Operations with Integers Class 7 Free Notes and Mind Map (Free PDF Download)

This chapter covers how to work with positive and negative numbers (integers) through addition, subtraction, multiplication, and division. You’ll learn patterns, rules, and real-life examples to make integer operations easier and more intuitive.

Rakesh’s Puzzle: A Number Game

Rakesh gives you a challenge: “I have thought of two numbers. Their sum is 25, and their difference is 11.”
You can find the two numbers by trying different pairs and checking if they add up to 25 and their difference is 11.
Method: Write your guesses in a table with columns for First Number, Second Number, Sum, Difference.

First Number	Second Number	Sum	Difference
10	15	25	-5
20	5	25	15
19	6	25	13
18	7	25	11

The correct pair is 18 and 7 (because 18 + 7 = 25 and 18 – 7 = 11).
Second challenge: “Think of two numbers whose sum is 25, but their difference is –11.”
If you swap the numbers from the first puzzle, you get the answer to the second puzzle: first number is 7 and second is 18 (7 + 18 = 25 and 7 – 18 = -11).

Question: Figure it Out

Find pairs of numbers from their sums and differences:

Question (a): Sum = 27, Difference = 9

Solution: First number = 18, Second number = 9 (18 + 9 = 27, 18 – 9 = 9).

Question (b): Sum = 4, Difference = 12

Solution: First number = 8, Second number = -4 (8 + (-4) = 4, 8 – (-4) = 12).

Question (c): Sum = 0, Difference = 10

Solution: First number = 5, Second number = -5 (5 + (-5) = 0, 5 – (-5) = 10).

Question (d): Sum = 0, Difference = −10

Solution: First number = -5, Second number = 5 (-5 + 5 = 0, -5 – 5 = -10).

Question (e): Sum = -7, Difference = −1

Solution: First number = -4, Second number = -3 (-4 + (-3) = -7, -4 – (-3) = -1).

Question (f): Sum = -7, Difference = −13

Solution: First number = -10, Second number = 3 (-10 + 3 = -7, -10 – 3 = -13).

Carrom Coin Integers

A carrom coin is struck to move it right or left on a number line, each strike moves the coin by some units based on force.
The coin starts at point 0, if struck twice with first strike moving it 4 units right and second strike moving it 3 units right, final position is 4 + 3 = 7 units from 0.
Formula: If first strike moves coin ‘a’ units right and second strike ‘b’ units right, final position P = a + b.
Now if the coin can move in either direction (left or right), we use positive for rightward and negative for leftward movement.
Suppose first strike moves coin 5 units right (movement = 5) and second strike 7 units left (movement = -7), final position = 5 + (-7) = -2 (coin is 2 units left of 0).

General formula: 
If first strike moves coin 'a' units (positive = right, negative = left)
and second strike 'b' units (positive = right, negative = left),
then final position P = a + b

Questions Based on Carrom Coin Model

Question 1: If the first movement is –4 and the final position is 5, what is the second movement?

Solution: Final position P = a + b, so 5 = -4 + b, therefore b = 5 – (-4) = 9. Second movement is 9.

Question 2: If there are multiple strikes causing movements in the order 1, – 2, 3, -4, …, – 10, what is the final position of the coin?

Solution: Add all movements: 1 + (-2) + 3 + (-4) + 5 + (-6) + 7 + (-8) + 9 + (-10) = (1 + 3 + 5 + 7 + 9) + (-2 – 4 – 6 – 8 – 10) = 25 + (-30) = -5. Final position is -5.

Token Model Recap

Green token (◉) represents positive 1, red token (◕) represents negative 1 (that is –1). Together they make zero as they cancel each other.
To find (+7) – (+18), we need to remove 18 positives from 7 positives, but there aren’t enough tokens.
We add 11 zero pairs so we have enough positives to remove 18, after removing 18 positives, 11 negatives remain, so 7 – 18 = −11.
Subtracting a number is same as adding its additive inverse.

Questions Using Tokens

Question (a): Argue 7-18 = 7 + (−18)

Solution: Subtracting 18 is same as adding additive inverse of 18 which is -18, so 7 – 18 = 7 + (-18).

Question (b): Argue 4-(-12) = 4 + 12

Solution: Subtracting -12 is same as adding additive inverse of -12 which is 12, so 4 – (-12) = 4 + 12.

Additive Inverse

Additive inverse of an integer a is represented as –a, so additive inverse of 18 is –(18) = −18, and additive inverse of –18 is –(-18) = 18.

Multiplication of Integers

Token Model for Multiplication

We use token model to understand multiplication of integers.
Putting 2 positive tokens into an empty bag 4 times means 4 × 2 = 8 (there are 8 positives in bag).
For 4 × (-2), we place 2 negatives (red tokens) into empty bag 4 times, result is 8 negatives in bag = -8, so 4 × (-2) = -8.
For (-4) × 2, we remove 2 positives from bag 4 times, but bag is empty so we first place 2 zero pairs and remove 2 positives, do this 4 times, 8 negatives remain in bag, so (−4) × 2 = -8.
For (-4) × (−2), we remove 2 negatives from bag 4 times, place 2 zero pairs and remove 2 negatives, do this 4 times, 8 positives remain in bag = +8, so (-4) × (-2) = 8.

Results established:
4 × 2 = 8
4 × (-2) = -8
(-4) × 2 = -8
(-4) × (-2) = 8

Questions: Figure it Out

Question 1: Using the token interpretation, find the values of:

(a) 3 × (-2)

Solution: Place 2 negatives into bag 3 times, result = 6 negatives = -6.

(b) (-5) × (-2)

Solution: Remove 2 negatives from bag 5 times (using zero pairs), result = 10 positives = 10.

(c) (-4) × (-1)

Solution: Remove 1 negative from bag 4 times (using zero pairs), result = 4 positives = 4.

(d) (-7)×3

Solution: Remove 3 positives from bag 7 times (using zero pairs), result = 21 negatives = -21.

Question 2: If 123 × 456 = 56088, without calculating, find the value of:

(a) (-123) × 456

Solution: One is negative and one is positive, so product is negative = -56088.

(b) (-123) × (-456)

Solution: Both are negative, so product is positive = 56088.

(c) (123) × (-456)

Solution: One is positive and one is negative, so product is negative = -56088.

Question 3: Try to frame a simple rule to multiply two integers.

Solution: When both integers have same sign (both positive or both negative), product is positive. When integers have different signs (one positive and one negative), product is negative. Magnitude of product depends on magnitudes of the two integers.

Patterns in Integer Multiplication

When multiplicand is positive, for every unit decrease in multiplier, product decreases by multiplicand.

Pattern 1 (multiplicand positive):
4 × 3 = 12
3 × 3 = 9 (decrease by 3)
2 × 3 = 6 (decrease by 3)
1 × 3 = 3 (decrease by 3)
0 × 3 = 0 (decrease by 3)
-1 × 3 = -3 (decrease by 3)
-2 × 3 = -6 (decrease by 3)
-3 × 3 = -9 (decrease by 3)

When multiplicand is negative, for every unit decrease in multiplier, product increases by multiplicand.

Pattern 2 (multiplicand negative):
4 × (-3) = -12
3 × (-3) = -9 (increase by 3)
2 × (-3) = -6 (increase by 3)
1 × (-3) = -3 (increase by 3)
0 × (-3) = 0 (increase by 3)
(-1) × (-3) = 3 (increase by 3)
(−2) × (-3) = 6 (increase by 3)
(−3) × (-3) = 9 (increase by 3)

Times 3 Tables (Positive and Negative)

Positive × Positive	Negative × Positive	Positive × Negative	Negative × Negative
1 × 3 = 3	-1 × 3 = -3	1 × -3 = -3	−1 × -3 = 3
2 × 3 = 6	-2 × 3 = -6	2 × −3 = -6	-2 × -3 = 6
3 × 3 = 9	-3 × 3 = -9	3 × -3 = -9	-3 × -3 = 9
4 × 3 = 12	-4 × 3 = -12	4 × -3 = −12	-4 × -3 = 12
5 × 3 = 15	-5 × 3 = -15	5 × -3 = -15	-5 × -3 = 15
6 × 3 = 18	-6 × 3 = -18	6 × -3 = -18	-6 × -3 = 18
7 × 3 = 21	−7 × 3 = −21	7 × -3 = −21	-7 × -3 = 21
8 × 3 = 24	-8 × 3 = -24	8 × −3 = −24	-8 × -3 = 24
9 × 3 = 27	-9 × 3 = -27	9 × -3 = -27	-9 × -3 = 27
10 × 3 = 30	-10 × 3 = -30	10 × -3 = -30	-10 × -3 = 30

Observations from Multiplication Tables

Magnitude of product does not change with change in signs of multiplier and multiplicand.
When both multiplier and multiplicand are positive, product is positive.
When both multiplier and multiplicand are negative, product is positive.
When one of multiplier or multiplicand is positive and other is negative, product is negative.

Questions: Figure it Out

Find the following products:

(a) 4× (-3)

Solution: 4 × (-3) = -12.

(b) (-6) × (-3)

Solution: (-6) × (-3) = 18.

(c) (-5) × (-1)

Solution: (-5) × (-1) = 5.

(d) (-8) × 4

Solution: (-8) × 4 = -32.

(e) (-9) × 10

Solution: (-9) × 10 = -90.

(f) 10 × (-17)

Solution: 10 × (-17) = -170.

Special Cases of Multiplication

For expression 1 × a, value is ‘a’ for all positive integers, also true for all negative integers.

1 × a = a (for all integers a, both positive and negative)

For expression –1 × a, product is additive inverse of multiplicand ‘a’.

text−1 × a = − a (for all integers a)

Commutative Property

Product is same when we swap multiplier and multiplicand.

Examples:

3 × -4 = -12 and -4 × 3 = -12
-30 × 12 = -360 and 12 × -30 = -360
-15 × -8 = 120 and -8 × -15 = 120
14 × − 5 = −70 and -5 × 14 = -70

Multiplication is commutative for integers.

For any two integers, a and b:
a × b = b × a

Brahmagupta’s Rules for Multiplication and Division

Brahmagupta in his Brāhmasphuṭasiddhānta (628 CE) gave explicit rules for integer multiplication and division using notions of fortune (dhana) for positive and debt (ṛṇa) for negative.
Rules: “The product or quotient of two fortunes is a fortune. The product or quotient of two debts is a fortune. The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt.”
This was first time rules for multiplication and division of positive and negative numbers were written, important step in development of arithmetic and algebra.

Example 1: Multiple Choice Questions

Question: An exam has 50 multiple choice questions. 5 marks are given for every correct answer and 2 negative marks for every wrong answer. What are Mala’s total marks if she had 30 correct answers and 20 wrong answers?

Solution:

Mark for each correct answer = positive integer 5, for each wrong answer = negative integer –2.
Marks for 30 correct answers = 30 × 5 = 150.
Marks for 20 wrong answers = 20 × (–2) = -40.
Total marks = 30 × 5 + 20 × (–2) = 150 + (–40) = 110.
Mala got 110 marks in exam.

Additional questions:

Maximum possible marks in exam = 50 × 5 = 250 (all correct).
Minimum possible marks = 50 × (-2) = -100 (all wrong).

Example 2: Elevator Problem

Question: There is an elevator in a mining shaft that moves above and below the ground. Elevator’s positions above ground are positive integers and positions below ground are negative integers.

(a) Elevator moves 3 metres per minute. If it descends into shaft from ground level (0), what will be its position after one hour?

Solution (Method 1):

Elevator moves 3 metres per minute, in one hour (60 minutes) it moves 60 × 3 = 180 metres.
Started at ground level (0 metres) and descended, so subtract 180 from 0: 0 – 180 = -180.
Elevator will reach (-180) metre position, which is 180 metres below ground.

Solution (Method 2):

Speed and direction represented by integer (metres per minute), +3 when moving up, (-3) when moving down.
Elevator moving down, speed is (-3) metres per minute for 60 minutes, so 60 × (-3) = (-180).
Position of elevator after 60 minutes is 180 metres below ground level.

(b) If it begins to descend from 15 m above the ground, what will be its position after 45 minutes?

Solution (Method 2):

Starting Position = 15.
Distance Travelled = Elevator moves down at 3 metres per minute for 45 minutes = 45 × (−3).
Ending Position = 15 + (45 × (-3)) = 15 + (−135) = (-120).
Elevator will be 120 metres below ground.

A Magic Grid of Integers

Grid contains numbers, you circle any number, strike out its row and column, circle any unstruck number, repeat until no numbers left, multiply all circled numbers.
No matter which numbers you choose, product remains same.
Magic is in the way numbers are arranged in grid.

Division of Integers

Division can be converted into multiplication.
For example, (–100) ÷ 25 can be reframed as “what should be multiplied to 25 to get (–100)?”, that is 25 × ? = (–100).
We know 25 × (−4) = (−100), therefore (−100) ÷ 25 = (−4).
Similarly, (-100) ÷ (− 4) can be reframed as “What should be multiplied to (-4) to get (–100)?”, that is (−4) × ? = (−100).
We know (−4) × 25 = (−100), therefore (-100) ÷ (-4) = 25.
Also, we know (-25) × (-2) = 50, therefore 50 ÷ (-25) = (-2).

Rules for integer division (where b ≠ 0):
a ÷ - b = -(a ÷ b)
− a ÷ b = − (a ÷ b)
-a ÷ -b = a ÷ b

Questions: Figure it Out

Question 1: Find the values of:

(a) 14 × (-15)

Solution: 14 × (-15) = -210.

(b) -16× (-5)

Solution: -16 × (-5) = 80.

(c) 36 ÷ (-18)

Solution: 36 ÷ (-18) = -2.

(d) (-46) ÷ (-23)

Solution: (-46) ÷ (-23) = 2.

Question 2: A freezing process requires that the room temperature be lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Solution:

Starting temperature = 32°C.
Rate of decrease = 5°C per hour.
After 10 hours, temperature decrease = 10 × 5 = 50°C.
Final temperature = 32 – 50 = -18°C.

Question 3: A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bag of grey cement sold.

(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?

Solution:

Profit from white cement = 3000 × 8 = ₹24,000.
Loss from grey cement = 5000 × (-5) = -₹25,000.
Total = 24,000 + (-25,000) = -₹1,000 (loss of ₹1,000).

(b) If the number of bags of grey cement sold is 6,400 bags, what is the number of bags of white cement the company must sell to have neither profit nor loss?

Solution:

Loss from 6,400 bags of grey cement = 6400 × (-5) = -₹32,000.
To have no profit or loss, profit from white cement should be ₹32,000.
Number of white cement bags = 32,000 ÷ 8 = 4,000 bags.

Question 4: Replace the blank with an integer to make a true statement.

(a) (-3) × ___ = 27

Solution: -3 × (-9) = 27, so blank = -9.

(b) 5 × ___ = (-35)

Solution: 5 × (-7) = -35, so blank = -7.

(c) ___ × (-8) = (-56)

Solution: 7 × (-8) = -56, so blank = 7.

(d) ___ × (-12) = 132

Solution: (-11) × (-12) = 132, so blank = -11.

(e) ___ ÷ (-8) = 7

Solution: (-56) ÷ (-8) = 7, so blank = -56.

(f) ___ ÷ 12 = -11

Solution: (-132) ÷ 12 = -11, so blank = -132.

Expressions Using Integers

Associative Property

Expression 5 × − 3 × 4 can be evaluated in different ways.

(5 × -3) × 4 = -15 × 4 = -60
5 × (-3 × 4) = 5 × -12 = -60

Product is same when we group multiplications in these two ways, so integer multiplication is associative.

For any three integers a, b, and c:
a × (b × c) = (a × b) × c

Product remains same when 3 or more numbers are multiplied in any order.

Patterns with -1 Multiplication

-1 × −1 = 1
-1 × −1 × −1 = -1
-1 × −1 × −1 × − 1 = 1
-1 × −1 × −1 × −1 × −1 = -1

When –1 is multiplied 2 or 4 times (even number of times), product is positive.
When it is multiplied 3 or 5 times (odd number of times), product is negative.

Rule to find sign of product of many integers:

If number of negative integers is even, product is positive.
If number of negative integers is odd, product is negative.

Distributive Property

Expression 5 × (4 + (−2)) equals 5 × 4 + 5 × (-2), this is distributive property.
Distributive property also holds for integers.

For any integers a, b, and c:
a × (b + c) = (a × b) + (a × c)

Can be visually shown using rectangular arrangement of tokens (green for positive, red for negative).

Pick the Pattern

Machine 1 operations:

Operation done by Machine 1 is (first number) + (second number) – (third number).
Written as expression: a + b − c, where a is first number, b is second number, c is third number.
Example: 5 + 8 − 3 = 10 and (−4) + (− 1) − (−6) = 1.
Result of (–10) + (−12) – (− 9) = -10 – 12 + 9 = -13.

Questions: Figure it Out

Question 1: Find the values of the following expressions:

(a) (-5) × (18 + (−3))

Solution: (-5) × (18 – 3) = (-5) × 15 = -75.

(b) (-7) × 4 × (-1)

Solution: (-7) × 4 × (-1) = -28 × (-1) = 28.

(c) (-2) × (-1) × (-5) × (-3)

Solution: Four negative numbers, product is positive = 2 × 1 × 5 × 3 = 30.

Question 2: Find the values of the following expressions:

(a) (-27)÷9

Solution: (-27) ÷ 9 = -3.

(b) (56) ÷ (-2)

Solution: 56 ÷ (-2) = -28.

Question 4: If 47 – 56 + 14 – 8 + 2 – 8 + 5 = −4, then find the value of −47 + 56 – 14 +8-2+8-5 without calculating the full expression.

Solution:

Second expression is negative of first expression, so if first = -4, then second = -(-4) = 4.

Question 5: Modified Collatz Conjecture with integers – start with any number, if even take half, if odd multiply by –3 and add 1, repeat.

Example sequence: -7 → 22 → 11 → -32 → -16 → -8 → -4 → -2 → -1 → 4 → 2 → 1 (pattern observed).

Question 6: In a test, (+4) marks for every correct answer and (−2) marks for every incorrect answer.

(a) Anita answered all questions. She scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? How many questions are in test?

Solution:

Marks from 15 correct = 15 × 4 = 60.
Total marks = 40, so marks from incorrect = 40 – 60 = -20.
Number of incorrect = -20 ÷ (-2) = 10.
Total questions = 15 + 10 = 25.

(b) Anil scored (–10) marks even though he had 5 correct answers. How many of his answers were incorrect? Did he leave any questions unanswered?

Solution:

Marks from 5 correct = 5 × 4 = 20.
Total marks = -10, so marks from incorrect = -10 – 20 = -30.
Number of incorrect = -30 ÷ (-2) = 15.
He answered 5 + 15 = 20 questions, if test has 25 questions, he left 5 unanswered.

Question 7: Pick the pattern — find the operations done by the machine shown.

Various input-output pairs given, need to identify pattern.

Question 8: Imagine you’re in a place where the temperature drops by 5°C each hour. If the temperature is currently at 8°C, write an expression which denotes the temperature after 4 hours.

Solution:

Expression: 8 + 4 × (-5) or 8 – 4 × 5.

Question 9: Find 3 consecutive numbers with a product of (a) – 6, (b) 120.

Solution:

(a) Product = -6: consecutive numbers are -3, -2, -1 (product = -6).
(b) Product = 120: try 4, 5, 6 (product = 120).

Question 10: Alien currency ‘pibs’ with two denominations: a+13 pibs coin and a −9 pibs coin. Is it possible to purchase an item that costs + 85 pibs?

Solution:

Yes, use 10 coins of +13 pibs and 5 coins of –9 pibs: 10 × 13 + 5 × (-9) = 130 – 45 = 85.

Try to get following totals:

(a) +20

Solution: Find combination using 13 and -9 multiples.

(b) +40

Solution: Find combination using 13 and -9 multiples.

(c) -50

Solution: Use more -9 coins than +13 coins.

(d) +8

Solution: Find combination.

(e) +10

Solution: Find combination.

(f) -2

Solution: Find combination.

(g) +1

Solution: Find combination using multiples of 13 and 9.

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

Solution: Check if 1568 can be expressed as 13a + (-9b) for some positive integers a and b.

Question 11: Find the values of:

(a) (32 × (-18)) ÷ ((−36))

Solution: (32 × (-18)) ÷ (-36) = -576 ÷ (-36) = 16.

(b) (32) ÷ ((−36) × (-18))

Solution: 32 ÷ ((-36) × (-18)) = 32 ÷ 648 = 0.049… (not integer).

(c) (25 × (-12)) ÷ ((45) × (-27))

Solution: (25 × (-12)) ÷ (45 × (-27)) = -300 ÷ (-1215) = 0.247… (not integer).

(d) (280 × (-7)) ÷ ((−8) × (−35))

Solution: (280 × (-7)) ÷ ((-8) × (-35)) = -1960 ÷ 280 = -7.

Question 12: Arrange the expressions given below in increasing order:

(a) (-348) + (−1064)

Value: -348 – 1064 = -1412.

(b) (-348) × (-1064)

Value: positive number = 370272.

(c) 348-(-1064)

Value: 348 + 1064 = 1412.

(d) (-348) × 964

Value: negative number = -335472.

(e) 348 × (-1064)

Value: negative number = -370272.

(f) 348 × 964

Value: positive number = 335472.

Increasing order: (e) < (d) < (a) < (c) < (f) < (b).

Question 13: Given that (-548) × 972 = – 532656, write the values of:

(a) (-547) × 972

Solution: (-547) × 972 = [(-548) + 1] × 972 = (-548) × 972 + 972 = -532656 + 972 = -531684.

(b) (-548) × 971

Solution: (-548) × 971 = (-548) × (972 – 1) = (-548) × 972 – (-548) = -532656 + 548 = -532108.

(c) (-547) × 971

Solution: (-547) × 971 = [(-548) + 1] × (972 – 1) = (-548) × 972 – (-548) + 972 – 1 = -532656 + 548 + 972 – 1 = -531137.

Question 14: Given that 207 × (−33 + 7) = -5382, write the value of – 207 × (33 – 7).

Solution:

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

Question 15: Use the numbers 3, – 2, 5, – 6 exactly once and the operations ‘+’, ‘-‘, and ‘x’ exactly once and brackets as necessary to write an expression such that:

(a) the result is the maximum possible

Solution: To maximize, multiply largest positive numbers and add, try (5 × 3) – (-6) – (-2) = 15 + 6 + 2 = 23.

(b) the result is the minimum possible

Solution: To minimize, multiply to get largest negative, try (-6) × 5 – 3 + (-2) = -30 – 3 – 2 = -35.

Question 16: Fill in the blanks in at least 5 different ways with integers:

(a) ___ + ___ = -36

Solutions: -20 + (-16) = -36, -30 + (-6) = -36, -18 + (-18) = -36, -40 + 4 = -36, -50 + 14 = -36.

(b) ___ × ___ = -1

Solutions: 1 × (-1) = -1, (-1) × 1 = -1, but only two integer pairs work.

(c) ___ × ___ = 12

Solutions: 3 × 4 = 12, 2 × 6 = 12, 1 × 12 = 12, (-3) × (-4) = 12, (-2) × (-6) = 12.

IT’S PUZZLE TIME!

Terhüchü

Terhüchü is a game played in Assam and Nagaland on a board with 16 squares and diagonals marked.
Usually scratched on large piece of stone or drawn on mud.
There are 2 players, each player has set of 9 coins placed as shown, coins in one set look different from other set.

Objective:

Goal is to capture all opponent’s coins, first player to do so is winner.
Player may also win by blocking any legal move by opponent.
If draw seems unavoidable, player with more coins wins.

Gameplay:

Starting position of game is as shown.
Players take turns, in each turn they can move a single coin along a line in any direction to a neighbouring vacant intersection.
Or, if opponent’s coin is at neighbouring intersection and there is vacant intersection just beyond it, they can jump over opponent’s coin and land in vacant intersection.
If player jumps over opponent’s coin, it is considered captured and removed from board.
Multiple captures in one move are allowed, direction can change after each jump.
Inside triangular corners (outside main square), coin may skip an intersection and move straight to next one, that is it can jump over empty intersection and go to one beyond it.

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