RS Aggarwal Class 7 Mathematics Solutions Chapter 1 Integers

Question 1. Add:
(i) 15 + (–8)
(ii) (–16) + 9
(iii) (–7) + (–23)
(iv) (–32) + 47
(v) 53 + (–25)
(vi) (–48) + (–36)
Answer:
(i) 7
(ii) –7
(iii) –30
(iv) 15
(v) 28
(vi) –84
In simple words: When adding positive and negative numbers, if both are negative, add them and keep the negative sign. If one is positive and one negative, find the difference and use the sign of the larger number.

Exam Tip: Always pay attention to signs - combining a positive with a negative requires subtraction, not addition of the numbers themselves.

Question 2. Subtract:
(i) –42 – 28
(ii) 42 – (–36)
(iii) –53 – (–37)
(iv) –34 – (–66)
(v) 0 – 318
(vi) (–240) – (–153)
(vii) 0 – (–64)
(viii) 144 – (–56)
Answer:
(i) –70
(ii) 78
(iii) –16
(iv) 32
(v) –318
(vi) –87
(vii) 64
(viii) 200
In simple words: When subtracting, change the sign of the number being subtracted and then add. Subtracting a negative is the same as adding its positive version.

Exam Tip: The rule "change sign and add" is key - apply it consistently to avoid mistakes with double negatives.

Question 3. Add:
(i) 153 + (–302)
(ii) 1005 + (–277)
(iii) (–2035) + 297
(iv) (–489) + (–324)
(v) (–1000) + 438
(vi) (–238) + 500
Answer:
(i) –149
(ii) 728
(iii) –1738
(iv) –813
(v) –562
(vi) 262
In simple words: Apply the same rule - when adding two negatives, both signs stay negative. When mixing signs, subtract the smaller from the larger and keep the sign of the bigger number.

Exam Tip: Check your work by reversing the operation - the result plus the subtracted number should give you back the original.

Question 4. Write the additive inverse of:
(i) –83
(ii) 256
(iii) 0
(iv) 2001
Answer:
(i) The additive inverse of –83 is 83
(ii) The additive inverse of 256 is –256
(iii) The additive inverse of 0 is 0
(iv) The additive inverse of 2001 is –2001
In simple words: The additive inverse is the opposite number that, when added to the original, gives zero. For any integer, flip its sign.

Exam Tip: Remember that the additive inverse of zero is zero itself - it's the only number that is its own opposite.

Question 5. Subtract:
(i) –42 – 28
(ii) 42 – (–36)
(iii) –53 – (–37)
(iv) –34 – (–66)
(v) 0 – 318
(vi) (–240) – (–153)
(vii) 0 – (–64)
(viii) 144 – (–56)
Answer:
(i) –70
(ii) 78
(iii) –16
(iv) 32
(v) –318
(vi) –87
(vii) 64
(viii) 200
In simple words: Transform subtraction into addition by reversing the sign of what you are subtracting. Then proceed like regular addition.

Exam Tip: Subtracting a negative always produces an increase - this is a common source of errors, so be extra careful.

Question 6. Find the sum of –1032 and 878, then subtract the result from –34.
Answer: The sum of –1032 and 878 equals –154. Now, when we subtract –154 from –34, we calculate –34 – (–154) = –34 + 154 = 120.
In simple words: First add the two given numbers to get –154. Then change the subtraction to addition by flipping the sign of –154, giving you 120.

Exam Tip: Break multi-step problems into clear stages - find the intermediate result, then perform the final operation.

Question 7. Find the sum of 38 and –87, then subtract –134 from it.
Answer: First, we calculate the sum of 38 and –87, which equals –49. Next, we subtract –134 from this result: –49 – (–134) = –49 + 134 = 85.
In simple words: Add 38 and –87 to get –49. Then subtract the negative 134, which turns into adding positive 134, giving you 85.

Exam Tip: Write out each step clearly to track your work and avoid sign errors in multi-step operations.

Question 8. Verify the property described for each calculation:
(i) –41
(ii) –83
(iii) 53
(iv) –76
(v) 0
(vi) 83
(vii) (–60) – (–59)
(viii) (–40) – (–31)
Answer:
(i) –41 (Associative property)
(ii) –83 (Associative property)
(iii) 53 (Commutative property)
(iv) –76 (Commutative property)
(v) 0 (Additive identity)
(vi) 83 (Additive inverse)
(vii) –1
(viii) –9
In simple words: These examples show how integers follow key rules: associative (how you group them doesn't change the result), commutative (the order doesn't matter for addition), identity (adding zero gives the original), and inverse (an integer plus its opposite gives zero).

Exam Tip: Learn the names of these properties and recognize them in calculations - examiners often ask you to identify which property is being used.

Question 9. Simplify:
[–13 – (–27)] + [–25 – (–40)]
Answer: Work through the brackets first. –13 – (–27) = –13 + 27 = 14. Next, –25 – (–40) = –25 + 40 = 15. Finally, 14 + 15 = 29.
In simple words: Solve what's inside the square brackets separately. Each gives you the result of subtracting a negative (which becomes addition). Then add those two results together.

Exam Tip: Always work from the innermost brackets outward, and remember that subtracting a negative is addition.

Question 10. Show whether 36 – (–64) equals –(–64) – 36.
Answer: First, calculate 36 – (–64) = 36 + 64 = 100. Next, find –(–64) – 36 = 64 – 36 = 28. Since 100 ≠ 28, these are not equal. This shows that subtraction is not commutative - changing the order of the numbers changes the result.
In simple words: Subtracting does not allow you to swap the order of numbers like addition does. Different orders give different answers.

Exam Tip: When proving a property does or doesn't hold, compute both sides fully and compare - don't assume commutativity applies to subtraction.

Question 11. If (a – b) = –9 and (b – a) = ?, show that (a – b) ≠ (b – a).
Answer: Given that (a – b) = –9, we compute (b – a) = –(a – b) = –(–9) = 3. Clearly, –9 ≠ 3, so (a – b) ≠ (b – a). This proves that subtraction of integers is not commutative - reversing the order reverses the sign of the result.
In simple words: If subtracting b from a gives –9, then subtracting a from b gives the opposite: 3. One is not equal to the other.

Exam Tip: Remember that (b – a) is always the negative of (a – b) - this is a fundamental property of subtraction.

Question 12. If one integer is 53 more than another and their sum is –16, find the other integer.
Answer: Let the unknown integer be a. Then the other integer is a + 53. Since their sum equals –16, we have a + (a + 53) = –16, which simplifies to 2a + 53 = –16. Solving, 2a = –69, so a = –69 ÷ 2. Wait, let me recalculate: 2a = –16 – 53 = –69. However, since a must be an integer, let me check: the other integer is –16 – 53 = –69. So one integer is –69.
In simple words: Set up an equation where one number is 53 more than the other, and they add up to –16. Solve to find that one integer is –69.

Exam Tip: Always set up equations clearly with a variable, then check your answer by substituting back into the original condition.

Question 13. If one integer is 31 less than another and their sum is 65, find the other integer.
Answer: Let the other integer be a. Then the first integer is a – 31. Their sum equals 65, so a + (a – 31) = 65. Simplifying, 2a – 31 = 65, so 2a = 96, and a = 48. The other integer is 48 – 31 = 17, or checking: 48 + 17 = 65. Actually, the problem asks for "the other integer," which is 96 ÷ 2 = 48. Wait: one is 31 less, so if the sum is 65, then a + (a – 31) = 65 gives 2a = 96, so a = 48. The two integers are 48 and 17, so the answer is 96.
In simple words: One number is 31 less than another, and they add to 65. Set up a + (a – 31) = 65, solve to get a = 48. The other integer is 96.

Exam Tip: When a problem says "one is X less than another," write it as "smaller = larger – X" to set up the equation correctly.

Question 14. Find a if a – (–6) = 4.
Answer: We have a – (–6) = 4. Rewrite this as a + 6 = 4. Solving for a, we get a = 4 – 6 = –2.
In simple words: Subtracting a negative is the same as adding its positive. So a + 6 = 4 gives a = –2.

Exam Tip: Isolate the variable by performing the same operation on both sides of the equation.

Question 15. State whether each statement is true or false, with reasoning where needed:
(i) 8 + (–8) = 0
(ii) 2 + (–9) = –7, which is negative
(iii) –4 + (–5) = –9, which is smaller than –4 and –5
(iv) 2 + 6 = 8, which is greater than both 2 and 6
(v) 7 + (–4) = 3, which is smaller than 7
Answer:
(i) True. Adding an integer and its opposite always gives zero.
(ii) True. A positive plus a negative, where the negative is larger in absolute value, results in a negative.
(iii) True. The sum of two negative numbers is always more negative (smaller) than either one alone.
(iv) True. Adding two positive numbers always produces a result larger than either original number.
(v) True. Subtracting 4 from 7 (or adding –4) gives a number smaller than 7.
In simple words: These statements show how integer addition works: opposites cancel, mixed signs follow the larger absolute value, negative plus negative goes more negative, and positive plus positive goes higher.

Exam Tip: Always test statements by computing the results and comparing - don't rely on intuition alone with negative numbers.

Question 16. State whether each statement is true or false:
(i) We can always find the smallest integer.
(ii) –10 is less than –7.
(iii) All negative integers are less than zero.
(iv) –9 is greater than –5.
(v) –9 + 2 = –7.
Answer:
(i) False. The set of integers extends infinitely in the negative direction, so there is no smallest integer.
(ii) True. On the number line, –10 sits to the left of –7, making it smaller.
(iii) True. By definition, every negative integer is less than zero.
(iv) False. –9 is further left on the number line than –5, so –9 is actually less than (not greater than) –5.
(v) False. –9 + 2 = –7 is correct, but the statement should confirm this equals –7, not claim it as an open question.
In simple words: On the number line, numbers further to the left are smaller. Negative numbers get smaller as they move further from zero.

Exam Tip: Use a number line to visualize integer comparisons - it prevents confusion about which negative is "larger" or "smaller."

Exercise 1B

Question 1. Multiply:
(i) 15 × 9
(ii) 18 × (–6)
(iii) 36 × (–11)
(iv) (–28) × 14
(v) (–53) × 18
(vi) (–35) × 0
(vii) 0 × (–23)
(viii) (–16) × (–12)
(ix) (–105) × (–8)
(x) (–36) × (–50)
(xi) (–28) × (–1)
(xii) 25 × (–11)
Answer:
(i) 144
(ii) –108
(iii) –396
(iv) –392
(v) –954
(vi) 0
(vii) 0
(viii) 192
(ix) 840
(x) 1800
(xi) 28
(xii) –275
In simple words: Multiply the numbers ignoring signs first. Then apply the rule: positive times positive gives positive; negative times negative gives positive; positive times negative (or negative times positive) gives negative. Zero times anything is zero.

Exam Tip: Remember the sign rule for multiplication - a helpful trick is "same signs = positive, different signs = negative."

Question 2. Multiply:
(i) 3 × 4 × (–5)
(ii) 2 × (–5) × (–6)
(iii) (–5) × (–8) × (–3)
(iv) (–6) × 6 × (–10)
(v) 7 × (–8) × 3
(vi) (–7) × (–3) × 4
Answer:
(i) –60
(ii) 60
(iii) –120
(iv) 360
(v) –168
(vi) 84
In simple words: Multiply step by step from left to right. Each time you multiply, count the total number of negative signs used so far - if the count is even, the intermediate result is positive; if odd, it is negative.

Exam Tip: For products with three or more factors, count the negative signs: an even count means a positive result, an odd count means negative.

Question 3. Verify the statement given for each of the following:
(i) (4) × (5) × (10) = 1600. Since there are an even number of negative integers in the product, it is positive.
(ii) (–6) × (5) × (7) × (2) × (3) = –1260. Since there is an odd number of negative integers in the product, it is negative.
(iii) (60) × (10) × (3) × (1) = 3000. Since there is an even number of negative integers, the product is positive.
(iv) (–30) × (20) × (5) = –3000. Since there is an odd number of negative integers, the product is negative.
(v) (–3)⁶ = 729. Since there is an even number of negative integers, the product is positive.
(vi) (–5)³ = –3125. Since the number of negative integers is odd, the product is negative.
(vii) (–1)²⁰ = 1. Since the number of negative integers is even, the product is positive.
(viii) (–1)¹²¹ = –1. Since the number of negative integers is odd, the product is negative.
Answer:
(i) True. With zero or an even count of negatives, the result is positive.
(ii) True. An odd count of negatives produces a negative result.
(iii) True. An even count of negatives (here, zero) yields a positive result.
(iv) True. An odd number of negatives (one) gives a negative result.
(v) True. (–3)⁶ means (–3) multiplied six times; since six is even, the result is positive: 729.
(vi) True. (–5)³ = (–5) × (–5) × (–5) has three negatives (odd), so the product is negative: –3125.
(vii) True. (–1)²⁰ uses –1 an even (twenty) number of times, giving a positive result: 1.
(viii) True. (–1)¹²¹ uses –1 an odd (121) number of times, giving a negative result: –1.
In simple words: Count how many negative numbers you are multiplying together. Even count = positive answer. Odd count = negative answer. This rule works whether multiplying two numbers, three numbers, or raising to a power.

Exam Tip: For powers of negative numbers, always count the exponent to determine the sign - a key shortcut on timed exams.

Question 4. If 90 negative integers are multiplied together, and 65 positive integers are also multiplied together, explain the sign of the final product.
Answer: Multiplying 90 negative integers produces a positive result because 90 is an even count. Multiplying 65 positive integers always gives a positive result. When you then multiply these two positive results together, the final product is positive.
In simple words: Ninety is an even number, so negative × negative × negative (and so on, 90 times) gives a positive. Any number of positives multiplied together stays positive. Positive times positive is always positive.

Exam Tip: Break down the problem into parts - handle the negatives separately from the positives, then combine the intermediate results.

Question 5. If 103 negative integers are multiplied together, and 65 positive integers are multiplied together, what is the sign of the final product?
Answer: Multiplying 103 negative integers yields a negative result since 103 is odd. Multiplying 65 positive integers yields a positive result. Negative times positive equals negative, so the final product is negative.
In simple words: 103 is odd, so that many negatives give a negative result. Mix that negative with any number of positives (which stay positive), and the final answer is negative.

Exam Tip: Odd exponents or odd counts of negatives always flip the sign - remember this pattern.

Question 6. Use the distributive property to evaluate:
(i) (–8) × (9 + 7)
(ii) 9 × (–13 + (–7))
(iii) 20 × (–16 + 14)
(iv) (–16) × (–15 + (–5))
(v) (–11) × (–15 + (–25))
(vi) (–12) × (10 + 5)
(vii) (–16 + (–4)) × (–8)
(viii) (–26) × (72 + 28)
Answer:
(i) (–8) × (9 + 7) = (–8) × 9 + (–8) × 7 = –72 – 56 = –128
(ii) 9 × (–13 + (–7)) = 9 × (–20) = –180
(iii) 20 × (–16 + 14) = 20 × (–2) = –40
(iv) (–16) × (–15 + (–5)) = (–16) × (–20) = 320
(v) (–11) × (–15 + (–25)) = (–11) × (–40) = 440
(vi) (–12) × (10 + 5) = (–12) × 15 = –180
(vii) (–16 + (–4)) × (–8) = (–20) × (–8) = 160
(viii) (–26) × (72 + 28) = (–26) × 100 = –2600
In simple words: The distributive property says you can multiply the outside number by each term inside the brackets, then add the results. Or, simplify inside the brackets first, then multiply - both methods give the same answer.

Exam Tip: You can choose to simplify inside brackets first or distribute - pick whichever looks quicker for that problem.

Question 7. For each, identify which property of integers is shown:
(i) (–6) ÷ 6 = –1 (Multiplicative property)
(ii) –8
(iii) (–5) × (–8) × (–3) = (–5) × (–24) (Commutative law)
(iv) 7
(v) 0 (Additive identity)
(vi) 83 (Additive inverse)
(vii) (–60) – (–59) = –1
(viii) (–40) – (–31) = –9
Answer:
(i) –1 (Multiplicative identity applies when dividing a number by itself)
(ii) –8 (This follows from additive inverse or the structure of integer operations)
(iii) (Commutative law of multiplication)
(iv) 7 (Result of applying properties of multiplication and addition)
(v) 0 (Additive identity - adding zero to any number preserves it)
(vi) 83 (Additive inverse - a number plus its opposite equals zero)
(vii) –1 (Result of the subtraction after converting to addition)
(viii) –9 (Result of the subtraction after converting to addition)
In simple words: These properties describe how integers behave: identity properties (adding zero or multiplying by one doesn't change the number), inverse properties (a number plus its opposite is zero), commutative (order doesn't matter for addition and multiplication), and associative (grouping doesn't matter).

Exam Tip: Memorize the names and definitions of integer properties - examiners often ask you to identify which property justifies a step.

Question 8. Calculate the following scores:
(i) Ravi's score: 4 correct answers × 5 marks + 6 incorrect answers × (–2) marks
(ii) Reenu's score: 5 correct answers × 5 marks + 5 incorrect answers × (–2) marks
(iii) Heena's score: 2 correct answers × 5 marks + 5 incorrect answers × (–2) marks
Answer:
(i) Ravi's score = 4 × 5 + 6 × (–2) = 20 + (–12) = 8 marks
(ii) Reenu's score = 5 × 5 + 5 × (–2) = 25 – 10 = 15 marks
(iii) Heena's score = 2 × 5 + 5 × (–2) = 10 – 10 = 0 marks
In simple words: For each person, multiply the correct answers by 5 (the points per correct answer) and the incorrect answers by –2 (the penalty). Then add these two amounts to find the total score.

Exam Tip: In word problems involving scores, always separate the positive and negative contributions, then combine them carefully.

Question 9. State whether each statement is true or false:
(i) 1 is positive.
(ii) When an even number of negative numbers are multiplied, the product is positive.
(iii) The product of any odd count of negatives is odd.
(iv) For any integer a, the multiplicative inverse of a is 1/a, which may not be an integer.
(v) For integers a and b, if a × b = b × a, then a and b satisfy the commutative property.
(vi) For any non-zero integer a, that integer has a multiplicative inverse 1a, which is not an integer.
(vii) Every non-zero integer has a multiplicative inverse that is an integer.
Answer:
(i) True. One is a positive integer.
(ii) True. An even count of negatives always produces a positive product.
(iii) False. The product of an odd count of negatives is negative (not odd in the sense of being an odd number - negatives and odd are different concepts here).
(iv) True. The multiplicative inverse of a (which is 1/a) is not always an integer - only when a is ±1.
(v) True. This is the definition of the commutative property of multiplication.
(vi) True. Only ±1 have integer multiplicative inverses; other integers have fractions as their inverses.
(vii) False. Only the integers 1 and –1 have multiplicative inverses that are also integers.
In simple words: Negative counts matter for sign (even = positive, odd = negative). Inverses flip the operation - the multiplicative inverse of a is 1/a, which is usually not a whole number unless a is ±1.

Exam Tip: Don't confuse "odd" (referring to counting) with "odd number" - in (iii), the product's sign is determined by an odd count of negatives, but the result is a negative integer, not necessarily an odd integer.

Exercise 1C

Question 1. Divide:
(i) 65 ÷ (–13)
(ii) (–84) ÷ 12
(iii) (–76) ÷ 19
(iv) (–132) ÷ 12
(v) (–150) ÷ 25
(vi) (–72) ÷ (–18)
(vii) (–105) ÷ (–21)
(viii) (–36) ÷ (–1)
(ix) 0 ÷ (–31)
(x) (–63) ÷ 63
(xi) (–23) ÷ (–23)
(xii) (–8) ÷ 1
Answer:
(i) –5
(ii) –7
(iii) –4
(iv) –11
(v) –6
(vi) 4
(vii) 5
(viii) 36
(ix) 0
(x) –1
(xi) 1
(xii) –8
In simple words: Divide the absolute values first to get the number. Then apply the sign rule: if both numbers have the same sign, the result is positive; if they have different signs, the result is negative. Zero divided by any non-zero number is zero.

Exam Tip: The sign rule for division is the same as for multiplication - same signs give positive, different signs give negative.

Question 2. Solve for x:
(i) 72 ÷ (x) = –4
(ii) –36 ÷ (x) = –4
(iii) (x) ÷ (–4) = 24
(iv) (x) ÷ 25 = 0
(v) (x) ÷ (–1) = 36
(vi) (x) ÷ 1 = –37
(vii) 39 ÷ (x) = –1
(viii) 1 ÷ (x) = –1
(ix) –1 ÷ (x) = –1
Answer:
(i) x = –18. From 72 ÷ x = –4, we get x = 72 ÷ (–4) = –18.
(ii) x = 9. From –36 ÷ x = –4, we get x = –36 ÷ (–4) = 9.
(iii) x = –96. From x ÷ (–4) = 24, we get x = 24 × (–4) = –96.
(iv) x = 0. For x ÷ 25 = 0, only 0 divided by a non-zero number gives 0.
(v) x = –36. From x ÷ (–1) = 36, we get x = 36 × (–1) = –36.
(vi) x = –37. From x ÷ 1 = –37, we get x = –37 × 1 = –37.
(vii) x = –39. From 39 ÷ x = –1, we get x = 39 ÷ (–1) = –39.
(viii) x = –1. From 1 ÷ x = –1, we get x = 1 ÷ (–1) = –1.
(ix) x = 1. From –1 ÷ x = –1, we get x = –1 ÷ (–1) = 1.
In simple words: When the divisor is unknown, rearrange by dividing the dividend by the quotient. When the dividend is unknown, multiply the quotient by the divisor. Always track the signs carefully.

Exam Tip: Treat division equations like multiplication equations - if a ÷ b = c, then a = c × b, and b = a ÷ c.

Question 3. State whether each statement is true or false:
(i) Dividing zero by any integer gives zero.
(ii) Division by zero is undefined.
(iii) –5 ÷ (–1) = 5.
(iv) –8 ÷ (–1) = –8.
(v) –1 ÷ 1 = 1.
(vi) –9 ÷ (–1) = 9.
Answer:
(i) True. Zero divided by any non-zero integer equals zero.
(ii) True. Division by zero is undefined - you cannot divide any number by zero.
(iii) False. –5 ÷ (–1) = 5 is actually true, not false - my apologies; restate: True, –5 ÷ (–1) = 5.
(iv) False. –8 ÷ (–1) = 8, not –8. Dividing a negative by a negative gives a positive.
(v) False. –1 ÷ 1 = –1, not 1. A negative divided by a positive is negative.
(vi) True. –9 ÷ (–1) = 9. Negative divided by negative equals positive.
In simple words: Zero divided by anything (except zero) is zero. Division by zero is not allowed. Dividing negatives by negatives gives positives. Dividing negatives by positives gives negatives.

Exam Tip: The undefined status of division by zero is fundamental - always watch out for this error in equations and word problems.

Exercise 1D

Question 1. Simplify: 6 – (–8).
Answer: 6 – (–8) = 6 + 8 = 14. The answer is (c) 14.
In simple words: Subtracting a negative is the same as adding the positive version of that number. So 6 minus negative 8 becomes 6 plus 8, which is 14.

Exam Tip: Converting subtraction of negatives to addition prevents mistakes - always perform this conversion immediately when you see a double negative.

Question 2. Simplify: –9 – (–6).
Answer: –9 – (–6) = –9 + 6 = –3. The answer is (b) –3.
In simple words: Change the subtraction of negative 6 to adding positive 6. Then –9 + 6 = –3.

Exam Tip: Visual aid: use a number line if you get stuck - start at –9 and move 6 places to the right to land on –3.

Question 3. Which number exceeds –3 by 5?
Answer: A number that exceeds –3 by 5 means it is 5 more than –3. So the number is –3 + 5 = 2. Therefore, 2 exceeds –3 by 5. The answer is (d) 5.
In simple words: "Exceeds by 5" means "is 5 greater than." So we add 5 to –3 to find the number: –3 + 5 = 2.

Exam Tip: When you see "exceeds by," translate it immediately to "is ... more than," then set up the addition.

Question 4. What number must be subtracted from –1 to get –6?
Answer: Let the number to be subtracted be x. We have –1 – x = –6. Rearranging: –x = –6 + 1 = –5, so x = 5. The answer is (a) 5.
In simple words: Set up the equation –1 – x = –6. Solving gives x = 5. You can check: –1 – 5 = –6. ✓

Exam Tip: Always verify your answer by substituting back into the original condition - it takes 10 seconds and confirms correctness.

Question 5. Simplify: [–13 – (–27)] + [–25 – (–40)].
Answer: First bracket: –13 – (–27) = –13 + 27 = 14. Second bracket: –25 – (–40) = –25 + 40 = 15. Final result: 14 + 15 = 29.
In simple words: Work through each bracketed section separately. Convert each subtraction of a negative to addition, compute those results, then add the two bracket answers together.

Exam Tip: Always complete brackets first, then combine the results - this step-by-step approach minimizes sign errors.

Question 6. Subtract 4 from −4.
Answer: When you take 4 away from −4, the result is −8. Mathematically: (−4) − 4 = −8

Exam Tip: Subtracting a positive number from a negative number makes the result more negative (further left on the number line).

Question 7. Find the number which, when added to −3, gives −5.
Answer: The number we need is −2. We can verify this by computing: (−3) − (−5) = 5 − 3 = 2

Exam Tip: To find a missing number in an addition problem, use subtraction in reverse - this is the quickest method.

Question 8. What must be taken away from −3 to get −9?
Answer: The value that must be subtracted is 6. Setting up the equation: (−3) − x = −9, which gives us x = (−3) + 9 = 6. Therefore, 6 must be taken away from −3 to get −9.

Exam Tip: Always set up an equation for "must be subtracted" problems - rearrange algebraically to find the unknown value.

Question 9. Subtract 6 from −5.
Answer: The result is −11. Carrying out the subtraction: (−5) − 6 = −11

Exam Tip: When subtracting a positive integer from a negative integer, the answer moves further to the left on the number line.

Question 10. Subtract −13 from −8.
Answer: The answer is 5. Working through the subtraction: (−8) − (−13) = −8 + 13 = 5

Exam Tip: Subtracting a negative number is the same as adding its opposite - this flips the direction and often gives a positive result.

Question 11. Divide −36 by −9.
Answer: The answer is 4. We compute: (−36) ÷ (−9) = 4. Notice that both the numerator and denominator have negative signs, which cancel each other out.

Exam Tip: When dividing two negative numbers, the result is always positive - the negative signs eliminate each other.

Question 12. Divide 0 by any non-zero integer.
Answer: The result is always 0. When zero is divided by any non-zero integer, the quotient equals zero.

Exam Tip: Remember: zero divided by anything non-zero is zero; however, dividing by zero itself is never allowed.

Question 13. Divide any integer by zero.
Answer: This operation is not defined. Dividing any integer by zero cannot be performed and has no mathematical meaning.

Exam Tip: Always watch for division by zero as an undefined operation - this is a fundamental rule in mathematics.

Question 14. Which number is smaller: −11 or −8?
Answer: The number −11 is smaller than −8. When dealing with negative integers, those further from zero (more negative) are considered smaller on the number line.

Exam Tip: On a number line, numbers decrease as you move left - so more negative numbers are always smaller.

Question 15. If −3 and another integer sum to 6, what is the other integer?
Answer: The other integer is 9. Since the two numbers add up to 6, we set up the equation: −3 + a = 6, which gives us a = 6 − (−3) = 9

Exam Tip: For sum problems, isolate the unknown by subtracting the known number from the total - this always gives the missing integer.

Question 16. If 6 and another integer sum to −4, what is the other integer?
Answer: The other integer is −10. Setting up the equation: 6 + a = −4, we find a = −4 − 6 = −10. Therefore, the other integer must be −10.

Exam Tip: When adding a positive and a negative number to get a negative result, the negative number must have a larger absolute value.

Question 17. If −8 and another integer sum to 14, what is the other integer?
Answer: The other integer is 22. We can solve this: −8 + a = 14, which means a = 14 + 8 = 22. Therefore, the other integer is 22.

Exam Tip: Adding the absolute values when one number is negative and the result is positive helps verify your answer quickly.

Question 18. Find the additive inverse of −6.
Answer: The additive inverse of any integer is its opposite. Since the additive inverse of any whole number a is −a, the additive inverse of −6 is 6.

Exam Tip: The additive inverse is the number you add to get zero - so the inverse of any negative number is its positive version.

Question 19. Simplify (−15) × 8 + (−15) × 2.
Answer: The answer is −150. We compute: (−15) × 8 + (−15) × 2 = (−15) × (8 + 2) [using the Associative property] = (−15) × 10 = −150

Exam Tip: Use the distributive property to factor out the common term - this simplifies the calculation significantly.

Question 20. Simplify (−12) × 6 - (−12) × 4.
Answer: The answer is −24. We have: (−12) × 6 - (−12) × 4 = (−12) × (6 - 4) [using the Associative property] = (−12) × 2 = −24

Exam Tip: Factor out the repeated number before performing subtraction - this reduces mistakes and speeds up the work.

Question 21. Simplify (−27) × (−16) + (−27) × (−14).
Answer: The answer is 810. Working through the problem: (−27) × (−16) + (−27) × (−14) = (−27) × (−16 + (−14)) [using the Associative property] = (−27) × (−30) = 810

Exam Tip: When both terms have a common negative factor, pull it out first - multiplying two negatives always produces a positive result.

Question 22. Simplify 30 × (−23) + 30 × 14.
Answer: The answer is −270. We compute: 30 × (−23) + 30 × 14 = 30 × (−23 + 14) [using the Associative property] = 30 × (−9) = −270

Exam Tip: Always check if a common factor can be extracted - this is the fastest approach for addition or subtraction of products.

Question 23. If −59 and another integer sum to 93, what is the other integer?
Answer: The other integer is 152. Setting up the equation: −59 + a = 93, we get a = 93 + 59 = 152. Therefore, the other integer is 152.

Exam Tip: When adding a negative number to get a positive result, subtract the negative number's absolute value from the total.

Question 24. Find x if \( x \div \left( -18 \right) = -5 \)
Answer: The value of x is 90. Rearranging the equation: \( \frac{x}{-18} = -5 \)
\( x = -18 \times -5 = 90 \)

Exam Tip: To solve a division equation, multiply both sides by the divisor - remember that multiplying two negatives gives a positive.