This MCQ module is based on: Addition and Subtraction of Integers
TOPIC 4 OF 23
Addition and Subtraction of Integers
🎓 Class 7
Mathematics
CBSE
Theory
Ch 2 — Integers
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Addition and Subtraction of Integers
Targeting Class 7 level in General Mathematics, with Basic difficulty.
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2.1 A Quick Recap of Integers
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." Can you tell me the two numbers?
You don't need any formula. Just try different pairs of numbers and check: (1) Do they add up to 25? (2) Is the difference between them 11? (Remember: difference means first number − second number.)
| First Number | Second Number | Sum | Difference |
|---|---|---|---|
| 10 | 15 | 25 | −5 |
| 20 | 5 | 25 | 15 |
| 19 | 6 | 25 | 13 |
| 18 | 7 | 25 | 11 |
The answer is 18 and 7.
Now Rakesh gives a 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: 7 and 18 (since 7 − 18 = −11).
Figure it Out
Find pairs of numbers from their sums and differences:
(a) Sum = 27, Difference = 9
Answer: Let the numbers be \(x\) and \(y\) with \(x > y\). Then \(x + y = 27\) and \(x - y = 9\). Adding: \(2x = 36\), so \(x = 18\), \(y = 9\). The numbers are 18 and 9.
(b) Sum = 4, Difference = 12
Answer: \(x + y = 4\), \(x - y = 12\). Adding: \(2x = 16\), \(x = 8\), \(y = -4\). The numbers are 8 and −4.
(c) Sum = 0, Difference = 10
Answer: \(x + y = 0\), \(x - y = 10\). Adding: \(2x = 10\), \(x = 5\), \(y = -5\). The numbers are 5 and −5.
(d) Sum = 0, Difference = −10
Answer: \(x + y = 0\), \(x - y = -10\). Adding: \(2x = -10\), \(x = -5\), \(y = 5\). The numbers are −5 and 5.
(e) Sum = −7, Difference = −1
Answer: \(x + y = -7\), \(x - y = -1\). Adding: \(2x = -8\), \(x = -4\), \(y = -3\). The numbers are −4 and −3.
(f) Sum = −7, Difference = −13
Answer: \(x + y = -7\), \(x - y = -13\). Adding: \(2x = -20\), \(x = -10\), \(y = 3\). The numbers are −10 and 3.
Carrom Coin Integers
A carrom coin? is struck to move it to the right. Each strike moves the coin a certain number of units rightward based on the force of the strike.
Number line with carrom coin starting at 0. Each strike moves it rightward.
🔵 If the coin is struck twice, with the first strike moving it by 4 units and the second moving it by 3 units, what will be the final position? It is clear that the coin will be at \(4 + 3 = 7\) units from 0.
Now, suppose the coin can be struck in either direction — left or right. We use positive numbers for rightward movement and negative numbers for leftward movement.
Number line with coin at 0. Rightward = positive movement, leftward = negative movement.
Suppose the first strike moves the coin rightward by 5 units (movement = +5), and the second strike moves it leftward by 7 units (movement = −7). The final position is:
\(P = 5 + (-7) = -2\)
The coin is at −2, which is 2 units to the left of 0.
🔵 If the first movement is −4 and the final position is 5, what is the second movement? Since \(P = a + b\), we get \(5 = (-4) + b\), so \(b = 9\). The second movement is +9 (rightward).
🔵 If there are multiple strikes causing movements 1, −2, 3, −4, ..., −10, what is the final position of the coin? Sum = \(1 + (-2) + 3 + (-4) + 5 + (-6) + 7 + (-8) + 9 + (-10) = (1-2) + (3-4) + (5-6) + (7-8) + (9-10) = (-1) \times 5 = -5\).
🔵 From the figures below, what can you conclude about the magnitudes and directions of a and b compared to each other? Remember to start from 0.
Three cases: (1) |a| > |b| and a is leftward → P < 0, (2) both rightward → P > 0, (3) equal magnitudes opposite directions → P = 0
Definition
Integers are the set of whole numbers and their negatives: \(\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\) Positive integers represent rightward/upward direction; negative integers represent leftward/downward direction.
By modeling the movements as numbers, both positive and negative, we capture two pieces of information — the distance (magnitude) and the direction (rightward or leftward). For example, when we say the movement is −4, the magnitude is 4 and the direction is leftward.
In general, if the first strike moves the coin \(a\) units and the second strike moves it \(b\) units, the final position is \(P = a + b\), where \(a\) and \(b\) can be positive or negative.
Interactive: Integer Number Line Explorer
Enter two movements (positive = right, negative = left) and watch the coin move!
Final Position: 5 + (−7) = −2
Token Model for Addition and Subtraction
In addition to the number line, we used the token model? to understand integers in Grade 6. We use green tokens + to represent positive 1 and red tokens − to represent negative 1 (i.e., −1). Together, one green and one red make zero — they cancel each other out.
Example: Find \((+7) - (+18)\)
To subtract 18 from 7, i.e., \((+7) - (+18)\), we need to remove 18 positives from 7 positives.
Problem: There are not enough positives! We only have 7, but need to remove 18.
Solution: Add 11 zero pairs (each pair is one green + one red). This doesn't change the value, but now we have 18 greens (7 original + 11 from pairs) and 11 reds.
+++++++
+
+−
+−
+−
... (11 zero pairs)
Now remove 18 positives. What is left? 11 negatives = −11.
\(7 - 18 = -11\)
Key Insight
Subtracting a number is the same as adding its additive inverse.\(7 - 18 = 7 + (-18) = -11\). The additive inverse? of 18 is −18, and the additive inverse of −18 is \(-(-18) = 18\).
🔵 Using tokens, argue out the following statements:
(a) \(7 - 18 = 7 + (-18)\) → The additive inverse of 18 is −18
(b) \(4 - (-12) = 4 + 12\) → The additive inverse of −12 is 12
(a) \(7 - 18 = 7 + (-18)\) → The additive inverse of 18 is −18
(b) \(4 - (-12) = 4 + 12\) → The additive inverse of −12 is 12
In general, the additive inverse of an integer \(a\) is represented as \(-a\). The additive inverse of −18 is \(-(-18) = 18\).
Activity: Integer Addition with Tokens
L3 Apply
Materials: 20 green counters (or coins heads-up) and 20 red counters (or coins tails-up)
Predict: If you have 5 green tokens and add 8 red tokens, will the result be positive or negative? What about 5 green and 5 red?
- Place 5 green tokens on the table. This represents +5.
- Add 8 red tokens. Now pair up each green with a red — these pairs are "zero pairs" (cancel out).
- Count the unpaired tokens. How many are left? What colour?
- Repeat with: (a) 3 green + 3 red, (b) 6 red + 2 green, (c) 4 green + 9 red
Observe: 5 green + 8 red: 5 zero pairs cancel, leaving 3 red = −3. So \(5 + (-8) = -3\).
(a) 3 green + 3 red = 3 zero pairs = 0. (b) 6 red + 2 green = 2 zero pairs + 4 red = −4. (c) 4 green + 9 red = 4 zero pairs + 5 red = −5.
Explain: The green and red tokens cancel in pairs. The result depends on which colour has more — that determines the sign, and the leftover count gives the magnitude.
Competency-Based Questions
Scenario: A submarine starts at sea level (position 0). It dives 150 metres below sea level, then rises 80 metres, and finally dives another 60 metres. Positions below sea level are represented as negative integers.
Q1. What is the submarine's final position?
L3 ApplyAnswer: (a) −130 m. \(0 + (-150) + 80 + (-60) = -150 + 80 - 60 = -130\). The submarine is 130 metres below sea level.
Q2. The submarine needs to reach −200 m. From its current position of −130 m, how much further must it dive?
L4 AnalyseAnswer: Additional dive needed: \(-200 - (-130) = -200 + 130 = -70\). The submarine must dive 70 metres more (movement = −70).
Q3. A second submarine starts at −50 m and makes the same three movements. Is its final position deeper or shallower than the first submarine? Evaluate.
L5 EvaluateAnswer: Second submarine: \(-50 + (-150) + 80 + (-60) = -50 - 130 = -180\). First submarine: −130. Since \(-180 < -130\), the second submarine is deeper (50 metres deeper, which makes sense since it started 50 m lower).
Q4. Create a sequence of exactly 4 movements that would bring the first submarine from −130 m back to sea level (0 m), with the constraint that no single movement exceeds 50 metres.
L6 CreateAnswer (one possible solution): The submarine needs to rise 130 metres total. With max 50 per movement: \(+40 + 40 + 30 + 20 = +130\). Starting from −130: \(-130 + 40 + 40 + 30 + 20 = 0\) ✓. Another option: \(+50 + 50 + 20 + 10 = +130\) ✓. Any four positive integers summing to 130, each ≤ 50, works.
Assertion–Reason Questions
Assertion (A): \(5 + (-8) = -3\)
Reason (R): When adding a positive and a negative integer, the result takes the sign of the integer with the larger absolute value.
Reason (R): When adding a positive and a negative integer, the result takes the sign of the integer with the larger absolute value.
Answer: (a) — Both true. \(|{-8}| = 8 > |5| = 5\), so the result is negative: \(-(8-5) = -3\). R correctly explains A.
Assertion (A): \((-7) - (-3) = -4\)
Reason (R): Subtracting a negative number is the same as adding its positive counterpart.
Reason (R): Subtracting a negative number is the same as adding its positive counterpart.
Answer: (d) — A is false: \((-7) - (-3) = -7 + 3 = -4\) ... wait, that is −4. Actually A is true. And R is true. R explains A: \(-7 - (-3) = -7 + 3 = -4\). Corrected answer: (a) — Both true, R correctly explains A.
Assertion (A): The sum of an integer and its additive inverse is always zero.
Reason (R): The additive inverse of \(a\) is \(-a\), and \(a + (-a) = 0\).
Reason (R): The additive inverse of \(a\) is \(-a\), and \(a + (-a) = 0\).
Answer: (a) — Both true. By definition, \(a + (-a) = 0\) for every integer \(a\). R is the mathematical proof of A.
Frequently Asked Questions
How do you add two positive integers?
Adding two positive integers works like regular addition. Simply add the numbers together. For example, 5 plus 8 equals 13 and 23 plus 47 equals 70. The result is always positive. This is the simplest integer addition case in NCERT Class 7 Ganita Prakash Part II Chapter 2.
How do you add a positive and negative integer?
When adding a positive and negative integer, subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value. For example, minus 8 plus 5: subtract 5 from 8 to get 3 and use the negative sign giving minus 3. NCERT Class 7 Chapter 2 teaches this rule.
What is the rule for subtracting integers?
To subtract an integer, change the subtraction to addition and flip the sign of the second number. So a minus b becomes a plus (minus b). For example, 7 minus (minus 3) becomes 7 plus 3 which equals 10. This is often remembered as the change-change rule in NCERT Class 7 Maths.
How does the number line help with integer operations?
On a number line, moving right means adding a positive number and moving left means adding a negative number. To add minus 3 to 5, start at 5 and move 3 steps left to reach 2. The number line gives a visual model for understanding integer operations in NCERT Class 7 Chapter 2.
What are the properties of integer addition?
Integer addition follows commutative property (a plus b equals b plus a), associative property, additive identity (a plus 0 equals a) and additive inverse (a plus minus a equals 0). NCERT Class 7 Ganita Prakash Part II explores these properties with examples.
Frequently Asked Questions — Chapter 2
What is Addition and Subtraction of Integers in NCERT Class 7 Mathematics?
Addition and Subtraction of Integers is a key concept covered in NCERT Class 7 Mathematics, Chapter 2: Chapter 2. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Addition and Subtraction of Integers step by step?
To solve problems on Addition and Subtraction of Integers, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 7 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 2: Chapter 2?
The essential formulas of Chapter 2 (Chapter 2) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Addition and Subtraction of Integers important for the Class 7 board exam?
Addition and Subtraction of Integers is part of the NCERT Class 7 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Addition and Subtraction of Integers?
Common mistakes in Addition and Subtraction of Integers include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Addition and Subtraction of Integers?
End-of-chapter NCERT exercises for Addition and Subtraction of Integers cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 2, and solve at least one previous-year board paper to consolidate your understanding.
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