Class 7 Maths Part 2 Chapter 2 Operations with Integers Questions

Concept: If the two numbers are x and y, then x + y is the sum and x - y is the difference. Adding the two equations gives 2x.

1. For (a), x + y = 27 and x - y = 9. Adding gives 2x = 36, so x = 18 and y = 9.
2. For (b), x + y = 4 and x - y = 12. Adding gives 2x = 16, so x = 8 and y = -4.
3. For (c), x + y = 0 and x - y = 10. Adding gives 2x = 10, so x = 5 and y = -5.
4. For (d), x + y = 0 and x - y = -10. Adding gives 2x = -10, so x = -5 and y = 5.
5. For (e), x + y = -7 and x - y = -1. Adding gives 2x = -8, so x = -4 and y = -3.
6. For (f), x + y = -7 and x - y = -13. Adding gives 2x = -20, so x = -10 and y = 3.

Final Answer: (a) 18 and 9; (b) 8 and -4; (c) 5 and -5; (d) -5 and 5; (e) -4 and -3; (f) -10 and 3.

Concept: The final position is the sum of all movements. For two strikes, P = a + b.

1. For the first part, use 5 = -4 + b.
2. Add 4 to both sides to get b = 9.
3. For the second part, pair the movements as (1 - 2), (3 - 4), (5 - 6), (7 - 8), and (9 - 10).
4. Each pair equals -1, and there are 5 pairs.
5. The total movement is 5 x (-1) = -5.

Final Answer: The second movement is 9. The final position after 1, -2, 3, -4, ..., -10 is -5.

Concept: Subtracting an integer means adding its additive inverse. Positive and negative token pairs make zero.

1. For 7 - 18, start with 7 positive tokens.
2. To remove 18 positives, add 11 zero pairs, then remove 18 positive tokens.
3. The 11 negative tokens left represent -11. This is the same as 7 + (-18).
4. For 4 - (-12), start with 4 positive tokens.
5. Removing 12 negative tokens is possible after adding 12 zero pairs.
6. When the 12 negative tokens are removed, 16 positive tokens remain. This is the same as 4 + 12.

Final Answer: 7 - 18 = -11 and 7 + (-18) = -11. Also 4 - (-12) = 16 and 4 + 12 = 16.

Concept: Same signs give a positive product. Different signs give a negative product. The magnitude is the product of the magnitudes.

1. 3 x (-2) means 3 groups of -2, so the answer is -6.
2. (-5) x (-2) has two negative signs, so the answer is positive. Its magnitude is 5 x 2 = 10.
3. (-4) x (-1) has two negative signs, so the answer is positive. Its magnitude is 4 x 1 = 4.
4. (-7) x 3 has one negative sign, so the answer is negative. Its magnitude is 7 x 3 = 21.

Final Answer: (a) -6; (b) 10; (c) 4; (d) -21.

Concept: Only the sign changes when signs of factors change. The magnitude remains 56088.

1. In (a), one factor is negative and one is positive, so the product is negative.
2. In (b), both factors are negative, so the product is positive.
3. In (c), one factor is positive and one is negative, so the product is negative.

Final Answer: (a) -56088; (b) 56088; (c) -56088.

Concept: The sign of the product depends on whether the two signs are the same or different.

1. Multiply the magnitudes of the two integers.
2. If both integers have the same sign, write a positive product.
3. If the integers have different signs, write a negative product.
4. This rule matches the token model and the multiplication patterns in the chapter.

Final Answer: Same signs give a positive product, and different signs give a negative product. Multiply magnitudes first, then decide the sign.

Concept: Use the integer multiplication sign rule and multiply magnitudes.

1. 4 x (-3) has different signs, so the product is -12.
2. (-6) x (-3) has same signs, so the product is 18.
3. (-5) x (-1) has same signs, so the product is 5.
4. (-8) x 4 has different signs, so the product is -32.
5. (-9) x 10 has different signs, so the product is -90.
6. 10 x (-17) has different signs, so the product is -170.

Final Answer: (a) -12; (b) 18; (c) 5; (d) -32; (e) -90; (f) -170.

Concept: Multiplication of integers is commutative, so swapping the factors does not change the product.

1. -30 x 12 has magnitude 30 x 12 = 360 and one negative factor, so the product is -360.
2. 12 x -30 is the same product after swapping factors, so it is also -360.
3. For -5 x blank = -70, divide the magnitudes: 70 divided by 5 is 14.
4. The product is negative, so the missing factor is positive 14.

Final Answer: -30 x 12 = -360, 12 x -30 = -360, and the missing factor is 14.

Concept: A downward movement is represented by subtraction from the starting height.

1. In 45 minutes, the elevator travels 45 x 3 = 135 metres downward.
2. Starting position is 15 metres above ground.
3. Using subtraction, ending position = 15 - 135.
4. 15 - 135 = -120.

Final Answer: The elevator will be at -120 m, which means 120 m below the ground.

Concept: For multiplication and division, same signs give a positive result and different signs give a negative result.

1. 14 x (-15) = -(14 x 15) = -210.
2. -16 x (-5) = +(16 x 5) = 80.
3. 36 divided by (-18) = -(36 divided by 18) = -2.
4. (-46) divided by (-23) = +(46 divided by 23) = 2.

Final Answer: (a) -210; (b) 80; (c) -2; (d) 2.

Concept: A fall in temperature is represented by a negative change.

1. The temperature change in 10 hours is 10 x (-5) = -50 degrees Celsius.
2. Starting temperature is 32 degrees Celsius.
3. Final temperature = 32 + (-50).
4. 32 - 50 = -18.

Final Answer: The room temperature after 10 hours will be -18 degrees Celsius.

Concept: Represent profit by a positive integer and loss by a negative integer.

1. Profit from 3000 white bags = 3000 x 8 = 24000.
2. Loss from 5000 grey bags = 5000 x (-5) = -25000.
3. Net result = 24000 + (-25000) = -1000, so there is a loss of Rs 1000.
4. For 6400 grey bags, loss = 6400 x 5 = Rs 32000.
5. To have no profit or loss, white cement profit must be Rs 32000.
6. Number of white bags = 32000 divided by 8 = 4000.

Final Answer: (a) Loss of Rs 1000. (b) The company must sell 4000 white cement bags.

Concept: Use inverse operations and then check the sign.

1. For (a), 27 divided by -3 is -9.
2. For (b), -35 divided by 5 is -7.
3. For (c), -56 divided by -8 is 7.
4. For (d), 132 divided by -12 is -11.
5. For (e), the number must be 7 x (-8) = -56.
6. For (f), the number must be -11 x 12 = -132.

Final Answer: (a) -9; (b) -7; (c) 7; (d) -11; (e) -56; (f) -132.

Concept: Simplify brackets first, and for many factors count the number of negative factors.

1. 18 + (-3) = 15, so (-5) x 15 = -75.
2. (-7) x 4 = -28, and -28 x (-1) = 28.
3. There are four negative factors in (-2) x (-1) x (-5) x (-3). An even number of negative factors gives a positive product.
4. The magnitude is 2 x 1 x 5 x 3 = 30.

Final Answer: (a) -75; (b) 28; (c) 30.

Concept: Use the integer division sign rule.

1. (-27) divided by 9 has different signs, so the answer is -3.
2. 84 divided by (-4) has different signs, so the answer is -21.
3. (-56) divided by (-2) has same signs, so the answer is 28.

Final Answer: (a) -3; (b) -21; (c) 28.

Concept: Multiplying by -1 gives the additive inverse of the integer.

1. If (-1) x n = 27, then n = -27.
2. If (-1) x n = -31, then n = 31.
3. If (-1) x n = -1, then n = 1.
4. If (-1) x n = 1, then n = -1.
5. If (-1) x n = 0, then n = 0.

Final Answer: (a) -27; (b) 31; (c) 1; (d) -1; (e) 0.

Concept: The second expression is the additive inverse of the first expression.

1. Compare term by term.
2. Each term in the second expression has the opposite sign of the matching term in the first expression.
3. So the value of the second expression is the opposite of -4.

Final Answer: The value is 4.

Concept: Follow the rule step by step. The sign may change when an odd number is used.

1. Starting with -21: -21 is odd, so (-3) x (-21) + 1 = 64. Then 64, 32, 16, 8, 4, 2, 1.
2. Starting with -6: -6, -3, 10, 5, -14, -7, 22, 11, -32, -16, -8, -4, -2, -1, 4, 2, 1.
3. Both examples reach 1 after several steps.
4. After reaching 1, the next terms are -2, -1, 4, 2, 1, so the sequence enters a repeating cycle.

Final Answer: The examples reach 1 and then repeat the cycle 1, -2, -1, 4, 2, 1.

Concept: Total score = 4 x correct answers + (-2) x incorrect answers.

1. For Anita, marks from correct answers = 15 x 4 = 60.
2. Let incorrect answers be x. Then 60 - 2x = 40.
3. So 2x = 20 and x = 10. Total questions = 15 + 10 = 25.
4. For Anil, marks from correct answers = 5 x 4 = 20.
5. Let incorrect answers be y. Then 20 - 2y = -10.
6. So 2y = 30 and y = 15. Total questions = 5 + 15 = 20.

Final Answer: (a) Anita had 10 incorrect answers and the test had 25 questions. (b) Anil had 15 incorrect answers and the test had 20 questions.