Chapter 2 exercise answers: Operations with Integers

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(a) Sum = 27, Difference = 9

1. Add: 27 + 9 = 36. Half of 36 is 18 — the first number.
2. Subtract: 27 − 9 = 18. Half of 18 is 9 — the second number.

AnswerFirst number 18, second number 9.

(b) Sum = 4, Difference = 12

1. Add: 4 + 12 = 16. Half of 16 is 8 — the first number.
2. Subtract: 4 − 12 = −8. Half of −8 is −4 — the second number.

AnswerFirst number 8, second number −4.

(c) Sum = 0, Difference = 10

1. Add: 0 + 10 = 10. Half of 10 is 5 — the first number.
2. Subtract: 0 − 10 = −10. Half of −10 is −5 — the second number.

AnswerFirst number 5, second number −5.

(d) Sum = 0, Difference = – 10

1. Add: 0 + (−10) = −10. Half of −10 is −5 — the first number.
2. Subtract: 0 − (−10) = 10. Half of 10 is 5 — the second number.

AnswerFirst number −5, second number 5.

(e) Sum = – 7, Difference = – 1

1. Add: −7 + (−1) = −8. Half of −8 is −4 — the first number.
2. Subtract: −7 − (−1) = −6. Half of −6 is −3 — the second number.

AnswerFirst number −4, second number −3.

(f) Sum = – 7, Difference = – 13

1. Add: −7 + (−13) = −20. Half of −20 is −10 — the first number.
2. Subtract: −7 − (−13) = 6. Half of 6 is 3 — the second number.

AnswerFirst number −10, second number 3.

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(a) 3 × (– 2)

1. The multiplier 3 is positive, so place 2 red tokens into the empty bag, 3 times.
2. That puts 6 red tokens in the bag, which is worth −6.

Answer−6

(b) (– 5) × (– 2)

1. The multiplier −5 is negative, so remove 2 red tokens from the bag, 5 times.
2. The bag is empty, so each time first put in 2 zero pairs (1 green + 1 red each), then remove the 2 red tokens. Each time, 2 green tokens are left behind.
3. After 5 times, 5 × 2 = 10 green tokens are left, which is worth 10.

Answer10

(c) (– 4) × (– 1)

1. The multiplier −4 is negative, so remove 1 red token from the bag, 4 times.
2. Each time first put in 1 zero pair (1 green + 1 red), then remove the red token, leaving 1 green token.
3. After 4 times, 4 green tokens are left, which is worth 4.

Answer4

(d) (– 7) × 3

1. The multiplier −7 is negative, so remove 3 green tokens from the bag, 7 times.
2. Each time first put in 3 zero pairs, then remove the 3 green tokens, leaving 3 red tokens.
3. After 7 times, 7 × 3 = 21 red tokens are left, which is worth −21.

Answer−21

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(a) (– 123) × 456

1. One factor turned negative (123 became −123), so exactly one sign flips.
2. The size stays 56088, so the product is −56088.

Answer−56088

(b) (– 123) × (– 456)

1. Both factors turned negative, so two sign flips cancel out.
2. The product keeps the same sign as 123 × 456, which is positive: 56088.

Answer56088

(c) (123) × (– 456)

1. One factor turned negative (456 became −456), so exactly one sign flips.
2. The product is −56088.

Answer−56088

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Open NCERT p. 31One way to think about it

1. Each of the three token sets is worth the same amount, −2: (a) two negative tokens; (b) two negatives sitting above two negatives and two positives (net 4 negatives, 2 positives = −2); (c) four negatives above two negatives and four positives (net 6 negatives, 4 positives = −2).
2. Placing any of these three sets into the bag 4 times just adds '−2, four times' — the total only depends on the value each set is worth, not on how that value is split into tokens. So all three give the same final answer, −8.
3. The same idea works for 5 × 4: represent 4 with different token sets — for example, 4 green tokens, or 6 green tokens together with 2 red tokens (still a net value of 4) — and place either set into the bag 5 times. Both give 20, because both sets are worth 4.
4. For −4 × 2 (remove 2 positive tokens, 4 times): removing a positive (green) token has the same effect as adding its opposite, a negative (red) token. So instead of removing 2 greens each time, we can just add 2 reds each time — done 4 times, this places 8 red tokens straight into the bag, with no removal at all, and gives the same answer, −8.

In shortA simple rule: the size of a product is the product of the sizes being multiplied, and the sign is positive when the two numbers have the same sign and negative when they differ — this holds no matter which token set is used to build a number, and any removal of tokens can always be rewritten as an addition of the opposite tokens.

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(a) 4 × (– 3)

1. Sizes: 4 × 3 = 12.
2. Signs differ (positive, negative), so the product is negative: −12.

Answer−12

(b) (– 6) × (– 3)

1. Sizes: 6 × 3 = 18.
2. Signs match (both negative), so the product is positive: 18.

Answer18

(c) (– 5) × (– 1)

1. Sizes: 5 × 1 = 5.
2. Signs match (both negative), so the product is positive: 5.

Answer5

(d) (– 8) × 4

1. Sizes: 8 × 4 = 32.
2. Signs differ, so the product is negative: −32.

Answer−32

(e) (– 9) × 10

1. Sizes: 9 × 10 = 90.
2. Signs differ, so the product is negative: −90.

Answer−90

(f) 10 × (– 17)

1. Sizes: 10 × 17 = 170.
2. Signs differ, so the product is negative: −170.

Answer−170

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(a) 14 × (– 15)

1. Sizes: 14 × 15 = 210.
2. Signs differ, so the product is negative: −210.

Answer−210

(b) – 16 × (– 5)

1. Sizes: 16 × 5 = 80.
2. Signs match (both negative), so the product is positive: 80.

Answer80

(c) 36 ÷ (– 18)

1. Sizes: 36 ÷ 18 = 2.
2. Signs differ, so the quotient is negative: −2.

Answer−2

(d) (– 46) ÷ (– 23)

1. Sizes: 46 ÷ 23 = 2.
2. Signs match (both negative), so the quotient is positive: 2.

Answer2

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1. Falling by 5°C every hour is a rate of −5°C per hour.
2. In 10 hours the temperature falls by 10 × 5 = 50°C.
3. Starting at 32°C: 32 − 50 = −18°C.

Answer−18°C

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(a) sells 3,000 bags of white cement and 5,000 bags of grey cement

1. Profit per bag of white cement is +₹8; loss per bag of grey cement is −₹5.
2. 3,000 bags of white cement: 3,000 × 8 = ₹24,000.
3. 5,000 bags of grey cement: 5,000 × (−5) = −₹25,000.
4. Total: 24,000 + (−25,000) = −₹1,000.

AnswerA loss of ₹1,000, written as the integer −1000.

(b) the number of bags of grey cement sold is 6,400 bags

1. Let x be the number of white cement bags sold.
2. Neither profit nor loss means the total is zero: 8x + 6,400 × (−5) = 0.
3. 8x = 32,000, so x = 4,000.

Answer4,000 bags of white cement.

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(a) (– 3) × _____ = 27

1. 27 ÷ (−3) = −9.

Answer−9

(b) 5 × _____ = (– 35)

1. (−35) ÷ 5 = −7.

Answer−7

(c) _____ × (– 8) = (– 56)

1. (−56) ÷ (−8) = 7.

Answer7

(d) _____ × (– 12) = 132

1. 132 ÷ (−12) = −11.

Answer−11

(e) _____ ÷ (– 8) = 7

1. 7 × (−8) = −56.

Answer−56

(f) _____ ÷ 12 = – 11

1. −11 × 12 = −132.

Answer−132

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(a) (– 5) × (18 + (– 3))

1. Inside the bracket: 18 + (−3) = 15.
2. (−5) × 15 = −75 (signs differ).

Answer−75

(b) (– 7) × 4 × (– 1)

1. Count the negatives: −7 and −1 are negative, that is 2 of them — an even count, so the product is positive.
2. Sizes: 7 × 4 × 1 = 28.

Answer28

(c) (– 2) × (– 1) × (– 5) × (– 3)

1. Count the negatives: all four factors are negative — an even count, so the product is positive.
2. Sizes: 2 × 1 × 5 × 3 = 30.

Answer30

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(a) (– 27) ÷ 9

1. 27 ÷ 9 = 3.
2. Signs differ, so the quotient is negative: −3.

Answer−3

(b) 84 ÷ (– 4)

1. 84 ÷ 4 = 21.
2. Signs differ, so the quotient is negative: −21.

Answer−21

(c) (– 56) ÷ (– 2)

1. 56 ÷ 2 = 28.
2. Signs match (both negative), so the quotient is positive: 28.

Answer28

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(a) 27

1. Multiplying by −1 flips the sign, so we need the opposite of 27.

Answer−27

(b) – 31

1. We need the opposite of −31.

Answer31

(c) – 1

1. We need the opposite of −1.

Answer1

(d) 1

1. We need the opposite of 1.

Answer−1

(e) 0

1. The opposite of 0 is 0 itself.

Answer0

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1. Every term of the second expression is the opposite of the matching term in the first expression.
2. So the second expression is the first expression with every sign flipped — the same as multiplying the whole thing by −1: −(47 − 56 + 14 − 8 + 2 − 8 + 5).
3. That is −(−4) = 4.

Answer4

Open NCERT p. 42One way to think about it

1. Follow the rule from −7: −7 is odd, so (−7) × (−3) + 1 = 22. 22 is even, so take half: 11. 11 is odd, so 11 × (−3) + 1 = −32. (The book's diagram prints 32 here, but the rule gives −32, and the next number, −16, is half of −32.)
2. Continue: −32 → −16 → −8 → −4 → −2 → −1. −1 is odd, so (−1) × (−3) + 1 = 4. Then 4 → 2 → 1, and 1 is odd, so 1 × (−3) + 1 = −2. From here −2 → −1 → 4 → 2 → 1 → −2 repeats for ever.
3. Start from −21: −21 → 64 → 32 → 16 → 8 → 4 → 2 → 1 → −2 → −1 → 4 → … It falls into the same loop.
4. Start from −6: −6 → −3 → 10 → 5 → −14 → −7 → 22 → 11 → −32 → −16 → −8 → −4 → −2 → −1 → 4 → 2 → 1 → −2 … The same loop again.
5. Pattern 1: an odd number is always followed by an even number, because an odd number times −3 is odd, and adding 1 makes it even.
6. Pattern 2: the sign changes only at an odd step — a negative odd number becomes positive and a positive odd number becomes negative. Taking half never changes the sign.
7. Not every start reaches this loop: 0 stays at 0 for ever, and 13 goes 13 → −38 → −19 → 58 → 29 → … and comes back to 13 after 31 steps without ever reaching 1.

In shortThere is more than one pattern to describe; here is one: −7, −21 and −6 all end in the same repeating loop −2 → −1 → 4 → 2 → 1 → −2, an odd number is always followed by an even number, and the sign flips at every odd step. Not every start does this: 0 stays at 0, and 13 falls into a different, longer loop.

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(a) She scored 40 marks even though 15 of her answers were correct

1. Anita answered every question, so incorrect + correct = total questions.
2. Score: 15 × 4 + incorrect × (−2) = 40, so 60 − 2×incorrect = 40.
3. 2×incorrect = 20, so incorrect = 10.
4. Total questions = 15 correct + 10 incorrect = 25.

Answer10 incorrect answers; 25 questions in the test.

(b) Anil scored (– 10) marks even though he had 5 correct answers

1. Score: 5 × 4 + incorrect × (−2) = −10, so 20 − 2×incorrect = −10.
2. 2×incorrect = 30, so incorrect = 15.
3. Anil answered 5 + 15 = 20 questions out of the 25 in the test, so he left 5 unanswered.

Answer15 incorrect answers; yes, he left 5 questions unanswered.

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1. Compare each row against the first number minus the product of the other two: 4 − (8×−3) = 4−(−24) = 28; 6 − (9×6) = 6−54 = −48; 2 − (3×−2) = 2−(−6) = 8; −9 − (5×−8) = −9−(−40) = 31; 7 − (−4×−6) = 7−24 = −17.
2. Every row fits, so the rule is: (first number) − (second number × third number).
3. For −16, −6, −9: −16 − ((−6)×(−9)) = −16 − 54 = −70.

AnswerThe machine computes a − b×c; for the row −16, −6, −9 the result is −70.

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1. Dropping by 5°C each hour is a rate of −5°C per hour.
2. After 4 hours the change is 4 × (−5) = −20°C.
3. Starting at 8°C: 8 + (−20) = −12°C.

Answer8 + 4×(−5) = −12°C

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(a) – 6

1. Try three consecutive negative integers close to zero: −3, −2, −1.
2. Product: (−3)×(−2)×(−1) = −6.

Answer−3, −2, −1

(b) 120

1. Try three consecutive positive integers: 4, 5, 6.
2. Product: 4×5×6 = 120.

Answer4, 5, 6

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Open NCERT p. 43Checked by computerAnswers can differ: one example

(a) +20

1. Multiples of 13: 13, 26, 39, 52, 65, … Multiples of 9: 9, 18, 27, 36, 45, …
2. 65 − 45 = 20: five +13 coins make 65 and five −9 coins make −45.

Answer5 coins of +13 pibs and 5 coins of −9 pibs: 5 × 13 − 5 × 9 = 20. This is one way; other combinations also work.

(b) +40

1. Double part (a): ten +13 coins make 130 and ten −9 coins make −90.
2. 130 − 90 = 40.

Answer10 coins of +13 pibs and 10 coins of −9 pibs: 10 × 13 − 10 × 9 = 40. This is one way; other combinations also work.

(c) – 50

1. One +13 coin makes 13 and seven −9 coins make −63.
2. 13 − 63 = −50.

Answer1 coin of +13 pibs and 7 coins of −9 pibs: 1 × 13 − 7 × 9 = −50. This is one way; other combinations also work.

(d) +8

1. Two +13 coins make 26 and two −9 coins make −18.
2. 26 − 18 = 8.

Answer2 coins of +13 pibs and 2 coins of −9 pibs: 2 × 13 − 2 × 9 = 8. This is one way; other combinations also work.

(e) +10

1. Seven +13 coins make 91 and nine −9 coins make −81.
2. 91 − 81 = 10.

Answer7 coins of +13 pibs and 9 coins of −9 pibs: 7 × 13 − 9 × 9 = 10. This is one way; other combinations also work.

(f) – 2

1. Four +13 coins make 52 and six −9 coins make −54.
2. 52 − 54 = −2.

Answer4 coins of +13 pibs and 6 coins of −9 pibs: 4 × 13 − 6 × 9 = −2. This is one way; other combinations also work.

(g) +1

1. Seven +13 coins make 91 and ten −9 coins make −90.
2. 91 − 90 = 1.

Answer7 coins of +13 pibs and 10 coins of −9 pibs: 7 × 13 − 10 × 9 = 1. This is one way; other combinations also work.

(h) Is it possible to purchase an item that costs 1568 pibs?

1. Look for a multiple of 13 that is 1568 more than a multiple of 9.
2. 122 coins of +13 make 122 × 13 = 1586, and 1586 − 1568 = 18, which is exactly 2 coins of −9.
3. So 1586 − 18 = 1568.

AnswerYes — for example, 122 coins of +13 pibs and 2 coins of −9 pibs make 122 × 13 − 2 × 9 = 1568 pibs. Other combinations also work.

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(a) (32 × (– 18)) ÷ ((– 36))

1. 32 × (−18) = −576 (signs differ, so negative).
2. (−576) ÷ (−36) = 16 (both negative, so positive).

Answer16

(b) (32) ÷ ((– 36) × (– 18))

1. (−36) × (−18) = 648 (both negative, so positive).
2. 32 ÷ 648: no integer times 648 gives 32, so the answer is not an integer.
3. As a fraction, 32/648 = 4/81 (divide top and bottom by 8).

Answer4/81

(c) (25 × (– 12)) ÷ ((45) × (– 27))

1. 25 × (−12) = −300.
2. 45 × (−27) = −1215.
3. (−300) ÷ (−1215) is positive, since both are negative. It is not an integer: 300/1215 = 20/81 (divide top and bottom by 15).

Answer20/81

(d) (280 × (– 7)) ÷ ((– 8) × (– 35))

1. 280 × (−7) = −1960.
2. (−8) × (−35) = 280.
3. (−1960) ÷ 280 = −7 (signs differ, so negative).

Answer−7

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1. Work out every value: (a) −348 + (−1064) = −1412; (b) −348 − (−1064) = 716; (c) 348 − (−1064) = 1412; (d) (−348) × (−1064) = 370272; (e) 348 × (−1064) = −370272; (f) 348 × 964 = 335472.
2. Sort the six results from smallest to largest: −370272 < −1412 < 716 < 1412 < 335472 < 370272.

Answer(e), (a), (b), (c), (f), (d)

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(a) (– 547) × 972

1. −547 is 1 more than −548, so (−547) × 972 = (−548) × 972 + 972.
2. −532656 + 972 = −531684.

Answer−531684

(b) (– 548) × 971

1. 971 is 1 less than 972, so (−548) × 971 = (−548) × 972 − (−548).
2. −532656 + 548 = −532108.

Answer−532108

(c) (– 547) × 971

1. Start from part (a): (−547) × 972 = −531684.
2. 971 is 1 less than 972, so (−547) × 971 = (−547) × 972 − (−547).
3. −531684 + 547 = −531137.

Answer−531137

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1. 33 − 7 is the exact opposite of −33 + 7, since −33 + 7 = −26 and 33 − 7 = 26 = −(−26).
2. So −207 × (33−7) = −207 × 26 = 207 × (−26), which is the same product as 207 × (−33+7).
3. That value is given as −5382.

Answer−5382

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Open NCERT p. 44Checked by computerAnswers can differ: one example

(a) the result is the maximum possible

1. The × changes the size the most, so multiply the number with the largest size, −6, by a bracket made from the other three numbers, 3, −2 and 5, using + and −.
2. For the largest result, that bracket should be as far below zero as possible: (−2) − (3 + 5) = −2 − 8 = −10.
3. (−6) × (−10) = 60.
4. Trying every order of the numbers, every placing of the three operations and every way of using brackets shows that no expression gives more than 60.

Answer(−6) × ((−2) − (3 + 5)) = 60

(b) the result is the minimum possible

1. For the smallest result, the bracket that multiplies −6 should be as far above zero as possible: 3 − (−2) + 5 = 3 + 2 + 5 = 10.
2. 10 × (−6) = −60.
3. Trying every arrangement in the same way shows that no expression gives less than −60. (Other orders, such as (3 + 5 − (−2)) × (−6), also give −60.)

Answer(3 − (−2) + 5) × (−6) = −60

Open NCERT p. 44Checked by computerAnswers can differ: one example

(a) □ + □ × □ = – 36

1. Choose the two numbers to multiply first, then pick the first number to make the total −36.
2. 6 × (−5) = −30, and −6 + (−30) = −36.
3. Four more ways: 0 + 6 × (−6) = −36; −36 + 5 × 0 = −36; 4 + (−8) × 5 = 4 − 40 = −36; −16 + 4 × (−5) = −16 − 20 = −36.

Answer−6, 6, −5 (in order): −6 + 6 × (−5) = −36. This is one way; there are many others.

(b) (□ – □) × □ = 12

1. Choose a difference and a multiplier whose product is 12, e.g. 4 × 3 = 12.
2. Make the difference 4 using 5 − 1: (5 − 1) × 3 = 4 × 3 = 12.
3. Four more ways: (8 − 2) × 2 = 12; (−1 − (−5)) × 3 = 4 × 3 = 12; (0 − 3) × (−4) = (−3) × (−4) = 12; (10 − (−2)) × 1 = 12.

Answer5, 1, 3 (in order): (5 − 1) × 3 = 12. This is one way; there are many others.

(c) (□ – (□ – □)) = – 1

1. The expression works out to (first) − (second) + (third), so choose three numbers with first − second + third = −1.
2. 2 − (5 − 2) = 2 − 3 = −1.
3. Four more ways: 0 − (1 − 0) = −1; −3 − (4 − 6) = −3 + 2 = −1; 5 − (10 − 4) = 5 − 6 = −1; −1 − (0 − 0) = −1.

Answer2, 5, 2 (in order): 2 − (5 − 2) = −1. This is one way; there are many others.