**Exercise 1.1 **

**Question 1: Determine each of the following products:**

**(i) 12 × 7**

**(ii) (-15) × 8**

**(iii) (-25) × (-9)**

**(iv) 125 × (-8)**** **

**Solution 1: **

(i) 12 × 7

For find the products of following numbers:

As the product of two integer values of opposite signs is similar to the additive inverse of the product of their absolute values.

** **

**Question 2. Find each of the following products:**

**(i) 3 × (-8) × 5**

**(ii) 9 × (-3) × (-6)**

**(iii) (-2) × 36 × (-5)**

**(iv) (-2) × (-4) × (-6) × (-8)**** **

**Solution 2: **

(i) 3 × (-8) ×5

For find the products of following numbers:

= 3 × (-8) × 5

= 3 × (-8 × 5)

= 3 × -40

= -120

The product of integers with opposite signs is equivalent to the additive inverse of the product of their absolute values.

(ii) 9 × (-3) × (-6)

For find the products of following numbers:

= 9 × (-3) × (-6)

= 9 × (-3 × -6)

= 9 × (+18)

= 162

[∵ The product of integers of like signs is equivalent to the product of their absolute values.]

(iii) (-2) × 36 × (-5)

For find the products of following numbers:

= (-2) × 36 × (-5)

= (-2 × 36) × (-5)

= -72 × -5 = +360

[∵ The product of integers of like signs is equivalent to the product of their absolute values.]

(iv) (-2) × (-4) × (-6) × (-8)

For find the products of following numbers:

= (-2) × (-4) × (-6) × (-8)

= (-2 × -4) × (-6 × -8)

= -8 × (-48) = +384

[∵ The product of integers of like signs is equivalent to the product of their absolute values.]

** **

**Question 3. Find the value of:**

**(i) 1487 × 327 + (-487) × 327**

**(ii) 28945 × 99 – (-28945)**** **

**Solution 3:**

(i) 1487 × 327 + (-487) × 327

According to the multiplication of integers,

1487 × 327 + (-487) × 327

= (1487 × 327) + (-487 × 327)

= 486249 – 159249

=327000

Since the product of integers of opposite signs is equivalent to the additive inverse of the product of their absolute values.

(ii) 28945 × 99 – (-28945)

According to the multiplication of integers,

28945 × 99 – (-28945)

= (28945 × 99) – (-28945)

= 2865555 + 28945

=2894500

Since the product of integers of like signs is equivalent to the product of their absolute values.

** **

**Question 4 : Complete the following multiplication table:**

**Is the multiplication table symmetrical about the diagonal joining the upper left corner to the lower right corner?**** **

**Solution 4:**

This is clear from the table that the table is symmetrical also with diagonal that links the upper left corner to the bottom right corner.

** **

**Question 5: Determine the integer whose product with ‘-1’ is**

**(i) 58**

**(ii) 0**

**(iii) -225**** **

**Solution 5: **

(i) 58

For find the integer of which is multiplied by -1

= 58 × (-1) = -58

Since the product of integers of opposite signs is equivalent to the additive inverse of the product of their absolute values.

(ii) 0

For find the integer of which is multiplied by -1

= 0 × -1 = 0

[because when any digit multiplied with 0 we get 0 as their result]

(iii) -225

For find the integer of which is multiplied by -1

= -225 × -1 = 225

Since the product of integers of like signs is equivalent to the product of their absolute values.

**Exercise 1.2**

**Question 1. Divide:**

**(i) 102 by 17**

**(ii) -85 by 5**

**(iii) -161 by -23**

**(iv) 76 by -19**

**(v) 17654 by -17654**

**(vi) (-729) by (-27)**

**(vii) 21590 by -10**

**(viii) 0 by -135**** **

**Solution 1 :**

**Exercise 1.3**

**Question 1: Find the value of **

**1. 36 ÷ 6 + 3**

**Solution 1:**

According to BODMAS rule

we have to division first: 36 ÷ 6 = 6

then we have to do addition: 6 + 3 = 9

so, the answer is 9.

**Question 2: 24 + 15 ÷ 3**

**Solution 2:**

According to BODMAS rule

we have to division first: 15 ÷ 3 = 5

then we have to do addition: 5 + 24 = 29

so, the answer is 29.** **

**Question: 3. 120 – 20 ÷ 4**

**Solution 3:**

According to BODMAS rule

we have to division first: 20 ÷ 4 = 5

then we have to do subtraction: 120 – 5 = 115

so, the answer is 115** **

**Question: 4. 32 – (3 × 5) + 4**

**Solution 4:**

According to BODMAS rule

we have to multiplication first: 3 × 5 =15

then addition: 15 _{}+ 4 = 19

then subtraction: 32 – 19 = 21

so, the answer is 21** **

**Question: 5. 3 – (5 – 6 ÷ 3)**

**Solution 5:**

According to BODMAS rule

we have to division first: 6 ÷ 3 = 2

then subtraction: (5 – 2) = 3

then subtraction:3 – 3 = 0

so, the answer is 0 ** **

**Question: 6. 21 – 12 ÷ 3 × 2**** **

**Solution 6:**

According to BODMAS rule

we have to division first: 12 ÷ 3= 4

then multiplication: 4 × 2 =

then subtraction: 21 – 8 = 13

so, the answer is 13.** **

**Question: 7. 16 + 8 ÷ 4 – 2 × 3**** **

**Solution 7:**

According to BODMAS rule

we have to division first: 8 ÷ 4 = 2

then multiplication: 2 × 3 = 6

then addition: 16 + 2= 18

then subtraction: 18 – 6 = 12

so, the answer is 12.** **

**Question: 8. 28 – 5 × 6 + 2**** **

**Solution 8:**

According to BODMAS rule

we have to multiplication first: 5 × 6= 30

then addition: -30 + 2 = -28

then subtraction: 28 – 28 = 0

So, the answer is 0.** **

**Question: 9. (-20) × (-1) + (-28) ÷ 7**** **

**Solution 9:**

According to BODMAS rule

we have to division first:

then multiplication:

then addition:

then subtraction:

Therefore, (-20) × (-1) + (-28) ÷ 7 = (-20) × (-1) – 4

= 20 – 4 = 16** **

**Question: 10. (-2) + (-8) ÷ (-4)**** **

**Solution 10:**

According to BODMAS rule

we have to division first: (-8) ÷ (-4) = 2

then addition: (-2) + 2 = 0

so, the answer is 0.** **

**Question: 11. (-15) + 4 ÷ (5 – 3)**** **

**Solution 11:**

we have to solve the bracket first: (5 – 3) = 2

then division: 4 ÷ 2= 2

then addition: ( -15) + 2 = -13

so, the answer is -13** **

**Question: 12. (-40) × (-1) + (-28) ÷ 7**** **

**Solution 12:**

According to BODMAS rule

we have to division first: (-28) ÷ 7= -4

then multiplication: (-40) × (-1)= 40

then addition: 40 + (-4) = 36

so, the answer is 36.

**Question: 13. (-3) + (-8) ÷ (-4) -2 × (-2)**

**Solution 13:**

According to BODMAS rule

we have to division first: (-8) ÷ (-4) = 2

then multiplication: -2 × (-2) = 4

then addition: 2 + 4 = 6

then subtraction: 6 -3 = 3

so, the answer is 3.

** **

**Question: 14. (-3) × (-4) ÷ (-2) + (-1)**

**Solution 14:**

(-3) × (-4) ÷ (-2) + (-1)

According to BODMAS rule

We have to division first: (-4) ÷ (-2) = 2

Then multiplication: (-3) × 2 = -6

Then addition: (-6) + (-1) = -6 – 1 = -7

**Exercise 1.4 **

** **

**Question 1: Simplify each of the following:**

**1. 3 – (5 – 6 ÷ 3)**** **

**Solution 1:**

According to BODMAS rule firstly, solve the bracket:

3 – (5 – 6 ÷ 3)

= 3 – (5 – 2)

= 3 – 3

= 0

** **

**Question 2: -25 + 14 ÷ (5 – 3)**** **

**Solution 2:**

According to BODMAS rule firstly, solve the bracket:

-25 + 14 ÷ (5 – 3)

= -25 + 14 ÷ 2

= -25 + 7

= -18

**Question 5: 36 - [18 - {14- (15 - 4 ÷ 2 x 2)}]**** **

**Solution 5:**

According to BODMAS rule firstly, solve the inner most bracket first:

= 36 – [18 – {14 – (11 ÷ 2 × 2)}]

= 36 – [18 – {14 – 11}]

Now removing the parentheses we get

= 36 – [18 – 3]

Now remove the braces we get

= 36 – 15

= 21

** **

**Question: 6. 45 – [38 – {60 ÷ 3 – (6 – 9 ÷ 3) ÷ 3}]**** **

**Solution 6:**

According to BODMAS rule firstly, solve the inner most bracket first:

= 45 – [38 – {20 – (6 – 3) ÷ 3}]

= 45 – [38 – {20 – 3 ÷ 3}]

Now remove the parentheses:

= 45 – [38 – 19]

Now remove the braces:

= 45 – 19

= 26

**Question 12: [29 – (-2) { 6 – (7 – 3)}] ÷ [ 3 × {5 + (-3) × (-2)]}**** **

**Solution 12:**

According to BODMAS rule inner move bracket

First we have to remove the innermost brackets,

= [29 – (-2) {6 – 4}] ÷ [3 × {5 + 6}]

Now remove the parentheses ,

= [29 + 2 (2)] ÷ [3 × 11]

Now remove all braces present,

= 33 ÷ 33

= 1

** **

**Question: 13.** **Using brackets, write a mathematical expression for each of the following:**

**(i) Nine multiplied by the sum of two and five.**

**(ii) Twelve divided by the sum of one and three.**

**(iii) Twenty divided by the difference of seven and two.**

**(iv) Eight subtracted from the product of two and three.**

**(v) Forty divided by one more than the sum of nine and ten.**

**(vi) Two multiplied by one less than the difference of nineteen and six.**** **

**Solution 13:**

(i) 9 × (2 + 5)

(ii) 12 ÷ (1 + 3)

(iii) 20 ÷ (7 – 2)

(iv) 2 × 3 -8

(v) 40 ÷ [1 + (9 + 10)]

(vi) 2 × [(19 -6) -1]