Operations with Integers Class 7 Solutions

(a) Sum = 27, Difference = 9 Hence correct pair is (18, 9). (b) Sum = 4, Difference = 12 Hence correct pair is (8, -4). (c) Sum = 0, Difference = 10 Hence correct pair is (5, -5). (d) Sum = 0, Difference = -10 Hence correct pair is (8, -8). (e) Sum = -7, Difference = -1 Hence, the correct pair is (-4, -3).

(f) Sum = -7, Difference = -13 Hence correct pair is (-10, 3).

(a) 3 × (-2) Two red tokens 3 times = -6 There are now 6 red tokens or 6 negatives in the box, i.e., -6 (b) (-5) × (-2) For (-5) × (-2), we need to remove 2 negatives from the box 5 times. Since there are no red tokens in the bag, we need to place 2 zero pairs and remove 2 negatives, and we need to do this 5 times.

Taking 2 red tokens from the zero pairs 5 times. Therefore, (-5) × (-2) = 10 (c) (-4) × (-1) For (-4) × (-1), we need to remove 1 negative from the box 4 times. Since there are no red tokens in the bag. We need to place 1 zero pair and remove 1 negative, and we need to do this 4 times. Taking 1 red token from the zero pairs 4 times.

Therefore, (-4) × (-1) = 4 (d) (-7) × 3 For (-7) × 3, we need to remove 3 positive numbers from the box 7 times. Taking 3 positives from the zero pairs 7 times = (-7) × 3 = -21.

(a) Given 123 × 456 = 56088 …..(i) Then(a) (-123) × 456 This is the product of a negative and a positive integer, so the result will be negative. i.e. -(123) × 456 = -(123 × 456) = -56088 (b) (-123) × (-456) This is the product of two negative integers, so the result will be positive. Therefore, (-123) × (-456) = 123 × 456 = 56088 (c) 123 × (-456) This is the product of a positive and a negative integer, so the result will be negative.

Therefore, 123 × (-456) = -(123 × 456) = -56088

Rule for multiplying two integers: Multiply their absolute values. If the integers have different signs, the product is negative. If both integers have the same sign, the product is positive.

(a) Here, 4 × (-3) = – [(4) × (3)] = -12 (because Multiplier is positive and the multiplicand is negative, their product is negative) (b) Here, (-6) × (-3) = 6 × 3 = 18 (because Both the multiplier and multiplicand are negative; their product is positive) (c) Here, (-5) × (-1) = 5 × 1 = 5 (because Both the multiplier and multiplicand are negative; their product is positive) (d) Here, (-8) × 4 = -(8 × 4) = -32 (because Multiplier is negative and multiplicand is positive, their product is negative) (e) Here, (-9) × 10 = -(9 × 10) = -90 (because Multiplier is negative and multiplicand is positive, their product is negative) (f) Here, 10 × (-17) = -(10 × 17) = -170 (because Multiplier is positive and multiplicand is negative, their product is negative)

(a) To find the value of 14 × (-15), we multiply the numbers and apply the rule that a positive number multiplied by a negative number results in a negative number. Therefore, 14 × (-15) = -(14 × 15) = -210 (b) To find the value of -16 × (-5), we multiply the numbers and apply the rule that a negative number multiplied by a negative number results in a positive number.

Therefore, -16 × (-5) = 80 (c) To find the value of 36 ÷ (-18), we divide the numbers and apply the rule that a positive number divided by a negative number results in a negative number. Therefore, 36 ÷ (-18) = -(36 ÷ 18) = -2 (d) To find the value of (-46) ÷ (-23), we divide the numbers and apply the rule that a negative number divided by a negative number results in a positive number.

Therefore, (-46) ÷ (-23) = 2

The temperature is lowered at the rate of 5°C every hour for 10 hours. Total temperature drop = 5°C/hour × 10 hour = 50°C The initial room temperature is 32°C. To find the final temperature, we have to subtract the initial temperature from the total temperature drop. Therefore, Final temperature = 32 – 50 = -18°C

(a) Profit on one white cement bag = ₹ 8 Loss on one grey cement bag = ₹ 5 Profit from selling white cement = 3000 bags × ₹ 8/bag = ₹ 24000 Loss from selling white cement = 5000 bags × ₹ 5/bag = ₹ 25,000 [because loss is more than profit, i.e., loss] Total loss = 25,000 – 24,000 = 1,000 Therefore, The company has a loss of ₹ 1000.

(b) Let x be the number of white cement bags sold. Total profit/loss is zero. Profit from white cement + Loss from grey cement = 0 ⇒ x × 8 + 6400 × (-5) = 0 ⇒ 8x – 32000 = 0 ⇒ x = 4000 Therefore, The company must sell 4000 bags of white cement to make neither profit nor loss.

(a) Let -3 × x = 27 ⇒ x = \frac{27}{-3}=-\frac{(27)}{3} = -9 (b) 5 × x = -35 ⇒ x = \frac{-35}{5}=-\frac{(35)}{5} = -7 (c) x × -8 = -56 ⇒ x = \frac {-56}{-8} = 7 (d) x × (-12) = 132 ⇒ x = \frac{132}{-12}=-\frac{-(132)}{12} = -11 (e) x ÷ (-8) = 7 ⇒ x × \frac {1}{-8} = 7 ⇒ x = 7 × (-8) = -56 (f) x ÷ 12 = -11 ⇒ x × \frac {1}{12} = -11 ⇒ x = -11 × 12 = -132

(a) Given (-5) × [(18 + (-3)] = -5 × 18 + (-5) × (-3) [because a(b + c) = ab + ac] = -90 + 15 = -75 (b) Here (-7) × 4 × (-1) = [(-7) × 4] × (-1) = (-28) × (-1) = 28 (c) Here [(-2) × (-1)] × [(-5) × (-3)] = 2 × (-5) × (-3) = 2 × 15 = 30

(a) (-27) ÷ 9 or 9 × ________ = -27 We know that, 9 × (-3) = -27 So, (-27) ÷ 9 = -3 (b) 84 ÷ (-4) or (-4) × ________ = 84 We know that, -4 × (-21) = 84 So, 84 ÷ (-4) = -21. (c) (-56) ÷ (-2) or (-2) × ________ = (-56) We know that, (-2) × 28 = -56 So, (-56) ÷ (-2) = 28

(a) Here, (-1) × (-27) = 27 Therefore, -27 is the required integer. (b) Here, (-1) × 31 = -31 Therefore, 31 is the required integer. (c) Here, (-1) × 1 = -1 Therefore, 1 is the required integer. (d) Here, (-1) × (-1) = 1 Therefore, (-1) is the required integer. (e) Here, (-1) × 0 = 1 Therefore, 0 is the required integer.

Given expression is 47 – 56 + 14 – 8 + 2 – 8 + 5 = -4 = E1 (Let) Here we see that the second expression is the negative of the first expression, then E2 = -47 + 56 – 14 + 8 – 2 + 8 – 5 = -1(47 – 56 + 14 – 8 + 2 – 8 + 5) (Taking -1 common) = -1(68 – 72) = -1(-4) = 4

(a) Following the rules for the sequence starting with -7. The rule is if even, take half, if odd, multiply by -3, and add 1. Start with -7 (odd) [(-7) × (-3)] + 1 = 21 + 1 = 22 (even) then 22 ÷ 2 = 11 (odd) then 11 × (-3) + 1 = -33 + 1 = -32 (even) then (-32) ÷ 2 = -16 (even) then (-16) ÷ 2 = -8 (even) then (-8) ÷ 2 = -4 (even) then (-4) ÷ 2 = -2 (even) then (-2) ÷ 2 = – 1 (odd) then [(-1) × (-3)] + 1 = 4 (even) then 4 ÷ 2 = 2 (even) then 2 ÷ 2 = 1 (odd) (b) (i) Now for the starting number -21 (odd) then [(-21) × (-3)] + 1 = 63 + 1 = 64 (even) then 64 ÷ 2 = 32 (even) then 32 ÷ 2 = 16 (even) then 16 + 2 = 8 (even) then 8 ÷ 2 = 4 (even) then 4 ÷ 2 = 2 (even) then 2 ÷ 2 = 1 (odd) then 1 × (-3) + 1 = -2 (even) then (-2) ÷ 2 = -1 (odd) then [-1 × (-3)] + 1 = 4 (even) then 4 ÷ 2 = 2 (even) then 2 ÷ 2 = 1 (odd) then 1 × (-3) + 1 = -2 (even) Hence the sequence for -21 is (ii) For the sequence, the starting number is -6 6 is even then -6 + 2 = -3 (odd) then [(-3) × (-3)] + 1 = 9 + 1 = 10 (even) then 10 ÷ 2 = 5 (odd) 5 × (-3) + 1 = -15 + 1 = -14 (even) then -14 ÷ 2 = -7 (odd) [(-7) × (-3)] + 1 = +21 + 1 = 22 (even) then 22 ÷ 2 = 11 (odd) then [11 × (-3)] + 1 = -33 + 1 = -32 (even) then -32 ÷ 2 = -16 (even) then -16 ÷ 2 = -8 (even) then -8 ÷ 2 = -4 (even) then -4 ÷ 2 = -2 (even) then (-2) ÷ 2 = -1 (odd) then [(-1) × (-3)] + 1 = 3 + 1 = 4 (even) then 4 ÷ 2 = 2 (even) then 2 ÷ 2 = 1 (odd) then 1 × (-3) + 1 = -2 Hence, the sequence is Observation: For numbers like -21, -6, etc., the sequences eventually reach a repeating loop of -2, -1, 4, 2, 1, -2.

All starting numbers end up in this cycle.

(a) Anita had 15 correct answers, and each correct answer is worth +4 marks. Marks from correct answers = 15 × 4 = 60 Therefore, Anita total score = 40 Marks lost = 60 – 40 = 20 Each incorrect answer is worth -2 marks Therefore, No. of incorrect answers = \frac {20}{2} = 10 Anita had 10 incorrect answers.

Total no. of questions = sum of correct and incorrect answers = 15 + 10 = 25 Hence, Anita had 10 incorrect answers out of 25 questions on the test. (b) Anil had 5 correct answers, and each correct answer is worth + 4 marks they 5 × 4 = 20 marks. From incorrect answers = -10 – 20 = -30. Each incorrect answer is worth -2 marks Therefore, \frac {-30}{-2} = 15 Hence, Anil had 15 incorrect answers.

Yes, he leaves some questions unanswered. Anil answered the question = (Total No. of Questions) – (Anil’s correct answers + Anil’s incorrect answers) = 25 – (5 + 15) = 25 – 20 = 5 Hence, Anil left 5 questions unanswered.

Operation done by machine is (First Number) – [Second number × Third Number] I row: 4 – [(8) × -3] = 4 – [-24] = 4 + 24 = 28. II row: 6 – [9 × 6] = 6 – [54] = 6 – 54 = -48. III row: 2 – [3 × (-2)] = 2 – (-6) = 2 + 6 = 8. IV row: -9 – [5 × (-8)] = -9 – (-40) = -9 + 40 = 31. V row: 7 – (-4 × -6) = 7 – (24) = 7 – 24 = -17.

VI row: -16 – (-6 × -9) = -16 – (54) = -70. Thus, we have found the pattern that satisfies all the values. Hence, the missing operation is -70.

Current temperature = 8°C The temperature drops by 5°C each hour Total drop in temperature = Temperature drop in 1 hour × no. of hours = 5 × 4 Temperature after 4 hours = 8 – 4 × 5 Hence required expression is 8 – (5 × 4).

(a) Let three consecutive numbers be n – 1, n, and n + 1. Then product = (n – 1) (n) (n + 1) = -6 The only consecutive integers whose product is -6 are -2, -1, and 0. But -2 × -1 × o = 0, not possible. Consecutive numbers are integers. The product is negative, so there must be an odd number of negative integers.

Integers are -2, -1, 1. Their product = – 2 × -1 × 1 = 2, not possible Integers are -3, -2, -1. Their product = [-3 × -2] × -1 = 6 × -1 = -6 Hence, consecutive integers are -3, -2, -1. (b) Let three consecutive numbers be n – 1, n, and n + 1. Therefore, (n – 1) (n) (n + 1) = 120 Now the cube root of 120 = 4.93 So n is likely to be 5.

Therefore, (5 – 1) × (5) × (5 + 1) = 4 × 5 × 6 = 120 Hence, consecutive integers are 4, 5, and 6.

The currency has two denominations: +13 pibs and -9 pibs. We need to determine if it is possible to make the given totals. This is a linear equation of the form 13x – 9y = total, where x and y are non-negative integers. (a) Now total = +20 Then we need to find integers 13x – 9y = 20, x, y > 0 such that If x = 1, then 13 – 9y = 20 ⇒ -9y = 7.

No solution. If x = 2, then 26 – 9y = 20 ⇒ -9y = -6. No solution. IF x = 3 then 39 – 9y = 20 ⇒ -9y = -19. No solution. If x = 4, then 52 – 9y = 20 ⇒ -9y = -32. No solution. If x = 5 then 65 – 9y = 20 ⇒ -9y = -45 ⇒ y = 5 Hence, x = 5, y = 5 is a valid solution. (b) +40 Take 10 coins of +13 and 10 coins of -9: 10 × 13 – 10 × 9 = 130 – 90 = +40 pibs.

(c) -50 Take 10 coins of +13 and 20 coins of -9: 10 × 13 – 20 × 9 = 130 – 180 = – 50 pibs. (d) +8 Take 2 coins of +13 and 2 coins of -9: 2 × 13 – 2 × 9 = 26 – 18 = +8 pibs. (e) +10 Take 7 coins of +13 and 9 coins of -9: 7 × 13 – 9 × 9 = 91 – 81 = +10 pibs. (f) -2 Take 13 coins of +13 and 19 coins of -9: 13 × 13 – 19 × 9 = 169 – 171 = -2 pibs.

(g) +1 Take 7 coins of +13 and 10 coins of -9: 7 × 13 – 10 × 9 = 91 – 90 = +1 pibs. (h) Yes, it is possible. Take 122 coins of +13 and 2 coins of -9: 122 × 13 – 2 × 9 = 1586 – 18 = 1568 pibs.

(a) Here 32 × (-18) ÷ (-36) = -576 ÷ (-36) = 16 (b) Here 32 ÷ [(-36) × (-18)] = (32) ÷ 648 = \frac {4}{81} (c) Here [25 × (-12)] ÷ [45 × (-27)] = (-300) ÷ (-1215) = \frac {20}{81} (d) Here [280 × (-7)] ÷ [(-8) × (-35)] = (-1960) ÷ 280 = -7

(a) (-348) + (-1064) Add two negative numbers → keep negative, add absolute values: 348 + (-1064) = -1412. (b) (-348) – (-1064) Subtracting a negative → same as adding positive: 348 – (-1064) = -348 + 1064 = 716. (c) 348 – (-1064) Subtracting a negative → add positive. 348 – (-1064) = 348 + 1064 = 1412.

(d) (-348) × (-1064) Negative × Negative = Positive (-348) × (-1064) = 370272 (e) 348 × (-1064) Positive × Negative = Negative 348 × (-1064) = -370272 (f) 348 × 964 Positive × Positive = Positive 348 × 964 = 335472 Arranging in increasing order: (e) 370272, (a) -1412, (b) 716, (c) 1412, (f) 335472, (d) 370272 Hence, (e) < (a) < (b) < (c) < (f) < (d).

(a) Given: (-547) × 972 = -532656 We need to find (-547) × 972 = (-548 + 1) × 972 = -548 × 972 + 972 = -532656 + 972 = -531684 (b) Now -548 × 971 = -548 × (972 – 1) = -548 × 972 – 548 × (-1) = -532656 + 548 = -532108 (c) Here -547 × 971 = (-548 + 1) × (972 – 1) = -548 × 972 + 548 + 972 – 1 = -532656 + 548 + 972 – 1 = -531137

Here 207 × (-33 + 7) = 207 × (-26) = -5382 Now -207 × (33 – 7) = -207 × 26 = -(207 × 26) = -5382

(a) Maximum possible value = [(-2) – (3 + 5)] × (-6) = (-2 – 8) × (-6) = 60 (b) Minimum possible value = [3 – (-2) + 5)] × (-6) = (3 + 2 + 5) × -6 = -60

(a) Here (i) 0 + -6 × 6 = -36 (ii) 4 + (-5 × 8) = -36 (iii) -4 + -8 × 4 = -36 (iv) 12 + -8 × 6 = -36 (v) -3 + 3 × (-11) = -36 (b) Here (i) (13 – 1) × 1 = 12 (ii) (10 – 4) × 2 = 12 (iii) (1 – 3) × (-6) = 12 (iv) (14 – 10) × 3 = 12 (v) (16 – 13) × 4 = 12 (c) Here (i) (5 – (10 – 4)) = -1 (ii) (0 – (3 – 2)) = -1 (iii) (-1 – (1 – 1)) = -1 (iv) (-5 – (0 – 4)) = -1 (v) (-10 – (-5 – 4)) = -1