Students often refer to Class 7 Ganita Prakash Solutions and NCERT Class 7 Maths Part 2 Chapter 2 Operations with Integers Question Answer Solutions to verify their answers.
Class 7 Maths Ganita Prakash Part 2 Chapter 2 Solutions
Ganita Prakash Class 7 Chapter 2 Solutions Operations with Integers
Class 7 Maths Ganita Prakash Part 2 Chapter 2 Operations with Integers Solutions Question
2.1 A Quick Recap of Integers
Figure it Out (Page 25)
Let us try to find a few more pairs of numbers from their sums and differences:
(a) Sum = 27, Difference = 9
(b) Sum = 4, Difference = 12
(c) Sum = 0, Difference = 10
(d) Sum = 0, Difference = -10
(e) Sum = -7, Difference = -1
(f) Sum = -7, Difference = -13
Solution:
| Sum | Difference | First Number | Second Number | |
| (a) | 27 | 9 | 18 | 9 |
| (b) | 4 | 12 | 8 | -4 |
| (c) | 0 | 10 | 5 | -5 |
| (d) | 0 | -10 | -5 | 5 |
| (e) | -7 | -1 | -4 | -3 |
| (f) | -7 | -13 | -10 | 3 |
Carrom Coin Integers (Page 25-27)
A carrom coin is struck … two strikes, is again P = a + b.
Based on this new model, answer the following questions:
Question 1.
If the first movement is -4 and the final position is 5, what is the second movement?
Solution:
Given, first movement = -4 and final position = 5
So, -4 + second movement = 5
[ ∵ P(final position) = a + b]
Therefore, second movement = 5 + 4 = 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:
The movements are: 1,-2, 3, -4, 5, -6, 7, -8, 9, -10
Here, Positive movements: 1+3 + 5 + 7 + 9 = 25
Negative movements: -2-4-6-8-10 = -30
Final position of the coin = 25 + (-30) = – 5
From the figures below, what can you conclude about the magnitudes of a and b compared to each other, and what are their directions? Remember to start from 0.
Question 1.
Solution:
Starting from 0, a goes to the right (positive direction) and then b goes to the left (negative direction). The final position P is to the left side of 0, so the magnitude of b is greater than that of a.
Question 2.
Solution:
Starting from O, movement of a is rightward (positive direction) and movement of b is leftward (negative direction). The final position P is to the right of O, so the magnitude of a is greater than b.
Question 3.
Solution:
Starting from O, a moves to leftward (negative direction) and b moves to rightward (positive direction). The final position P is at O, so the magnitude of a is equal to the magnitude of b.
2.2 Multiplication of Integers (Page 29)
Similarly find the values of 4 x (-6) and 9 x (-7)
Solution:
4 x (-6); 4 x (-6) can be interpreted as placing 6 negatives in the empty bag 4 times. So, we place 6 negatives into the bag 4 times.
There are now 24 negatives in the bag, meaning -24.
∴ 4 x (-6) = -24
9 x (-7); We place 7 negatives into the bag 9 times that is 63 negatives in the bag, meaning -63.
∴ 9 x (-7) = -63
Figure it Out (Page 31)
Question 1.
Using the token interpretation, find the values of:
(a) 3 x (-2)
(b) (-5) x (-2)
(c) (-4) x (-1)
(d) (-7) x 3
Solution:
(a) On adding 2 negative tokens 3 times, we get
i.e. 3 x (-2) = -6.
(b) Removing 2 negative tokens from zero pairs 5 times = adding 10 positives.
i.e., (-5) x (-2) 10.
(c) Removing 1 negative token 4 times from zero pairs
= adding 4 positives
i.e., (-4) x(-1)=4
(d) Removing 3 positive tokens 7 times from zero pairs, we get
i.e., (-7) x 3=-21.
Question 2.
If 123 x 456 = 56088, without calculating, find the value of:
(a) (-123) x 456
(b) (-123) x (-456)
(c) (123) x (-456)
Solution:
(a) (-123) x 456 = -56088
(b) (-123)x (-456) = 56088
(c) (123) x (-456) =-56088
Question 3.
Try to frame a simple rule to multiply two integers.
Solution:
Do it yourself.
(Page 31)
What integer do we get as the final answer in each case? Do we get different answers because the sets look different, or the same answer because they all represent – 2?
Solution:
If we take 4 times each of these sets, we get 4 x (-2) = -8 in each case. We get the same answer because all the initial sets of tokens represent the same integer, i.e., -2. The physical arrangement or the inclusion of zero pairs (which sum to zero) does not alter the actual value of the integer; thus, the result of the multiplication must be consistent.
Figure it Out (Pages 33-34)
Find the following products:
(a) 4 x (-3)
(b) (-6) x (-3)
(c) (-5) x (4)
(d) (-8) x 4
(e) (-9) x 10
(f) 10 x (-17)
Solution:
(a) 4 x (-3) = -12
(b) (-6) x (-3) = 18
(c) (-5) x (-1) = 5
(d) (-8) x 4 = -32
(e) (-9) x 10 = -90
(f) 10 x (-17) = -170
In the case of integers, is the product the same when we swap the multiplier and the multiplicand? Try this for some numbers.
Observe the following pairs of multiplications (fill in the blanks where needed):
Solution :
| 3 x-4 = -42 | -4 x 3=-12 |
| -30 x 12 = –360 | 12 x -30 = –360 |
| -15 x-8= 120 | -8x-15=120 |
| 14 x-5 =-70 | -5 x 14 =-70 |
(Pages 35-37)
Question:
Just like for addition …. in the exam
What are the maximum possible marks in the exam? What are the minimum possible marks?
Solution:
Total questions: 50, For every correct answer: +5 marks
For every wrong answer: -2 marks
If a student answers all 50 questions correctly, then the highest score = 50 x 5 = 250
Maximum marks =250
If a student answers all 50 questions incorrectly, then the lowest score =50 ×(-2)=-100
Minimum marks =-100
Question:
Find the solution to part (b) using Method 1 described above.
Solution:
The elevator begins to descend from 15 metres above the ground. It moves at 3 metres per minute, and descends for 45 minutes. So,
Total distance descended =45 × 3=135 metres
Starting position =15 metres
Since it is descending, so we subtract 135 from 15
i.e., 15-135=-120
So, after 45 minutes, the elevator will be 120 metres below the ground.
A Magic Grid of Integers (Page 38)
Play the same game with the grid below. What answer do you get?
Solution:
No matter which number we choose first the final product will alway be the same as -30240.
Question :
What is so special about these grids? Is the magic in the number or the way they are arranged or both? Can you make more such grids?
Do it yourself.
Division of Integers
Figure it Out (Page 39)
Question 1.
Find the values of:
(a) 14 × (-15)
(b) -16 × (-5)
(c) 36 ÷ (-18)
(d) (-46) ÷ (-23)
Solution:
(a) 14 × (-15)=-210
(b) -16× (-5)=80
(c) 36 ÷ (-18)=-2
(d) (-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:
Initial temperature = 32°C
Rate of decrease = 5°C per hour
Time = 10 hours
Temperature decrease in 10 hours = 10 x 5°C = 50°C
New temperature = 32°C – 50°C = -18°C
Room temperature after 10 hours = -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. [Represent the profit/loss as integers.]
(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₹
(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:
Given, Profit on white cement = ₹8 per bag and Loss on grey cement = -₹5 per bag
(a) Number of white cement bags sold = 3,000
bags Profit = 3,000 x ₹8 = ₹24,000
Number of grey cement bags sold = 5,000
bags Loss = 5,000 x ₹5 = ₹25,000
Net profit/loss = + ₹24,000 + (- ₹25,000) = – ₹1,000
Loss of ₹ 1,000
(b) Let the number of white cement bags sold = x
Profit = 8x
Number of grey cement bags sold = 6400
Loss = 6,400 x ₹5 = ₹32,000
For no profit, no loss: 8x = 32,000
⇒ x = 4,000
The company must sell 4,000 white cement bags to have neither profit nor loss.
Question 4.
Replace the blank with an integer to make a true statement.
(a) (-3) ×___= 27
(b) 5 x = ___ (-35)
(c) ___ × (-8) = (-56)
(d) ___ × (-12)= 132
(e) ___ ÷ (-8) = = 7
(f) ___ ÷ 12 = -11
Solution:
(a) (-3) ×(-9)= 27
(b) 5 x = (-7) = (-35)
(c) (-7)× (-8) = (-56)
(d) (-11) × (-12)= 132
(e) (-56) ÷ (-8) = = 7
(f) (-132) ÷ 12 = -11
Expressions Using Integers (Pages 40-41)
“Are there orders in which 5 x (-3) x 4 can be evaluated? Will the product be the same in all these cases?”
Solution:
Yes, there are different orders in which 5 x (-3) x 4 can be evaluated. However, multiplication of integers is commutative as well as associative, the product will always be the same no matter how we group or order the numbers.
Question.
Multiply the expression 25 x -6 x 12 in all the different orders and check if the product is the same in all cases.
Solution:
Order 1: [25 x (-6)] x 12 = (-150) x 12 = -1800 Order 2: 25 x [(-6) x 12] = 25 x (-72) = -1800 Order 3: [(25 x 12) x (-6)] = 300 x (-6) = -1800 Hence, the product remains the same, no matter how the numbers are grouped.
Question.
Using this understanding of multiplication of many integers, can you give a simple rule to find the sign of the product of many integers?
Solution:
Since the product of each pair of negative numbers gives a positive result. We can say that:
- if the number of negative integers being multiplied is even, the product is positive.
- if the number of negative integers being multiplied is odd, the product is negative.
Question.
Check if the distributive property holds for (-2)x (4 + (-3)) (that is, if this expression equals (-2) x 4 + (-2) x (-3)), and for a few other such expressions of your choice. What do you observe? We see that the distributive property seems to hold for integers, as well. Will this always happen?
Solution:
(-2) x [4 + (-3)] = (-2) x 1 = -2 ………..(i)
(-2) x 4 + (-2) x (-3) = -8 + 6 = -2 ………(ii)
Both (i) and (ii) gives same result, i.e., -2, so the distributive property holds.
Yes, the distributive property always holds for integers whether the numbers are positive, negative, or zero.
Question.
Can you visually show the distributive property for an expression like -4 x (2 + (-3))?
[Hint: Use the fact that multiplying a number by -4 is adding the inverse of the number 4 times.]
Solution:
Pick the Pattern (Pages 41-42)
Question.
Two pattern machines … out the result.
Find the operations being done by Machine 2 and fill in the blank.
Solution:
Operation performed by Machine 2 is – (first number x second number + third number), i.e., multiply the first two inputs, add the third, then take the negative of that sum.
Figure it Out (Pages 42-44)
Question 1.
Find the values of the following expressions:
(a) (-5) x [18 +(-3)]
(b)(-7)x4x(-l)
(c) (-2) x (-1) x (-5) x (-3)
Solution:
(a) (-5) x [18 + (-3)] = (-5)x 15 = -75
(b) (-7) x 4 x (-1) = (-28) x (-1) = 28
(c) (-2) x (-1) x (-5) x (-3) = 2 x 15 = 30
Question 2.
Find the values of the following expressions:
(a) (-27) ÷ 9
(b) 84 ÷ (-4)
(c) (-56) ÷ (-2)
Solution:
(a) (-27) ÷ 9 = -3
(b) 84 ÷ (-4) = -21
(c) (-56) ÷ (-2) = 28
Question 3.
Find the integer whose product with (-1) is:
(a) 27
(b) -31
(c) -1
(d) 1
(e) 0
Solution:
Integer x (-1) = Product
⇒ Integer = Product = ÷ (-1)
(a) Integer = 27 ÷ (-1) = -27
(b) Integer = (-31) ÷ (-1) = 31
(c) Integer = (-1) ÷ (-1) = 1
(d) Integer = 1 ÷ (-1) = -1
(e) Integer = 0 ÷ (-1) = 0
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:
We have 47-56+14-8 + 2- 8 + 5 =-4 …(i)
Now, every term of -47 + 56- 14 + 8- 2 + 8- 5 is the negative of corresponding term in the first expression.
So, -47 + 56 – 14 + 8-2 + 8-5 = -(47-56+14-8 + 2-8 + 5)
= – (-4) = +4 [Using (i)]
Question 5.
Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is – start with any number; if the number is even, take half of it; if the number is odd, multiply it by -3 and add 1; repeat till you get 1. An example sequence is shown below.
Try this with different starting numbers: -21, -6, and so on. Describe the patterns you observe.
Solution:
Question 6.
In a test, (+4) marks are given for every correct answer and ( -2 ) marks are given for every incorrect answer.
(a) Anita answered all the questions in the test. She scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? How many questions are in the test?
(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:
(a) Anita scored =40 marks
Correct answers = 15
Then: 4(15)+(-2) x No. of incorrect answers =40 marks
Number of incorrect answers =\(\frac{60-40}{2}\)=10
Total questions in the test =15+10=25
(b) Anil scored =(-10) marks
Correct answers =5
Then: (+4) x 5+(-2) x No. of incorrect answer =-10
Number of incorrect answers =\(\frac{20+10}{2}\)=15
Total questions =5+15=20
So, Anil left 5 questions unanswered.
Question 7.
Pick the pattern – find the operations done by the machine shown below.
Solution:
Output = First number – (Second number × Third number)
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:
Given current temperature = 8°C and it drops by 5°C each hour.
Temperature after h hours: 8-5 × h
Temperature after 4 hours: 8 – 5(4) = 8 – 20 = -12°C.
Question 9.
Find 3 consecutive numbers with a product of
(a) -6, (b) 120.
Solution:
(a) The three consecutive numbers are -3, -2, and -1.
Since, (-3)(-2)(-1) = -6
(b) The three consecutive numbers are 4, 5, and 6.
Since, 4 x 5 x 6 = 120
Question 10.
An alien society uses a peculiar currency called ‘pibs’ with just two denominations of coins – a +13 pibs coin and a -9 pibs coin. You have several of these coins. Is it possible to purchase an item that costs +85 pibs? Yes, we can use 10 coins of +13 pibs and 5 coins of -9 pibs to make a total of +85. Using the two denominations, try to get the following totals:
(a) +20
(b) +40
(c) -50
(d) +8
(e) +10
(f) -2
(g) +1
[Hint: Writing down a few multiples of 13 and 9 can help.]
(h) Is it possible to purchase an item that costs 1568 pibs?
Solution:
(a) +20 = 5 coins of+13 and 5 coins of-9
(b) +40 = 10 coins of+13 and 10 coins of-9
(c) -50 = 1 coin of + 13 and 7 coins of-9
(d) +8 = 2 coins of+13 and 2 coins of-9
(e) +10 = 7 coins of+13 and 9 coins of-9
(f) -2 = 4 coins of+13 and 6 coins of-9
(g) +1 = 7 coins of+13 and 10 coins of-9
(h) Yes, it is possible to purchase an item costing 1568 pibs.
1568 = 122 coins of+13 and 2 coins of-9.
Question 11.
Find the values of:
(a) [32 x (-18)]+ (-36)
(b) (32) + [(-36) x (-18)]
(c) [25 x (-12)] + [(45) x (-27)]
(d) [280 x (-7)] + [(-8) x (-35)]
Solution:
(a) [32 x (-18)] + (-36) = -576 ÷ (-36) = 16
(b) (32) + [(-36) x (-18)] = 32 ÷ 648
=\(\frac{32}{648}=\frac{4}{81}\)
(c) [25 x (-12)] + [(45) x (-27)] = (-300) + (-1215)
= \(\frac{300}{1215}=\frac{20}{81}\)
(d) [280 x (-7)] + [(-8) x (-35)] = -1960 + 280 = -7
Question 12
Arrange the expressions given below in increasing order.
(a) (-348) + (-1064)
(b) (-348) – (-1064)
(c) 348 – (-1064)
(d) (-348) x (-1064)
(e) 348 x (-1064)
(f) 348 x 964
Solution:
(a) (-348) + (-1064) = -1412
(b) (-348)-(-1064) = 716
(c) 348-(-1064)= 1412
(d) (-348) x (-1064) = 370272
(e) 348 x (-1064) =-370272
(f) 348 x 964 = 335472
The above expressions in increasing order:
(e)<(a)<(b)<(c)<(f)<(d)
Question 13
Given that (-548) x 972 = -532656, write the values of:
(a) (-547) x 972
(b) (-548) x 971
(c) (-547) x 971
Solution:
(a) (-547) x 972 = [(-548) + 1] x 972
= (-548 x 972) + (1 x 972)
= -532656+ 972 = -531684
(b) (-548) x 971 = (-548) x (972 – 1)
= (-548 x 972)-(-548 x 1)
= -532656+ 548 = -532108
(c) (-547) x 971 = (-547) x (972 – 1)
= (-547) x 972 – (-547) [From (a)]
= -531684 + 547 = -531137
Question 14.
Given that 207 x (-33 + 7) = -5382, write the value of -207 x (33 – 7) = _____.
Solution:
Given: 207 x (-33 + 7) = -5382
207 x (-26) =-5382 …(i)
We have to find: -207 x (33 – 7)
-207 x 26 = -(207 x 26)
From (i), 207 x 26 = 5382
So, -207 x 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
(b) the result is the minimum possible
Solution:
(a) The expression (using 3, -2, 5, -6 once and using +, -, x once) is: -6 x [-2 – (3 + 5)] = 60.
Maximum possible value = 60
(b) The expression is: [(3 + 5) – (-2)] x (-6) = -60.
Minimum possible value = -60
Question 16.
Fill in the blanks in at least 5 different ways with integers:
Solution:
(a) (i) 0 + (-6) x 6 = -36
(ii) (-4) + (-8) x 4 = -36
(iii) 12 + (-6) x 8 = -36
(iv) (-6) + 3 x (-10) = -36
(v) 9 + (-9) x 5 = -36 (Answer may vary)
(b) (i) (8 – 5) x 4 = 12
(ii) (10 – 7) x 4 = 12
(iii) (6 – 2) x 3 = 12
(iv) (15 – 9) x 2 = 12
(v) [0 – (-3)] x 4 = 12 (Answer may vary)
(c) (i) [(2 – (5 – 2)] = -1
(ii) [2 – (4 – 1)] = -1
(iii) 0 [1 – (3 – 1)] = -1
(iv) [0 – (2 – 1)] = -1
(v) [3 – (5 – 1)] = -1 (Answer may vary)