses 2 marks per incorrect answer, and she lost 26 marks. Therefore, she answered -26 marks / -2 marks/incorrect answer = 13 questions incorrectly.

Therefore, Mohini attempted 13 questions incorrectly.

Question 7. 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?

Answer

• Step 1: Calculate the total distance. The elevator needs to travel from 10m above ground to 350m below ground, a total of 10 m + 350 m = 360 m.

• Step 2: Calculate the descent time. The elevator descends at a rate of 6 m/minute. To descend 360 m, it will take 360 m / 6 m/minute = 60 minutes.

Therefore, it will take the elevator 60 minutes (or 1 hour) to reach -350 m.