Here’s a brief explanation about what’s in the exercise:
- Multiplication Practice: You will multiply numbers to see what happens when you mix positive and negative numbers, like 3 times -1, or a negative times a negative. You’ll also see what happens when you multiply with zero.
- Rules: You will check if multiplication distributes over addition. It means you’ll see if multiplying a number with a group of numbers added together is the same as multiplying each one and then adding.
- Exploring -1’s Effect: You’ll find out what happens when you multiply any number by -1. You will also figure out which number becomes -22, 37, or 0 when multiplied by -1.
- Patterns with -1: Starting with -1 times 5, you’ll make a list of multiplications to discover a pattern and show that -1 times -1 equals 1.
| Class: | 7 |
| Chapter and Exercise: | 1 – 1.2 |
| Chapter Name: | Integers |
| Academic Session: | 2023-24 (CBSE) |
| Medium: | English |
| Book Name: | NCERT: Mathematics (Textbook for Class VII) |
| Edition: | December 2022 Agrahayana 1944 (latest) |
| Page Number: | 14 |
Presenting a brief note for the chapter. Have a glance on the concept that will be required for solving the questions. I have tried making it as simple as possible for better understanding –
- Multiplying Integers:
- When multiplying two integers with the same sign, the result is positive.
- Example: (-3) × (-5) = 15 (negative times negative is positive).
- When multiplying two integers with different signs, the result is negative.
- Example: (-7) × 4 = -28 (negative times positive is negative).
- Multiplying by Zero: Any number multiplied by zero is zero.
- Example: 5 × 0 = 0.
- Distributive Property: a × (b + c) = (a × b) + (a × c).
Example 1: Find the product of 3 and -4.
Step 1: Identify signs. 3 is positive, -4 is negative.
Step 2: Multiply absolute values. 3 × 4 = 12.
Step 3: Determine the sign. Positive × Negative = Negative. Answer: -12.
Example 2: Verify 5 × (6 + -2) = (5 × 6) + (5 × -2).
Step 1: Solve inside brackets. 6 + -2 = 4.
Step 2: Multiply. 5 × 4 = 20.
Step 3: Distribute. (5 × 6) + (5 × -2) = 30 + -10 = 20.
Both sides are equal.
Example 3: What is the product of -1 and 7?
Step 1: Multiply absolute values. 1 × 7 = 7.
Step 2: Determine the sign. Negative × Positive = Negative. Answer: -7.
Example 4: Demonstrate that (-1) × (-1) = 1.
Step 1: Multiply absolute values. 1 × 1 = 1.
Step 2: Determine the sign. Negative × Negative = Positive. Answer: 1.
NCERT Class 7 Maths Chapter 1 Other Exercises
NCERT Solutions Class 7 Maths Chapter 1 Exercise 1.2 Integers
Question 1: Find each of the following products:
(a) 3 × (–1):
3 × (–1) = –3 (A positive times a negative is negative)
(b) (–1) × 225:
(–1) × 225 = –225 (Multiplying a positive number by -1 gives its negative)
(c) (–21) × (–30):
(–21) × (–30) = 630 (A negative times a negative is positive)
(d) (–316) × (–1):
(–316) × (–1) = 316 (Multiplying a negative number by -1 gives its positive)
(e) (–15) × 0 × (–18):
(–15) × 0 × (–18) = 0 (Anything times 0 is always 0, regardless of other numbers)
(f) (–12) × (–11) × 10:
(–12) × (–11) × 10 = 1320 (Two negatives make a positive, then multiply by the positive number)
(g) 9 × (–3) × (–6):
9 × (–3) × (–6) = 162 (Two negatives make a positive, then multiply by the positive number)
(h) (–18) × (–5) × (–4):
(–18) × (–5) × (–4) = –360 (Two negatives make a positive, but three negatives make a negative)
(i) (–1) × (–2) × (–3) × 4:
(–1) × (–2) × (–3) × 4 = –24 (Three negatives make a negative, then multiply by the positive number)
(j) (–3) × (–6) × (–2) × (–1):
(–3) × (–6) × (–2) × (–1) = –36 (An even number of negatives make a positive, but an odd number of negatives make a negative)
Question 2: Verify the following:
(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]:
18 × [7 + (–3)] = 18 × 4 = 72
[18 × 7] + [18 × (–3)] = 126 + (–54) = 72
Both sides are equal.
(b) (–21) × [(–4) + (–6)] = [(–21) × (–4)] + [(–21) × (–6)]:
(–21) × [(–4) + (–6)] = (–21) × (–10) = 210
[(–21) × (–4)] + [(–21) × (–6)] = 84 + 126 = 210
Both sides are equal.
Question 3:
(i) For any integer a, what is (–1) × a equal to?
(–1) × a is equal to the negative of a.
(ii) Determine the integer whose product with (–1) is:
(a) –22: The integer is 22 because (–1) × 22 = –22.
(b) 37: The integer is –37 because (–1) × (–37) = 37.
(c) 0: The integer is 0 because (–1) × 0 = 0.
Question 4: Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1.
(–1) × 5 = –5
(–1) × (–1) × 5 = 1 × 5 = 5
(–1) × (–1) × (–1) × 5 = –1 × 5 = –5
(–1) × (–1) × (–1) × (–1) × 5 = 1 × 5 = 5
The pattern shows that the product of an even number of (–1)s is 1, and an odd number of (–1)s is –1, thus demonstrating (–1) × (–1) = 1.
Additional Questions similar to Exercise 1.2 of Integers Chapter
Question 1: Find each of the following products:
(a) -7 × 3
(b) 4 × (-5)
(c) (-15) × (-10)
(d) (-20) × 1
(e) (-10) × 0 × 5
(f) (-9) × (-7) × 8
(g) 6 × (-2) × (-7)
(h) (-12) × (-3) × (-2)
(i) (-1) × (-3) × (-4) × 5
(j) (-4) × (-5) × (-1) × 2
(a) -7 × 3
= -21 (A negative times a positive is negative)
(b) 4 × (-5)
= -20 (A positive times a negative is negative)
(c) (-15) × (-10)
= 150 (A negative times a negative is positive)
(d) (-20) × 1
= -20 (Any number times 1 is the number itself, keeping the sign)
(e) (-10) × 0 × 5
= 0 (Anything times 0 is always 0, regardless of other numbers)
(f) (-9) × (-7) × 8
= 504 (Two negatives make a positive, then multiply by the positive number)
(g) 6 × (-2) × (-7)
= 84 (Two negatives make a positive, then multiply by the positive number)
(h) (-12) × (-3) × (-2)
= -72 (Two negatives make a positive, but three negatives make a negative)
(i) (-1) × (-3) × (-4) × 5
= -60 (Three negatives make a negative, then multiply by the positive number)
(j) (-4) × (-5) × (-1) × 2
= -40 (Two negatives make a positive, but three negatives make a negative)
Question 2: Verify the following:
(a) 15 × [8 + (-4)] = [15 × 8] + [15 × (-4)]
(b) (-12) × [(-5) + (-3)] = [(-12) × (-5)] + [(-12) × (-3)]
(a) 15 × [8 + (-4)]
= 15 × 4 = 60 (First add inside the bracket, then multiply)
[15 × 8] + [15 × (-4)] = 120 – 60 = 60 (Distribute 15 into each term inside brackets, then add)
Both sides are equal.
(b) (-12) × [(-5) + (-3)]
= (-12) × (-8) = 96 (First add inside the bracket, then multiply)
[(-12) × (-5)] + [(-12) × (-3)] = 60 + 36 = 96 (Distribute -12 into each term inside brackets, then add)
Both sides are equal.
Question 3:
(i) For any integer a, what is (-1) × a equal to?
(ii) Determine the integer whose product with (-1) is:
(a) -33
(b) 28
(c) 0
(i) For any integer a, what is (-1) × a equal to?
(-1) × a is equal to -a. (Multiplying by -1 changes the sign of a)
(ii) Determine the integer whose product with (-1) is:
(a) -33
The integer is 33 because (-1) × 33 = -33. (The opposite sign of -33 is 33)
(b) 28
The integer is -28 because (-1) × (-28) = 28. (The opposite sign of 28 is -28)
(c) 0
The integer is 0 because (-1) × 0 = 0. (Zero times any number is always zero)
Question 4: Starting from (-1) × 4, write various products showing some pattern to show (-1) × (-1) = 1.
(-1) × 4 = -4
(-1) × (-1) × 4 = 1 × 4 = 4
(-1) × (-1) × (-1) × 4 = -1 × 4 = -4
(-1) × (-1) × (-1) × (-1) × 4 = 1 × 4 = 4
This pattern demonstrates that the product of an even number of (-1)s is 1, and an odd number of (-1)s is -1, thus (-1) × (-1) = 1.
Additional Questions for Practice
Question 1: Find each of the following products:
(a) -5 × 4
(b) 3 × (-6)
(c) (-14) × (-12)
(d) (-25) × 2
(e) (-8) × 0 × 7
(f) (-5) × (-4) × 9
(g) 7 × (-3) × (-5)
(h) (-10) × (-2) × (-6)
(i) (-2) × (-3) × (-4) × 6
(j) (-4) × (-7) × (-3) × (-2)
Question 2: Verify the following:
(a) 12 × [9 + (-4)] = [12 × 9] + [12 × (-4)]
(b) (-15) × [(-3) + (-7)] = [(-15) × (-3)] + [(-15) × (-7)]
Question 3:
(i) What is the product of (-1) and any integer a?
(ii) Determine the integer whose product with (-1) is:
(a) -45
(b) 19
(c) 0
