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CBSE · Class 7 · Mathematics

Chapter 10: Operations with Integers

5 concepts · 40 practice questions with answers · Free to read

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Adding and subtracting integers

From daily life

A lift at floor 3 goes down 5 floors. A bank balance of ₹100 falls by ₹150. Moves up and down are positive and negative numbers.

Simple definition

Integers are …, –2, –1, 0, 1, 2, … To add a positive number move right on the number line; to add a negative number move left. Subtracting a number is the same as adding its opposite.

–8–6–4–2024683–23 + (–5) = –2

In the figure: Start at 3, move 5 left, land at –2.

Live figure

Adding and subtracting integers

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Rule 1: Same signs: add and keep the sign — Add the sizes and keep the common sign.

  • Easy: (–4) + (–9) = –13.
  • Medium: 6 + 8 = 14.
  • Exam-type: (–15) + (–25) = –40.

Rule 2: Different signs: subtract and keep the bigger one's sign — Subtract the smaller size from the larger; use the sign of the larger.

  • Easy: (–7) + 12 = 5.
  • Medium: 8 + (–13) = –5.
  • Exam-type: (–20) + 20 = 0.

Rule 3: Subtract = add the opposite — a – b becomes a + (opposite of b). Two minus signs make a plus.

  • Easy: 5 – 9 = 5 + (–9) = –4.
  • Medium: (–5) – (–8) = (–5) + 8 = 3.
  • Exam-type: 14 – (–9) = 14 + 9 = 23.
Remember

Minus a negative is a plus.

Common mistake

Writing 8 + (–13) = 5. Subtract 8 from 13 and give the sign of 13: –5.

Shortcut

Draw a quick number line and walk the steps. Memory clue: "Two minuses make a plus."

Spot the difference

CaseResult sign
(–) + (–)–
(+) + (+)+
(–) + (+)sign of bigger size

Practice questions

Easy
  1. 1Find (–6) + 9.
  2. 2Find 5 + (–11).
  3. 3Find (–8) + (–7).
Medium
  1. 4Find 12 – 20.
  2. 5Find (–4) – (–10).
  3. 6Find 15 – (–8).
Challenging
  1. 7Find (–30) + 18 – (–5).
  2. 8Find 25 – (–12) + (–40).
Show answers (try them yourself first)
  1. 13
  2. 2–6
  3. 3–15
  4. 4–8
  5. 56
  6. 623
  7. 7–7
  8. 8–3

Multiplying integers

From daily life

Losing ₹5 each day for 3 days is a total change of 3 × (–5) = –15. A repeated loss is repeated addition of a negative number.

Simple definition

Multiply the sizes, then decide the sign: same signs give +, different signs give –. Anything times 0 is 0. With many factors, count the negative ones: an even count gives +, an odd count gives –.

(+) × (+) = +(–) × (–) = +(+) × (–) = –(–) × (+) = –

In the figure: Four sign combinations for multiplication.

Live figure

Multiplying integers

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Rule 1: Two numbers — Multiply the sizes; apply the sign rule.

  • Easy: (–6) × 7 = –42.
  • Medium: (–4) × (–9) = 36.
  • Exam-type: 8 × (–5) = –40.

Rule 2: Many factors — Count the negatives: even count +, odd count –.

  • Easy: (–2) × (–3) × (–4) = –24 (three negatives).
  • Medium: (–5) × 6 × (–2) = 60 (two negatives).
  • Exam-type: (–1) × (–1) × (–1) × (–1) = 1.

Rule 3: Zero and one — 0 times anything is 0; 1 times a number leaves it unchanged.

  • Easy: 0 × (–12) = 0.
  • Medium: (–9) × 1 = –9.
  • Exam-type: (–3) × 0 × (–50) = 0.
Remember

Same signs +, different signs –.

Common mistake

Thinking (–3) × (–3) = –9. Two negatives give a positive: +9.

Shortcut

Count the minus signs first, then multiply the sizes. Memory clue: "Friends' friend is a friend; enemy's enemy is a friend."

Spot the difference

Negatives in the productSign
0, 2, 4+
1, 3, 5–

Practice questions

Easy
  1. 1Find (–5) × 8.
  2. 2Find (–7) × (–6).
  3. 3Find 9 × (–4).
Medium
  1. 4Find (–12) × (–3).
  2. 5Find (–2) × (–5) × (–3).
  3. 6Find 6 × (–4) × (–2).
Challenging
  1. 7Find (–3) × (–3) × (–3) × (–3).
  2. 8Find (–1) × (–1) × (–1) × (–1) × (–1) × (–1) × (–1).
Show answers (try them yourself first)
  1. 1–40
  2. 242
  3. 3–36
  4. 436
  5. 5–30
  6. 648
  7. 781
  8. 8–1

Dividing integers

From daily life

A debt of ₹96 shared equally by 8 friends is –96 ÷ 8 = –12 each. Sharing is the reverse of multiplying.

Simple definition

Division follows the same sign rule as multiplication: same signs give +, different signs give –. This works because division undoes multiplication: (–9) × 8 = –72, so (–72) ÷ 8 = –9. Dividing by 0 is not allowed.

(–72) ÷ 8 = –9(–72) ÷ (–8) = 972 ÷ (–8) = –972 ÷ 8 = 9

In the figure: Division uses the same four sign combinations.

Live figure

Dividing integers

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Rule 1: Sign rule — Divide the sizes, then use the sign rule.

  • Easy: (–48) ÷ 6 = –8.
  • Medium: 54 ÷ (–9) = –6.
  • Exam-type: (–35) ÷ (–7) = 5.

Rule 2: Check by multiplying — Answer × divisor must give the dividend back.

  • Easy: (–72) ÷ 8 = –9, check (–9) × 8 = –72.
  • Medium: (–100) ÷ 25 = –4, check (–4) × 25 = –100.
  • Exam-type: (–90) ÷ (–15) = 6, check 6 × (–15) = –90.

Rule 3: Zero in division — 0 ÷ (any non-zero number) = 0, but nothing can be divided by 0.

  • Easy: 0 ÷ (–8) = 0.
  • Medium: (–6) ÷ 0 is not defined.
  • Exam-type: Left to right: (–3) × (–8) ÷ (–4) = 24 ÷ (–4) = –6.
Remember

Division follows the multiplication sign rule.

Common mistake

Saying (–20) ÷ (–4) = –5. Same signs give +5.

Shortcut

Do the sign first, then the numbers. Memory clue: "Divide like you multiply."

Spot the difference

DivisionResult
0 ÷ 50
5 ÷ 0not defined

Practice questions

Easy
  1. 1Find (–56) ÷ 7.
  2. 2Find 63 ÷ (–9).
  3. 3Find (–45) ÷ (–5).
Medium
  1. 4Find (–84) ÷ 12.
  2. 5Find 0 ÷ (–15).
  3. 6Find (–6) × (–10) ÷ (–4).
Challenging
  1. 7Find (–120) ÷ 8 ÷ (–3).
  2. 8Find (–18) × (–4) ÷ (–9) × (–2).
Show answers (try them yourself first)
  1. 1–8
  2. 2–7
  3. 39
  4. 4–7
  5. 50
  6. 6–15
  7. 75
  8. 816

Properties of integer operations

From daily life

Splitting a bill: 4 × (3 + 5) gives the same as 4 × 3 + 4 × 5. Smart splitting makes mental sums easy.

Simple definition

Addition and multiplication of integers are commutative (order does not matter) and associative (grouping does not matter). Multiplication distributes over addition. 1 is the multiplicative identity and 0 the additive identity. Subtraction and division are not commutative.

(–4) × (3 + 5)= (–4)×3 + (–4)×5= –12 + (–20) = –32

In the figure: Distribute the multiplier over each term inside.

Live figure

Properties of integer operations

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Rule 1: Commutative and associative — Move or regroup to make easy products.

  • Easy: (–6) × 4 = 4 × (–6) = –24.
  • Medium: (–4) × 17 × 25 = 17 × (–100) = –1700.
  • Exam-type: But 8 – 3 ≠ 3 – 8, and 6 ÷ 2 ≠ 2 ÷ 6.

Rule 2: Distributive property — a × (b + c) = a × b + a × c, also with negative numbers.

  • Easy: (–5) × (–2 + 7) = (–5) × 5 = –25.
  • Medium: (–8) × 5 + (–8) × 3 = (–8) × 8 = –64.
  • Exam-type: (–9) × 102 = (–9) × 100 + (–9) × 2 = –918.

Rule 3: Identities and (–1) — n × 1 = n, n + 0 = n, and n × (–1) = –n (the opposite).

  • Easy: (–1) × 14 = –14.
  • Medium: (–1) × (–31) = 31.
  • Exam-type: (–45) × 1 = –45.
Remember

Use the distributive property to split a hard number: 99 = 100 – 1.

Common mistake

Assuming subtraction is commutative: 8 – 3 is not 3 – 8.

Shortcut

Round numbers like 10, 100, 25 × 4 make products easy. Memory clue: "Spread the multiplier to every term."

Spot the difference

OperationCommutative?
+ , ×Yes
– , ÷No

Practice questions

Easy
  1. 1Find (–1) × (–31).
  2. 2Find (–45) × 1.
  3. 3Find (–7) × (–3) = (–3) × □, □ =.
Medium
  1. 4Find (–6) × (5 + 4).
  2. 5Find (–9) × 13 + (–9) × 7.
  3. 6Find (–8) × 27 × (–5).
Challenging
  1. 7Find 98 × (–7).
  2. 8Find (–25) × 43 × (–4).
Show answers (try them yourself first)
  1. 131
  2. 2–45
  3. 3–7
  4. 4–54
  5. 5–180
  6. 61080
  7. 7–686
  8. 84300

Integer word problems and order of operations

From daily life

A quiz gives +4 for a right answer and –2 for a wrong one. Your score is a mix of products and sums of integers.

Simple definition

Turn the story into an expression, then work in order: brackets, then multiplication and division (left to right), then addition and subtraction (left to right). Gains are positive, losses are negative.

Lift at floor 3, goes down 53 + (–5) = –2= basement level 2

In the figure: A lift story written as an integer sum.

Live figure

Integer word problems and order of operations

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Rule 1: Change problems — Start value plus (change × times).

  • Easy: 9°C falls by 14: 9 – 14 = –5°C.
  • Medium: 4°C falls 3° per hour for 5 hours: 4 + (–3) × 5 = –11°C.
  • Exam-type: Balance –150 plus deposit 400 = 250.

Rule 2: Order of operations — Multiply and divide before adding and subtracting.

  • Easy: (–6) + 4 × (–3) = –6 + (–12) = –18.
  • Medium: (–20) – (–5) × 3 = –20 + 15 = –5.
  • Exam-type: (–36) ÷ 4 – (–7) = –9 + 7 = –2.

Rule 3: Score problems — Score = (right × gain) + (wrong × loss).

  • Easy: +4 right, –2 wrong; 9 right, 6 wrong: 36 – 12 = 24.
  • Medium: Debt –96 shared by 8: –96 ÷ 8 = –12 each.
  • Exam-type: Loss ₹30 a day for 6 days: (–30) × 6 = –180.
Remember

Write the story as an expression, then use BODMAS order.

Common mistake

Adding first: (–6) + 4 × (–3) is not (–2) × (–3). Multiply first.

Shortcut

Underline the multiplications before you start. Memory clue: "Multiply first, then add."

Spot the difference

WordsInteger
rise, profit, deposit+
fall, loss, withdrawal–

Practice questions

Easy
  1. 1Temperature 6°C falls by 11. New temperature (°C)?
  2. 2A lift at floor –2 goes up 7 floors. Floor?
  3. 3A shop loses ₹45 a day for 4 days. Total change (₹)?
Medium
  1. 4+3 for right, –1 for wrong: 12 right and 5 wrong. Score?
  2. 5Find (–10) + 3 × (–4).
  3. 6A debt of ₹240 (–240) is shared equally by 6 people. Each share?
Challenging
  1. 7Find (–36) ÷ 4 – (–7).
  2. 820 questions all attempted: +4 right, –2 wrong, score 38. How many right?
Show answers (try them yourself first)
  1. 1–5
  2. 25
  3. 3–180
  4. 431
  5. 5–22
  6. 6–40
  7. 7–2
  8. 813