NCERT Solutions: Operations with Integers

Page No. 25

Figure it Out 

Q1: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
Ans:
(a) Sum = 27, Difference = 9

Let the two numbers be x and y with x + y = 27 and x – y = 9. Add the two equations to eliminate y:
(x + y) + (x – y) = 27 + 9 → 2x = 36 → x = 18.
Substitute back: 18 + y = 27 → y = 9.
Therefore, the numbers are 18 and 9.

(b) Sum = 4, Difference = 12

Let x + y = 4 and x – y = 12. Add: 2x = 16 → x = 8.
Then 8 + y = 4 → y = -4.
Therefore, the numbers are 8 and -4.

(c) Sum = 0, Difference = 10

Let x + y = 0 and x – y = 10. Add: 2x = 10 → x = 5.
Then 5 + y = 0 → y = -5.
Therefore, the numbers are 5 and -5.

(d) Sum = 0, Difference = – 10

Let x + y = 0 and x – y = -10. Add: 2x = -10 → x = -5.
Then -5 + y = 0 → y = 5.
Therefore, the numbers are -5 and 5.

(e) Sum = – 7, Difference = – 1

Let x + y = -7 and x – y = -1. Add: 2x = -8 → x = -4.
Then -4 + y = -7 → y = -3.
Therefore, the numbers are -4 and -3.

(f) Sum = – 7, Difference = – 13

Let x + y = -7 and x – y = -13. Add: 2x = -20 → x = -10.
Then -10 + y = -7 → y = 3.
Therefore, the numbers are -10 and 3.

Page No. 26

Based on this new model, answer the following questions:

1. If   the    first    movement    is    - 4    and    the    final    position    is    5,    what    is    the     second movement?
Ans: P = a + b

Given:

• a = -4
• P = 5
• b = ?

Substitute:

5 = -4 + b

Add 4 to both sides:

b = 5 + 4
b = 9

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

Ans: Movements:

1, -2, 3, -4, 5, -6, 7, -8, 9, -10

Group them in pairs:

• (1 – 2) = -1
• (3 – 4) = -1
• (5 – 6) = -1
• (7 – 8) = -1
• (9 – 10) = -1

There are 5 such pairs, each equal to -1.

Final Position:

Total = 5 × (-1) = -5

The final position of the coin is -5.

Page No. 27

Q: 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. 

1. 

• The yellow coin starts at 0 and is moved rightward by movement a to a point right of 0 (the dotted coin).
• Then movement b moves the coin leftward, crossing 0 and stopping at point P, which lies to the left of 0.

Conclusion:

• a is positive (movement to the right)
• b is negative (movement to the left)
• The coin ends up to the left of 0, so: |b| > |a|
  (the leftward movement is larger in magnitude)

a > 0, b < 0, and |b| > |a|.

2. 

• Movement b takes the coin from 0 to a point right of 0 (toward P).
• Then movement a moves the coin even further right to a point beyond P, where the dotted coin is.

Conclusion:

• Both a and b are positive (rightward movements)
• Since the final position is farther to the right after a, we get: a > b

a > 0, b > 0, and a > b.

3. 

• The dotted coin begins on the left side (negative side).
• Movement a moves the coin leftward.
• Movement b moves the coin further right and ends at point P, which is right of 0.

Conclusion

• a < 0 and b > 0
• Since after applying b, the coin travels farther right than after a, we get: b > a

a < 0, b > 0, and b > a

Q: Using tokens, argue out the following statements. 

(a) 7    - 18    =    7    +    (- 18)    (additive    inverse    of    18    is    - 18)
 (b) 4    - (- 12)    =    4    +    12    (additive    inverse    of    - 12    is    12)

Ans:
(a) Start with 7 positives

To subtract 18 positives, we must remove 18 positive tokens, but we only have 7.
So we add zero pairs to get enough positives.

We need 11 more positives, so add 11 zero pairs:

Now remove 18 positive tokens.

What is left? Only 11 negatives remain:

This represents -11.

So, using tokens:

7 – 18 gives the same result as 7 + (-18).
Both give -11.

(b) Start with 4 positive tokens:

We must subtract -12, meaning we need to remove 12 negative tokens.
But there are no negative tokens yet.

So we add 12 zero pairs (each pair is + and – together):

Now we have 12 negative tokens available.

Remove 12 negative tokens.

What is left?
The remaining tokens are:

That is 4 + 12 = 16 positives.

So, using tokens:

4 – (-12) is the same as 4 + 12, because subtracting a negative is the same as adding its additive inverse.

Page No. 31

Figure it Out 

Q1: Using the token interpretation, find the values of:
(a) 3 × (- 2)
(b) (- 5) × (- 2)
(c) (- 4) × (- 1)
(d) (- 7) × 3
Ans:
(a) 3 × (- 2)
Two red tokens 3 times = (-6)
So, 3 × (- 2) = (- 6).

(b) (- 5) × (- 2)
Remove 2 red tokens from the zero pairs, 5 times.
So, (- 5) × (- 2) = 10.

(c) (- 4) × (- 1)

Remove 1 red token from the zero pair, 4 times.

So, (- 4) × (- 1) = 4.

(d) (- 7) × 3
Remove 3 green tokens from the zero pairs, 7 times.
So, (- 7) × 3 = (-21).

Q2: If 123 × 456 = 56088, without calculating, find the value of:
(a) (- 123) × 456
(b) (- 123) × (- 456)
(c) (123) × (- 456)
Ans: (a) Given 123 × 456 = 56088 …..(i)
Then(a) (-123) × 456
This is the product of a negative and a positive integer, so the result will be negative.
i.e. -(123) × 456 = -(123 × 456) = -56088

(b) (-123) × (-456)
This is the product of two negative integers, so the result will be positive.
∴ (-123) × (-456) = 123 × 456 = 56088

(c) 123 × (-456)
This is the product of a positive and a negative integer, so the result will be negative.
∴ 123 × (-456) = -(123 × 456) = -56088

Q3: Try to frame a simple rule to multiply two integers.
Ans: Rule for multiplying two integers:
(i) Multiply their absolute values.
(ii) If the integers have different signs, the product is negative.
(iii) If both integers have the same sign, the product is positive.

Page No. 31

Consider the numbers represented by the following tokens:

Each token represents:

• + = +1
• – = -1
• A zero pair (+ and – together) = 0

We examine each set:

(a) 

Ans: This is 2 negatives, so the value is:

-1 + (-1) = -2

(b) 

Ans: Count positives and negatives:

• Negatives = 4
• Positives = 2

Value = (-4) + (+2) = -2

So (b) also represents -2.

(c)

Ans: Count them:

• Negatives = 6
• Positives = 4

Value = (-6) + (+4) = -2
So (c) also represents -2.
We can see that all of them represent the number (- 2). Now, take 4  times each of these token sets. That is, place each set into the empty bag  4 times.

Q: 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?
Ans: If we take 4 times each token set, meaning we place each set into a bag 4 times, we get:

4 × (-2) = -8

So the bag now represents -8.

Q: Check this for 5 × 4, by taking different token sets corresponding to 4.
We have seen that -4 × 2 is the number obtained by removing 2 positive tokens from the empty bag 4 times. We know that removing or subtracting a number is the same as adding its inverse.
Ans: Different token sets (such as four positives, or combinations of zero pairs that still equal 4) will all give the same final answer, because they all represent the number 4.
Taking each set 5 times gives the integer:

5 × 4 = 20

So, even if the token sets look different, the product is the same, because all the sets represent 4.

Using this, can -4 × 2 be defined through a process of addition of tokens instead of removal of tokens?

Yes. We have already seen that -4 × 2 can be obtained by removing 2 positive tokens from the empty bag 4 times.

Since removing 2 positive tokens is the same as adding 2 negative tokens, we can rewrite the process entirely using addition:

• Instead of removing 2 positives 4 times,
• We add 2 negative tokens to the empty bag 4 times.

Each time we add:

Doing this 4 times produces:

This is 8 negative tokens, representing – 8.

Thus: -4 × 2 = -8

Page No. 33

Figure it Out 

Q1: Find the following products.
(a) 4 × (- 3)
(b) (- 6) × (- 3)
(c) (- 5) × (- 1)
(d) (- 8) × 4
(e) (- 9) × 10
(f) 10 × (- 17)
Ans:
(a) 4 × (- 3)
Multiplier is positive, multiplicand is negative → product is negative.
∴ 4 × (-3) = -12.

(b) (- 6) × (- 3)
Both numbers are negative → product is positive.
∴ (-6) × (-3) = 18.

(c) (- 5) × (- 1)
Both numbers are negative → product is positive.
∴ (- 5) × (- 1) = 5.

(d) (- 8) × 4
Multiplier is negative, multiplicand is positive → product is negative.
∴ (- 8) × 4 = -32.

(e) (- 9) × 10
Multiplier is negative, multiplicand is positive → product is negative.
∴ (- 9) × 10 = -90.

(f) 10 × (- 17)
Multiplier is positive, multiplicand is negative → product is negative.
∴ 10 × (- 17) = -170.

Page No. 34

Q: In the case of integers, is the product the same when we swap the multiplier and the multiplicand? Try this for some numbers.
Ans: Yes, the product remains the same even if we swap the multiplier and the multiplicand.
This means:

a × b = b × a

Let us check with a few examples:

1. 4 × (-3) = -12,    (-3) × 4 = -12

2. (-5) × 7 = -35,    7 × (-5) = -35

3. (-6) × (-2) = 12,    (-2) × (-6) = 12

In all cases, the product is the same.

Conclusion:

Yes, multiplication of integers is commutative.
Swapping the multiplier and the multiplicand does not change the product.

Q: Observe the following pairs of multiplications (fill in the blanks where needed):

Ans:

Page No. 39

Figure it Out

Q1: Find the values of:
(a) 14 × (- 15)
(b) (- 16) × (- 5)
(c) 36 ÷ (- 18)
(d) (- 46) ÷ (- 23)
Ans:
(a) 14 × (- 15)
Multiplier is positive, multiplicand is negative → product is negative.
∴ 14 × (- 15) = -210.

(b) (- 16) × (- 5)
Both numbers are negative → product is positive.
∴ (- 16) × (- 5) = 80.
Multiplication Practice Software

(c) 36 ÷ (- 18)
or (-18) × ____ = 36
We know that (-18) × (- 2) = 36.
Therefore, 36 ÷ (- 18) = (- 2).

(d) (- 46) ÷ (- 23)
or (- 23) × ____ = (- 46)
We know that (- 23) × 2 = (- 46).
Therefore, (- 46) ÷ (- 23) = 2.
Q2: 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?
Ans: Initial temperature = 32°C
Temperature decreases at = 5°C per hour
Time = 10 hours
Decrease in temperature after 10 hours = 10 × (5°C) = 50°C
Final temperature = 32°C – 50°C = -18°C
∴ The room temperature after 10 hours will be -18°C.
Q3: 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.
Ans:
Profit on one white cement bag = ₹8
Loss on one grey cement bag = ₹5
(Profit is positive, loss is negative.)
(a)
White cement sold = 3000 bags
Grey cement sold = 5000 bags
Profit from white cement = 3000 × 8 = 24,000.
Loss from grey cement = 5000 × (-5) = -25,000.
Total profit/loss = 24,000 + (-25,000) = -1,000.
∴ The company has a loss of ₹1,000.

(b)
Grey cement sold = 6400 bags
Loss = 6400 × (-5) = -32,000
Let the number of white cement bags to be sold = x
Profit from white cement = x × 8
For no profit and no loss,
8x + (-32,000) = 0 → 8x = 32,000 → x = 4,000
∴ The company must sell 4,000 bags of white cement.

Q4: Replace the blank with an integer to make a true statement.
(a) (- 3) × _____ = 27
(b) 5 × _____ = (- 35)
(c) _____ × (- 8) = (- 56)
(d) _____ × (- 12) = 132
(e) _____ ÷ (- 8) = 7
(f) _____ ÷ 12 = – 11
Ans:
(a) (-3) × ___ = 27 → 27 ÷ (-3) = -9 → (-3) × (-9) = 27

(b) 5 × _____ = (- 35) → -35 ÷ 5 = -7 → 5 × (-7) = (- 35)

(c) _____ × (- 8) = (- 56) → -56 ÷ (-8) = 7 → 7 × (- 8) = (- 56)

(d) _____ × (- 12) = 132 → 132 ÷ (-12) = -11 → (-11) × (- 12) = 132

(e) _____ ÷ (- 8) = 7 → 7 × (-8) = -56 → (-56) ÷ (- 8) = 7

(f) _____ ÷ 12 = – 11 → -11 × 12 = -132 → (-132) ÷ 12 = – 11

Page No. 41

Q: Can you visually show the distributive property for an expression like -4 × (2 + (-3))?    [Hint: Use the fact that multiplying a number by -4 is adding the inverse of the number 4 times.] 
Ans:  -4 × (2 + (-3))
= (-4 × 2) + (-4 × (-3))
= -8 + 12
= 4

Page No. 42

Figure it Out

Q1:Find the values of the following expressions:
(a) (- 5) × (18 + (- 3))
(b) (- 7) × 4 × (- 1)
(c) (- 2) × (- 1) × (- 5) × (- 3)
Sol
(a) (- 5) × (18 + (- 3))
= (- 5) × 18 + (- 5) × (- 3) (by distributivity)
= (- 90) + 15
= (- 75).

(b) (- 7) × 4 × (- 1)
= (- 28) × (- 1)
= 28.

(c) (- 2) × (- 1) × (- 5) × (- 3)
= 2 × (- 5) × (- 3) (since (-2) × (-1) = 2)
= 2 × 15
= 30.
Q2: Find the values of the following expressions:
(a) (- 27) ÷ 9
(b) 84 ÷ (- 4)
(c) (- 56) ÷ (- 2)
Ans:
(a) (- 27) ÷ 9 → 9 × (-3) = (-27) → (-27) ÷ 9 = (-3).

(b) 84 ÷ (- 4) → (-4) × (-21) = 84 → 84 ÷ (-4) = (-21).

(c) (- 56) ÷ (- 2) → (-2) × 28 = (-56) → (-56) ÷ (-2) = 28.
Q3: Find the integer whose product with (- 1) is:
(a) 27
(b) – 31
(c) – 1
(d) 1
(e) 0
Ans:
(a) 27 → (-1) × (-27) = 27 → required integer: (-27).

(b) -31 → (-1) × 31 = -31 → required integer: 31.

(c) -1 → (-1) × 1 = -1 → required integer: 1.

(d) 1 → (-1) × (-1) = 1 → required integer: (-1).

(e) 0 → (-1) × 0 = 0 → required integer: 0.
Q4: 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.
Ans: Let E₁ = 47 – 56 + 14 – 8 + 2 – 8 + 5 = -4.
The second expression E₂ is the negative of E₁:
E₂ = -1 × E₁ = -1 × (-4) = 4.
Q5: 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. An example sequence is shown below.

Try this with different starting numbers: (- 21), (- 6), and so on. Describe the patterns you observe.

Ans:(a) Following the rules for the sequence starting with -7.

The rule is if even, take half; if odd, multiply by -3 and add 1.

Start with -7 (odd): (-7) × (-3) + 1 = 21 + 1 = 22 (even)

22 ÷ 2 = 11 (odd)

11 × (-3) + 1 = -33 + 1 = -32 (even)

-32 ÷ 2 = -16 → -8 → -4 → -2 → -1 (then -1 leads to 4 and the sequence cycles)

then 2 ÷ 2 = 1 (odd)

(b) (i) Starting number -21: (-21) is odd → (-21) × (-3) + 1 = 64, then halving repeatedly reaches 1, then the small cycle -2, -1, 4, 2, 1 repeats.

Hence the sequence for -21 is

(ii) Starting number -6: -6 is even → -3 (odd) → then follows the same pattern as above and eventually reaches the repeating loop -2, -1, 4, 2, 1.

Hence, the sequence is

Observation: For several negative starting numbers tried (for example -21, -6, -7), the sequence eventually enters the small repeating cycle: -2, -1, 4, 2, 1, -2. Many starting numbers reach this cycle.

Q6: 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?
Ans:
(a) Anita’s score = 40
Correct answers = 15 → marks from correct = 15 × 4 = 60
Total marks = marks from correct + marks from incorrect
So marks from incorrect = 40 – 60 = -20
Each incorrect answer gives -2 marks → number of incorrect answers = 20 ÷ 2 = 10
Total questions = correct + incorrect = 15 + 10 = 25.

(b) Anil’s score = -10
Correct answers = 5 → marks from correct = 5 × 4 = 20
Let incorrect answers = k → marks from incorrect = k × (-2) = -2k
Total score: 20 – 2k = -10 → -2k = -30 → k = 15 incorrect answers.
If the test has 25 questions (as in part (a)), questions answered = 5 + 15 = 20 → unanswered = 25 – 20 = 5.
Thus, Anil had 15 incorrect answers and left 5 questions unanswered.

Q7: Pick the pattern – find the operations done by the machine shown below.

Ans: Operation done by machine is (First Number) – [Second Number × Third Number].

Check the rows:

I row: 4 – [8 × (-3)] = 4 – (-24) = 4 + 24 = 28.

II row: 6 – [9 × 6] = 6 – 54 = -48.

III row: 2 – [3 × (-2)] = 2 – (-6) = 2 + 6 = 8.

IV row: -9 – [5 × (-8)] = -9 – (-40) = -9 + 40 = 31.

V row: 7 – [(-4) × (-6)] = 7 – 24 = -17.

VI row: -16 – [(-6) × (-9)] = -16 – 54 = -70.

The missing output (VI row) is -70.

Q8: 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.
Ans: Current temperature = 8°C
Drop each hour = 5°C → total drop in 4 hours = 4 × 5°C = 20°C
Expression: 8 – (4 × 5) and value = 8 – 20 = -12°C.

Q9: Find 3 consecutive numbers with a product of (a) – 6, (b) 120.
Ans:
(a) Let three consecutive integers be n – 1, n, n + 1. Try small integers. The consecutive integers -3, -2, -1 give product (-3) × (-2) × (-1) = -6. Hence the numbers are -3, -2, -1.

(b) Need (n – 1) n (n + 1) = 120. Try n = 5: 4 × 5 × 6 = 120. Hence the numbers are 4, 5, 6.

Q10: 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
(h) Is it possible to purchase an item that costs 1568 pibs?
Ans: The currency has two denominations: +13 pibs and -9 pibs.
We need to determine if it is possible to make the given totals. Each total is solved by finding non-negative integers x, y satisfying 13x – 9y = total.

(a) +20
One solution: x = 5, y = 5 → 13×5 – 9×5 = 65 – 45 = 20.

(b) +40
x = 10, y = 10 → 130 – 90 = 40.

(c) -50
x = 10, y = 20 → 130 – 180 = -50.

(d) +8
x = 2, y = 2 → 26 – 18 = 8.

(e) +10
x = 7, y = 9 → 91 – 81 = 10.

(f) -2
x = 13, y = 19 → 169 – 171 = -2.

(g) +1
x = 7, y = 10 → 91 – 90 = 1.

(h) 1568
Yes. Example: x = 122, y = 2 → 122×13 – 2×9 = 1586 – 18 = 1568.

Q11:Find the values of:
(a) (32 × (- 18)) ÷ ((- 36))
(b) (32 ) ÷ ((- 36) × (- 18))
(c) (25 × (- 12)) ÷ ((45) × (- 27))
(d) (280 × (- 7)) ÷ ((- 8) × (- 35))
Ans:
(a) (32 × (- 18)) ÷ (- 36) = (-576) ÷ (-36) = 16.

(b) 32 ÷ ((-36) × (-18)) = 32 ÷ 648 = 32/648 = 4/81 after reducing by 8.

(c) (25 × (-12)) ÷ (45 × (-27)) = (-300) ÷ (-1215) = 300/1215 = 20/81 after reducing by 15.

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

Q12: Arrange the expressions given below in increasing order.
(a) (- 348) + (- 1064)
(b) (- 348) – (- 1064)
(c) 348 – (- 1064)
(d) (- 348) × (- 1064)
(e) 348 × (- 1064)
(f ) 348 × 964
Ans: Compute each value:
(a) (-348) + (-1064) = -1412.
(b) (-348) – (-1064) = -348 + 1064 = 716.
(c) 348 – (-1064) = 348 + 1064 = 1412.
(d) (-348) × (-1064) = 348 × 1064 = 370272 (positive).
(e) 348 × (-1064) = -370272.
(f) 348 × 964 = 335472.
Now arrange in increasing order (smallest to largest):
(e) -370272 < (a) -1412 < (b) 716 < (c) 1412 < (f) 335472 < (d) 370272.

Q13: Given that (- 548) × 972 = – 532656, write the values of:
(a) (- 547) × 972
(b) (- 548) × 971
(c) (- 547) × 971
Ans:
(a) (-547) × 972 = (-548 + 1) × 972 = (-548 × 972) + (1 × 972) = -532656 + 972 = -531684.

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

(c) (-547) × 971 = (-547 × 972) – (-547 × 1) = -531684 + 547 = -531137.

Q14:Given that 207 × (- 33 + 7) = – 5382, write the value of – 207 × (33 – 7) = _________.
Ans:
207 × (-33 + 7) = 207 × (-26) = -5382.
-207 × (33 – 7) = -207 × 26 = -5382.

Q15:Use the numbers 3, – 2, 5, – 6 exactly once and the operations ‘+’, ‘-‘, and ‘×’ 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
Ans:

Here is the complete, correct answer with explanations in clear, simple plain text for you to copy directly into your document:

(a) Maximum Possible Value

• Correct Expression: (3 – (-6)) × 5 + (-2)
• Explanation: To maximise the result, make the expression inside brackets as large as possible before multiplying. Subtracting a negative adds its value, so (3 – (-6)) = 3 + 6 = 9. Multiplying by 5 gives 45. Adding the remaining number -2 gives 45 – 2 = 43.
• Calculation: (3 + 6) × 5 – 2 = 9 × 5 – 2 = 45 – 2 = 43
• Final Answer: 43

(b) Minimum Possible Value

• Correct Expression: (-6 – 3) × 5 + (-2)
• Explanation: To obtain the smallest result, make the bracketed value as large a negative as possible before multiplying. (-6 – 3) = -9. Multiplying by 5 gives -45. Adding -2 makes the result even smaller: -45 – 2 = -47.
• Calculation: (-9) × 5 – 2 = -45 – 2 = -47
• Final Answer: -47

Q16: Fill in the blanks in at least 5 different ways with integers:

Ans:

(a) Here

(i) 0 + (-6 × 6) = -36
(ii) 4 + (-5 × 8) = 4 + (-40) = -36
(iii) -4 + (-8 × 4) = -4 + (-32) = -36
(iv) 12 + (-8 × 6) = 12 + (-48) = -36
(v) -3 + 3 × (-11) = -3 + (-33) = -36

(b) Here

(i) (13 – 1) × 1 = 12
(ii) (10 – 4) × 2 = 12
(iii) (1 – 3) × (-6) = (-2) × (-6) = 12
(iv) (14 – 10) × 3 = 4 × 3 = 12
(v) (16 – 13) × 4 = 3 × 4 = 12

(c) Here

(i) 5 – (10 – 4) = 5 – 6 = -1
(ii) 0 – (3 – 2) = 0 – 1 = -1
(iii) -1 – (1 – 1) = -1 – 0 = -1
(iv) -5 – (0 – 4) = -5 – (-4) = -1
(v) -10 – (-5 – 4) = -10 – (-9) = -1