Integers — Questions and Answers
Welcome to HSLC Guru. This page gives the complete, step-by-step solution of ASSEB Class 7 New Mathematics Chapter 10 Integers — every “Work it Out” box, all twelve worked Examples and the whole of Exercise 10 — together with extra questions and a key-terms table for revision.
Summary
Integers are the positive numbers, the negative numbers and zero taken together (…, −3, −2, −1, 0, 1, 2, 3, …). On the number line, positive integers lie to the right of zero and negative integers to the left. In this chapter we revise the addition and subtraction of integers and see that integers are closed under both; they obey the commutative and associative properties under addition but not under subtraction.
Zero is the additive identity and the additive inverse of a is −a. The sign rules for multiplication are: two integers with the same sign give a positive product, and two integers with different signs give a negative product. Integers are closed under multiplication and obey the commutative, associative and distributive properties; 1 is the multiplicative identity.
Division is the inverse operation of multiplication. Integers are not closed under division, nor is division commutative or associative. For any non-zero integer a, a ÷ 0 is not defined, while 0 ÷ a = 0. The Indian mathematician Brahmagupta first recorded the sign rules for multiplication and division using dhana (fortune) for positive and rna (debt) for negative numbers.
Summary: This ASSEB Class 7 New Mathematics Chapter 10 “Integers” (অখণ্ড সংখ্যা) English-medium solution revises addition and subtraction of integers, then covers the closure, commutative, associative and distributive properties, additive identity and additive inverse, the sign rules for multiplication, multiplication and division of integers, operations with zero, Brahmagupta’s rules and real-life applications. Every Work it Out 10.1–10.5 box, all twelve worked Examples and Exercise 10 are solved step by step for HSLC Guru.
Integers on the Number Line
The number line below shows zero (0) in the middle, positive integers (1, 2, 3, …) to the right and negative integers (−1, −2, −3, …) to the left. To add we move right and to subtract we move left. For example, starting at −2 and moving 5 steps to the right lands on 3, that is, (−2) + 5 = 3.
Textbook Questions and Answers
Filling the Tables (Addition and Subtraction)
Table I — Add each vertical integer to each horizontal integer and fill the empty cells (for example, 3 + (−2) = 1).
Answer: Each cell holds (row number + column number) —
| + | 11 | −14 | −2 | 16 | 17 | −8 |
|---|---|---|---|---|---|---|
| −12 | −1 | −26 | −14 | 4 | 5 | −20 |
| 3 | 14 | −11 | 1 | 19 | 20 | −5 |
| 0 | 11 | −14 | −2 | 16 | 17 | −8 |
| −19 | −8 | −33 | −21 | −3 | −2 | −27 |
| 15 | 26 | 1 | 13 | 31 | 32 | 7 |
Every sum is an integer — this demonstrates the closure property of integers under addition.
Table II — Subtract each horizontal integer from each vertical integer and fill the empty cells (for example, (−8) − (−4) = −4).
Answer: Each cell holds (row number − column number) —
| − | 20 | 12 | −4 | 0 | 9 | 6 |
|---|---|---|---|---|---|---|
| 14 | −6 | 2 | 18 | 14 | 5 | 8 |
| 0 | −20 | −12 | 4 | 0 | −9 | −6 |
| −8 | −28 | −20 | −4 | −8 | −17 | −14 |
| −4 | −24 | −16 | 0 | −4 | −13 | −10 |
| 22 | 2 | 10 | 26 | 22 | 13 | 16 |
Every difference is also an integer — this demonstrates the closure property of integers under subtraction.
Work it Out 10.1
1. Fill in the blanks.
Answer:
(i) (−19) + (−9) = (−9) + (−19) (commutative property of addition).
(ii) (−1) + {(−8) + 7} = {(−1) + (−8)} + 7 (associative property of addition).
(iii) (−16) + 0 = −16 = 0 + (−16) (additive identity).
(iv) (27) + (−27) = 0 = (−27) + 27 (additive inverse).
2. State whether the following statements are True or False.
Answer:
(i) 4 + (−15) + (−6) = (−6) + 4 + (−15) — both sides equal −17, so True.
(ii) (16 − 15) − (−7) = 1 + 7 = 8 and 16 − {15 + (−7)} = 16 − 8 = 8 — True.
(iii) Integers are closed with respect to subtraction — True (the difference of two integers is always an integer).
(iv) The commutative and associative properties are not satisfied with respect to subtraction — True.
Work it Out 10.2
1. Find the products using tokens — (a) 5 × (−3) (b) (−4) × (−6) (c) (−4) × 2 (d) 4 × (−2).
Answer:
(a) 5 × (−3) = −15 (positive × negative = negative).
(b) (−4) × (−6) = 24 (negative × negative = positive).
(c) (−4) × 2 = −8 (negative × positive = negative).
(d) 4 × (−2) = −8 (positive × negative = negative).
2. Are the products (−6) × 2 and 6 × (−2) the same? Explain using tokens.
Answer: Yes, both equal −12. (−6) × 2 means 2 red (negative) tokens in each of 6 bottles = 12 negative tokens = value −12; and 6 × (−2) means 2 red tokens put into each of 6 bottles = 12 negative tokens = value −12. Hence (−6) × 2 = 6 × (−2) = −12.
Work it Out 10.3
1. Find the products — (a) (−5) × 9 (b) (−6) × (−13) (c) 4 × (−8) (d) (−15) × (−13) (e) 6 × (−13).
Answer:
(a) (−5) × 9 = −45.
(b) (−6) × (−13) = 78.
(c) 4 × (−8) = −32.
(d) (−15) × (−13) = 195.
(e) 6 × (−13) = −78.
2. Using 13 × 43 = 559, find the products — (a) (−13) × 43 (b) (−13) × (−43) (c) 13 × (−43).
Answer: First multiply the numbers ignoring signs (13 × 43 = 559), then apply the sign rules —
(a) (−13) × 43 = −559.
(b) (−13) × (−43) = 559.
(c) 13 × (−43) = −559.
3. Find the products — (i) 5 × (−3) × 6 (ii) 7 × 5 × (−6) (iii) (−4) × (−6) × 5 (iv) 3 × (−7) × 8 × (−5) (v) 5 × (−6) × (−3) × (−8).
Answer:
(i) 5 × (−3) × 6 = (−15) × 6 = −90.
(ii) 7 × 5 × (−6) = 35 × (−6) = −210.
(iii) (−4) × (−6) × 5 = 24 × 5 = 120.
(iv) 3 × (−7) × 8 × (−5) = (−21) × 8 × (−5) = (−168) × (−5) = 840 (two negative signs → positive).
(v) 5 × (−6) × (−3) × (−8) = (−30) × (−3) × (−8) = 90 × (−8) = −720 (three negative signs → negative).
Work it Out 10.4
1. Find the product — (i) (−26) × (13) (ii) (−40) × (−50) × 20.
Answer:
(i) (−26) × 13 = −338.
(ii) (−40) × (−50) × 20 = 2000 × 20 = 40000.
2. (i) For any integer a, what is the value of (−1) × a? (ii) The product of two integers is 67. If one of them is −1, what is the other?
Answer: (i) (−1) × a = −a (multiplying any integer by −1 gives its additive inverse). (ii) The other integer = 67 ÷ (−1) = −67 (since (−1) × (−67) = 67).
3. Based on the properties, check the correctness — (i) 27 × {(−5) + 10} = 27 × (−5) + 27 × 10 (ii) (−44) × {(−33) + (−11)} = (−44) × (−33) × (−11).
Answer: (i) LHS = 27 × 5 = 135; RHS = (−135) + 270 = 135. Both equal, so the statement is correct (distributive property). (ii) LHS = (−44) × (−44) = 1936; RHS = (−44) × (−33) × (−11) = 1452 × (−11) = −15972. They are unequal, so the statement is incorrect (the distributive property requires the inner numbers to be added, not multiplied).
4. Use the Commutative and Associative properties to find the value — (i) 125 × (−54) × 8 (ii) 15 × (−25) × (−4) × (−10).
Answer:
(i) 125 × (−54) × 8 = (125 × 8) × (−54) = 1000 × (−54) = −54000.
(ii) 15 × (−25) × (−4) × (−10) = 15 × {(−25) × (−4)} × (−10) = 15 × 100 × (−10) = 1500 × (−10) = −15000.
5. Find the value using the Distributive property — (i) 999 × 99 + 99 (ii) (−159) × 82 + (−159) × 16 + (−159) × 2.
Answer:
(i) 999 × 99 + 99 = 99 × (999 + 1) = 99 × 1000 = 99000.
(ii) (−159) × 82 + (−159) × 16 + (−159) × 2 = (−159) × (82 + 16 + 2) = (−159) × 100 = −15900.
6. (i) Is the value of 7 × (−5 + 4) the same as 7 × (−5) + 7 × 4? (ii) Is the value of 14 × {3 + (−8)} the same as 14 × 3 − 14 × 8?
Answer: (i) 7 × (−5 + 4) = 7 × (−1) = −7 and 7 × (−5) + 7 × 4 = −35 + 28 = −7 — the same. (ii) 14 × {3 + (−8)} = 14 × (−5) = −70 and 14 × 3 − 14 × 8 = 42 − 112 = −70 — the same. (Both illustrate the distributive property.)
Work it Out 10.5
1. Find the quotient — (i) 15 ÷ (−3) (ii) (−144) ÷ 12 (iii) 0 ÷ (−22) (iv) {(−6) ÷ (−6)} ÷ (−1).
Answer:
(i) 15 ÷ (−3) = −5.
(ii) (−144) ÷ 12 = −12.
(iii) 0 ÷ (−22) = 0.
(iv) {(−6) ÷ (−6)} ÷ (−1) = 1 ÷ (−1) = −1.
2. Fill in the blanks — (i) 625 ÷ (−25) = ____ (ii) {6 × (−12)} ÷ ____ = −9 (iii) ____ ÷ (5 − 6) = −20.
Answer:
(i) 625 ÷ (−25) = −25.
(ii) (−72) ÷ ____ = −9, so the blank is 8 (since (−72) ÷ 8 = −9).
(iii) ____ ÷ (−1) = −20, so the blank is 20 (since 20 ÷ (−1) = −20).
Worked Examples
Example 1. (i) Write some pairs of integers whose sum is −13. (ii) Write some pairs whose difference is −4. (iii) Find the pairs of integers whose sum is 28 and difference is 4, and whose sum is 25 and difference is 1.
Answer: (i) Sum −13: (−6) + (−7), (−4) + (−9), (−20) + 7, etc. (ii) Difference −4 (that is, a − b = −4): 3 − 7, (−5) − (−1), 0 − 4, etc. (iii) Sum 28, difference 4 → the numbers are 16 and 12 (16 + 12 = 28, 16 − 12 = 4). Sum 25, difference 1 → the numbers are 13 and 12 (13 + 12 = 25, 13 − 12 = 1).
Example 2. Use the multiplication properties to find the value — (i) 50 × (−123) × 2 (ii) (−62) × 98 + (−62) × 2 (iii) (−96) × 98.
Answer:
(i) 50 × (−123) × 2 = 50 × {2 × (−123)} = (50 × 2) × (−123) = 100 × (−123) = −12300 (commutative and associative properties).
(ii) (−62) × 98 + (−62) × 2 = (−62) × (98 + 2) = (−62) × 100 = −6200 (distributive property).
(iii) (−96) × 98 = (−96) × (100 − 2) = {(−96) × 100} − {(−96) × 2} = (−9600) − (−192) = −9600 + 192 = −9408 (distributive property).
Example 3. In a quiz, 10 marks were awarded for each correct answer and −5 marks for each wrong answer. (i) The first participant answered every question, got 9 correct and scored 35 marks. (ii) The second answered every question, got 6 correct and scored −10 marks. How many wrong answers did each give?
Answer: (i) Marks for 9 correct answers = 9 × 10 = 90; but the score is 35. Marks from wrong answers = 35 − 90 = −55; each wrong answer gives −5, so wrong answers = (−55) ÷ (−5) = 11. (ii) Marks for 6 correct answers = 6 × 10 = 60; but the score is −10. Marks from wrong answers = (−10) − 60 = −70; wrong answers = (−70) ÷ (−5) = 14. So the first gave 11 and the second gave 14 wrong answers.
Example 4. The average temperature of a place on a day was 5°C. Because of a cold wave it falls 3°C every day, but on the fifth day it rose by 5°C. What was the average temperature on the fifth day?
Answer: Temperature on day 1 = 5°C. After 4 days = (5 − 4 × 3)°C = (5 − 12)°C = −7°C. On the fifth day it rose 5°C, so the average temperature = (−7 + 5)°C = −2°C.
Example 5. Gautam stands on a bridge 15 m above the water level of a river. If the depth of the river there is 25 m, how high is Gautam above the river bed? (a) 10 m (b) 15 m (c) 25 m (d) 40 m.
Answer: Taking the water level as 0, the height of Gautam’s feet is h₁ = 15 − 0 = 15 m, and the depth of the river bed is h₂ = 0 − (−25) = 25 m. So the distance from Gautam’s feet to the river bed = (15 + 25) = 40 m. Correct answer (d) 40 metres.
Example 6. The value of {(−12) × 16} + (−13) is — (a) −41 (b) −179 (c) −205 (d) −220.
Answer: {(−12) × 16} + (−13) = (−192) + (−13) = −205. Correct answer (c) −205.
Example 7. Which options have the same value as (−20) ÷ 5? (i) −4 (ii) −(20 ÷ 5) (iii) 20 ÷ (−5) (iv) (−5) ÷ 20. Options: (a) i, ii (b) ii, iii (c) iii, iv (d) i, ii, iii.
Answer: (−20) ÷ 5 = −(20 ÷ 5) = −4. Now (i) −4 (same), (ii) −(20 ÷ 5) = −4 (same), (iii) 20 ÷ (−5) = −4 (same), but (iv) (−5) ÷ 20 = $-\frac{1}{4} \ne -4$. So the correct answer is (d) i, ii, iii.
Example 8. Select the odd pair — (a) 2, −30 (b) −12, 5 (c) 4, −12 (d) 3, −20.
Answer: 2 × (−30) = −60, (−12) × 5 = −60, 3 × (−20) = −60; but 4 × (−12) = −48. So the odd pair is (c) 4, −12.
Example 9. Find the pair of integers whose product is (−48) and whose difference is 16.
Answer: Pairs with product (−48): (−1)×48, 1×(−48), (−2)×24, 2×(−24), (−3)×16, 3×(−16), (−4)×12, 4×(−12), (−6)×8, 6×(−8). Among these the pairs with difference 16 are (−4), 12 and 4, (−12) (since 12 − (−4) = 16 and 4 − (−12) = 16).
Example 10. Assertion (A): The product of the integers (−16) and (−9) is 144. Reason (R): The product of two negative integers is always a positive integer.
Answer: (−16) × (−9) = 144 is true, and the product of two negatives being positive is true, and the reason correctly explains the assertion. So the answer is (c) — both (A) and (R) are true and (R) is the correct explanation of (A).
Example 11. For two integers a and b, Δ is an operation such that a Δ b = a × b + a × a + b × b. Find the value of (−5) Δ 2.
Answer: (−5) Δ 2 = (−5) × 2 + (−5) × (−5) + 2 × 2 = −10 + 25 + 4 = −10 + 29 = 19.
Example 12. A computing machine gives (6, 8, −3 → −18), ((−2), 7, 4 → 26), ((−7), (−6), (−9) → 47). Find the rule and fill the blanks — (4, −8, 5 → ?), ((−11), −3, 11 → ?), (9, 4, 17 → ?).
Answer: The machine computes first number + second number × third number (that is, a + b × c) — check 6 + 8 × (−3) = 6 − 24 = −18 and (−2) + 7 × 4 = −2 + 28 = 26. So —
4 + (−8) × 5 = 4 − 40 = −36.
(−11) + (−3) × 11 = −11 − 33 = −44.
9 + 4 × 17 = 9 + 68 = 77.
Fun with Integers (Magic Hexagon and Magic Multiplication)
Magic Hexagon: The hexagon has circles at its vertices and boxes on its sides. What is the relation between the circle numbers and the box numbers, and how do you fill the empty circles and boxes?
Answer: The rule is: the number in each middle box equals the product of the two circle numbers at the ends of that side. For example, between the circles −4 and 10 the box is −40, because (−4) × 10 = −40. So if one circle and the box on a side are known, dividing the box by the known circle gives the other circle (for example, box 72 with one circle −8 gives the other circle 72 ÷ (−8) = −9). Applying this sign rule to each side fills every empty circle and box.
Magic Multiplication: From a 4 × 4 square, encircle a number and strike out its row and column, repeat, and finally multiply the four encircled numbers. Why does everyone get the same product?
Answer: Each cell of the square is the product of a top (horizontal) number and a side (vertical) number. Because you must pick exactly one cell from each row and one from each column, the product of the four encircled cells is always (product of all four horizontal numbers) × (product of all four vertical numbers). This value does not depend on which cells were encircled, so everyone gets the same “magic product” (−6912 in the example).
Exercise 10
1. Find the value of the following — (a) (−987) + (−13) + 864 (b) (−4) × (−5) + (−4) × 7 (c) 225 × (−17) × 4 (d) (−8) × {12 + (−5)} (e) 0 ÷ (−187).
Answer:
(a) (−987) + (−13) + 864 = (−1000) + 864 = −136.
(b) (−4) × (−5) + (−4) × 7 = (−4) × (−5 + 7) = (−4) × 2 = −8.
(c) 225 × (−17) × 4 = (225 × 4) × (−17) = 900 × (−17) = −15300.
(d) (−8) × {12 + (−5)} = (−8) × 7 = −56.
(e) 0 ÷ (−187) = 0.
2. State true or false — (a) If a + b = 0 then a and b are additive inverses of each other. (b) Subtraction of integers obeys the commutative law. (c) For two integers a and b, a × b = b × a, therefore a ÷ b = b ÷ a. (d) (−1) × (−1) × (−1) × (−1) × (−1) × (−1) = 1. (e) For a non-zero integer a, 0 ÷ a = 0, therefore a ÷ 0 = 0.
Answer:
(a) True (a + b = 0 means b = −a, so they are additive inverses).
(b) False (subtraction is not commutative).
(c) False (multiplication is commutative but division is not, a ÷ b ≠ b ÷ a).
(d) True (an even number of (−1) factors gives 1; here there are six).
(e) False (a ÷ 0 is not defined).
3. The product of two negative integers is 2025. If one of them is −9, what is the other?
Answer: The other integer = 2025 ÷ (−9) = −225 (since (−9) × (−225) = 2025, and both are negative).
4. Observe the pattern and fill in the blanks — −4 × 3 = −12; −4 × 2 = −8 = −12 − (−4); −4 × 1 = (−4) = −8 − (−4); −4 × 0 = 0 = (−4) − (−4); −4 × (−1) = ____; −4 × (−2) = ____; −4 × (−3) = ____.
Answer: At each step the product increases by 4 (subtracting −4 each time) —
−4 × (−1) = 0 − (−4) = 4.
−4 × (−2) = 4 − (−4) = 8.
−4 × (−3) = 8 − (−4) = 12.
5. Write two different integers whose product is smaller than each of the integers.
Answer: Taking one positive and one negative integer gives a negative product that can be smaller than both. For example, 3 and (−2): the product 3 × (−2) = −6 is smaller than 3 and also smaller than (−2).
6. Write two different integers such that one is less than −11 and the other is greater than −11, but their difference is 11.
Answer: −16 and −5: here −16 < −11 and −5 > −11, and the difference = (−5) − (−16) = −5 + 16 = 11.
7. For two integers a and b there is an operation ¶ such that a ¶ b = a + b + a × b. Find — (i) (−12) ¶ (−11) (ii) (−24) ¶ (9).
Answer:
(i) (−12) ¶ (−11) = (−12) + (−11) + (−12) × (−11) = −23 + 132 = 109.
(ii) (−24) ¶ 9 = (−24) + 9 + (−24) × 9 = −15 + (−216) = −231.
8. In a test, 5 marks are given for every correct answer and (−2) for every wrong answer. (i) Rabiya answered all questions, got 16 correct and scored 64 — how many did she get wrong? (ii) Sahil got 5 correct and scored −3 — how many did he get wrong and how many did he not attempt?
Answer: (i) Marks for 16 correct = 16 × 5 = 80; but she scored 64. Marks from wrong answers = 64 − 80 = −16; each wrong gives −2, so wrong answers = (−16) ÷ (−2) = 8. Total questions = 16 + 8 = 24 (she answered all). (ii) Sahil’s marks for 5 correct = 5 × 5 = 25; but he scored −3. Marks from wrong answers = (−3) − 25 = −28; wrong answers = (−28) ÷ (−2) = 14. He attempted 5 + 14 = 19, so questions not attempted = 24 − 19 = 5.
9. A calculating machine shows results with an arrow (→): (−6, 12, 19 → −91), (17, −9, −25 → −128), (11, 14, −28 → 182). Find the operation and complete — ((−3), −32, −41 → ?), ((−13), −21, 31 → ?), (15, −18, ? → −286).
Answer: The machine computes first number × second number − third number — check (−6) × 12 − 19 = −72 − 19 = −91; 17 × (−9) − (−25) = −153 + 25 = −128; 11 × 14 − (−28) = 154 + 28 = 182. So —
(−3) × (−32) − (−41) = 96 + 41 = 137.
(−13) × (−21) − 31 = 273 − 31 = 242.
15 × (−18) − ? = −286 → −270 − ? = −286 → ? = 16.
10. Find the value — (a) {18 × (−15)} ÷ (−30) (b) (−315) ÷ {(−9) × (−7)} (c) {(−51) × 8} ÷ {24 × (−17)}.
Answer:
(a) {18 × (−15)} ÷ (−30) = (−270) ÷ (−30) = 9.
(b) (−315) ÷ {(−9) × (−7)} = (−315) ÷ 63 = −5.
(c) {(−51) × 8} ÷ {24 × (−17)} = (−408) ÷ (−408) = 1.
11. Which of the following has the same value as {(−5) × (−4)} + 7? (i) (5 × 4) + 7 (ii) 7 + {(−5) × (−4)} (iii) {(−4) × (−5)} + 7 (iv) (5 + 4) × 3. Options: (A) (i),(v) (B) (i),(iv) (C) (i),(ii),(iii) (D) (i),(ii),(iii),(iv).
Answer: {(−5) × (−4)} + 7 = 20 + 7 = 27. Now (i) (5 × 4) + 7 = 27; (ii) 7 + 20 = 27; (iii) 20 + 7 = 27; (iv) (5 + 4) × 3 = 9 × 3 = 27 — all four equal 27. So the correct answer is (D) (i), (ii), (iii), (iv).
12. Which statements are true or false? (P) The multiplicative identity for integers is 1. (Q) The additive inverse of a is −a. Options: (A) (P) True, (Q) False (B) (P) False, (Q) True (C) Both True (D) Both False.
Answer: (P) is true (the multiplicative identity is 1) and (Q) is true (the additive inverse of a is −a). So the correct answer is (C) Both (P) and (Q) are True.
13. Assertion (A): 5 × (4 + 6) = 5 × 4 + 5 × 6, both sides equal 50. Reason (R): For any integers a, b, c, a × (b + c) = a × b + a × c — the distributive property.
Answer: 5 × 10 = 50 and 20 + 30 = 50, so (A) is true; (R) is also true and correctly explains (A). So the correct answer is (A) — (A) true, (R) true, and (R) is the correct explanation of (A).
14. Match Column I with Column II — (P) a + b = b + a; (Q) (a × b) × c = a × (b × c); (R) a × (b + c) = a × b + a × c; (S) (a + b) is an integer. [(A) Distributive Property; (B) Closure Property of Addition; (C) Commutative Property of Addition; (D) Associative Property of Multiplication].
Answer: (P) → (C) Commutative Property of Addition; (Q) → (D) Associative Property of Multiplication; (R) → (A) Distributive Property; (S) → (B) Closure Property of Addition. So the correct option is (A): (P)→(C), (Q)→(D), (R)→(A), (S)→(B).
15. (P) {5 + (−8)} + 9; (Q) (−3) − (−6); (R) 4 × {(−1) × (−2)}; (S) (−20) ÷ (−5). Which option arranges them in increasing order of value? (A) R,P,Q,S (B) S,Q,R,P (C) P,R,S,Q (D) Q,S,P,R.
Answer: P = (−3) + 9 = 6; Q = −3 + 6 = 3; R = 4 × 2 = 8; S = 4. Increasing order: Q(3), S(4), P(6), R(8). So the correct answer is (D) Q, S, P, R.
16. Put the sign <, = or > in the box — (i) 180 × (−5) __ (−180) × 5 (ii) −660 ÷ (10) __ 600 ÷ (−66) (iii) {(−8) × 9} × (−1) __ (8) × {9 × (−1)} (iv) (−13) × (−1) × (−11) __ 13 × 1 × 11.
Answer:
(i) 180 × (−5) = −900 and (−180) × 5 = −900 → =.
(ii) −660 ÷ 10 = −66 and 600 ÷ (−66) ≈ −9.09; −66 < −9.09 → <.
(iii) {(−8) × 9} × (−1) = 72 and (8) × {9 × (−1)} = −72; 72 > −72 → >.
(iv) (−13) × (−1) × (−11) = −143 and 13 × 1 × 11 = 143; −143 < 143 → <.
17. Fill in the two squares so that each becomes a magic square (the sum of the numbers in every row, column and diagonal is the same).
Answer: Method for a magic square — add a completed row (or column or diagonal) to get the magic sum S; in a 3 × 3 square the centre cell = S ÷ 3. Then for any line with two known entries, the third = S − (sum of the two known). For instance, one completed magic square with magic sum −6 is: rows (−6, −4, 4), (8, −2, −12), (−8, 0, 2) — here every row, column and diagonal adds to −6. Using the same method the magic sum is found from the given entries and every empty cell of both squares is filled.
18. Follow the direction of the arrows and, using the given operations, fill in the blank boxes.
Answer: This is a flow chart — start from the given number and apply, in order, the operation written on each arrow (such as +(−5), ×(−4), ÷3, ÷(−5), …) to get the value in the next box. For example, starting from 40 and applying +(−5) gives 40 + (−5) = 35, and the next arrow’s operation is then applied to 35. Following the sign rules and the rules of multiplication and division, each blank box is filled in turn.
19. The temperature of a tea garden’s cold-storage room must be reduced from 36°C at 3°C per hour. What is the temperature 7 hours after the process begins?
Answer: Total fall in 7 hours = 7 × 3 = 21°C. So the temperature = 36 − 21 = 15°C.
20. A ceramic industry makes ₹7 profit on each white clay pot and ₹2 loss on each red clay pot. (a) In a month it sells 2500 white and 3200 red pots — what is its net profit or loss? (b) If 3500 red pots are sold, how many white pots must be sold for no profit and no loss?
Answer: (a) Profit on white pots = 2500 × 7 = ₹17500; loss on red pots = 3200 × (−2) = −₹6400. Net = 17500 + (−6400) = ₹11100 profit. (b) Loss on 3500 red pots = 3500 × 2 = ₹7000; for no profit or loss the white-pot profit must also be ₹7000, so white pots = 7000 ÷ 7 = 1000.
21. Using the clues, find the values of E, I, L, N, S, T — L × I = −6, I × 9 = −18, S ÷ I = 3, T × E = −35, 45 ÷ E = −9, E × N = −20. Then find the sum and product of the letters for each word — (i) LIST (ii) LINES (iii) SILENT.
Answer: I × 9 = −18 → I = −2; L × (−2) = −6 → L = 3; S ÷ (−2) = 3 → S = −6; 45 ÷ E = −9 → E = −5; T × (−5) = −35 → T = 7; (−5) × N = −20 → N = 4.
- (i) LIST = 3, (−2), (−6), 7 → sum = 3 − 2 − 6 + 7 = 2; product = 3 × (−2) × (−6) × 7 = 252.
- (ii) LINES = 3, (−2), 4, (−5), (−6) → sum = 3 − 2 + 4 − 5 − 6 = −6; product = 3 × (−2) × 4 × (−5) × (−6) = −720.
- (iii) SILENT = (−6), (−2), 3, (−5), 4, 7 → sum = −6 − 2 + 3 − 5 + 4 + 7 = 1; product = (−6) × (−2) × 3 × (−5) × 4 × 7 = −5040.
22. If 56 − 87 + 15 − 8 − 40 = −64, then without calculating, write the value of −56 + 87 − 15 + 8 + 40.
Answer: −56 + 87 − 15 + 8 + 40 is obtained by changing the sign of every term of (56 − 87 + 15 − 8 − 40), so it is its additive inverse. Hence the value = −(−64) = 64.
23. Given 108 × (−765) = −82620, find — (a) 109 × (−765) (b) 108 × (−764) (c) (−108) × (−766).
Answer:
(a) 109 × (−765) = 108 × (−765) + (−765) = −82620 − 765 = −83385.
(b) 108 × (−764) = 108 × (−765) + 108 = −82620 + 108 = −82512.
(c) (−108) × (−766) = 108 × 766 = 108 × 765 + 108 = 82620 + 108 = 82728.
24. Given 309 × (−46 + 4) = −12978, then (−309) × (46 − 4) = ?
Answer: 309 × (−46 + 4) = 309 × (−42) = −12978, so 309 × 42 = 12978. Now (−309) × (46 − 4) = (−309) × 42 = −(309 × 42) = −12978.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. What is the value of (−7) × 6? (a) 42 (b) −42 (c) −13 (d) 13
Answer: (b) −42.
2. What is the additive identity for integers? (a) 1 (b) −1 (c) 0 (d) 10
Answer: (c) 0.
3. What is the multiplicative identity for integers? (a) 0 (b) 1 (c) −1 (d) 2
Answer: (b) 1.
4. What is the value of (−1) × (−1) × (−1) × (−1) × (−1)? (a) 1 (b) −1 (c) 5 (d) −5
Answer: (b) −1 (an odd number of (−1) factors gives −1).
5. Under which operation are integers NOT closed? (a) Addition (b) Subtraction (c) Multiplication (d) Division
Answer: (d) Division.
6. What is the additive inverse of a? (a) a (b) 1/a (c) −a (d) 0
Answer: (c) −a.
7. For any non-zero integer a, what is a ÷ 0? (a) 0 (b) 1 (c) a (d) not defined
Answer: (d) not defined.
8. What is the value of (−48) ÷ (−4)? (a) 12 (b) −12 (c) 44 (d) −44
Answer: (a) 12 (a negative divided by a negative is positive).
9. What did Brahmagupta call positive numbers? (a) rna (debt) (b) dhana (fortune) (c) zero (d) root
Answer: (b) dhana (fortune).
10. a × (b + c) = a × b + a × c is which property? (a) Closure (b) Commutative (c) Associative (d) Distributive
Answer: (d) Distributive property.
Fill in the Blanks
1. The product of two integers with the same sign is ____.
Answer: positive.
2. For any non-zero integer a, 0 ÷ a = ____.
Answer: 0.
3. For integers, 1 (one) is called the ____ identity.
Answer: multiplicative.
4. Division is the inverse operation of ____.
Answer: multiplication.
5. (−a) × (−b) = ____ (where a, b are positive integers).
Answer: a × b.
True or False
1. Integers obey the commutative property under addition.
Answer: True.
2. Integers obey the associative property under subtraction.
Answer: False.
3. Any integer multiplied by 0 gives the product 0.
Answer: True.
4. Dividing a positive integer by a negative integer gives a positive quotient.
Answer: False (the quotient is negative).
5. Integers are closed under division.
Answer: False (for example, 8 ÷ (−16) is not an integer).
Short Answer Questions
1. State the sign rules for the multiplication of integers.
Answer: Positive × Positive = Positive; Positive × Negative = Negative; Negative × Positive = Negative; Negative × Negative = Positive. In short — the product of two integers with the same sign is positive, and with different signs is negative.
2. Show with an example that subtraction of integers is not commutative.
Answer: (−8) − (−4) = −8 + 4 = −4, but (−4) − (−8) = −4 + 8 = 4. Since −4 ≠ 4, we have (−8) − (−4) ≠ (−4) − (−8), so subtraction is not commutative.
3. Why is a ÷ 0 not defined?
Answer: Suppose a ÷ 0 = b (a ≠ 0). Then 0 × b would have to equal a. But any number multiplied by 0 is 0, so 0 × b can never equal a non-zero a. Because of this contradiction, a ÷ 0 is not defined.
4. Use the distributive property to find (−25) × 98.
Answer: (−25) × 98 = (−25) × (100 − 2) = (−25) × 100 − (−25) × 2 = −2500 + 50 = −2450.
Key Terms
| Term | Meaning |
|---|---|
| Integer | Positive numbers, negative numbers and zero taken together |
| Positive number | A number greater than zero (right of 0 on the number line) |
| Negative number | A number less than zero (left of 0 on the number line) |
| Closure property | The result of the operation is again the same type of number |
| Commutative property | Changing the order does not change the result |
| Associative property | Changing the grouping does not change the result |
| Distributive property | a × (b + c) = a × b + a × c |
| Additive identity | 0 — adding it leaves the number unchanged |
| Additive inverse | For a it is −a; their sum is 0 |
| Multiplicative identity | 1 — multiplying by it leaves the number unchanged |
| Quotient | The result obtained on dividing |
| Brahmagupta | Indian mathematician who first stated the sign rules (dhana/rna) |