Exercise 1.1

1.Name the properties involved in the following examples.

(i) 85 + 0 = 85 = 0 + 85

solution:

AdditiveInverse Identity

(ii) 2(35 + 12) = 2(35) + 2(12)

Solution:

law

(iii) 37 × 1 = 37 = 1 × 37

Solution:

MultiplicativeAdditive Identity

(iv) (−25) × 1 = 25 = 1 × (25)

Solution:

MultiplicativeAdditive Identity

(v) 25 + 13 = 13 + 25

Solution:

law of addition

(vi) 52 × 37 = 1514

Solution:

law in multiplication

(vii) 7a = (-7a) = 0

Solution:

AdditiveMultipicative inverse law

(viii) x × 1x = 1 (x=!0)

Solution:

MultiplicativeAdditive Inverse law

(ix) (2 × x) + (2×6) = 2 × (x+6)

Solution:

law

2. Write the additive and multiplicative inverse of the following.

(i) −35

Solution: 3553 , −5353

(ii) 1

Solution: ,

(iii) 0

Solution: Does not exist0

(iv) 79

_Solution: −79−97 , 97−97

(v) -1

Solution: ,

3. Fill in the blanks:

i. (−117) + ___ = (−125) + (−117)

Solution:

To find the missing value, we can subtract from both sides of the equation:

(−117) - (−117) = (−125) + (−117) - (−117)

on both sides, +(−117) - (−117) = , so we're left with:

The missing value is: (−125)

(ii) −23 + ___ =−23

Solution:

To isolate the blank, subtract from both sides:

−23 - −23 =−23 - −23

Now simplify both sides: −23 - −23 =

The value that fills the blank:

iii. 1 × ___ = 911

Solution: To find the missing value (represented by the blank), we can see that multiplying by any number will give that number.

So, the missing value must be .

(iv) -12 + (12 + 13) = (34 × 12) + (34 × ? )

Solution:

First, let's simplify the left side, 12 + 13. To add these fractions, we need a common . The least common denominator (LCD) of 2 and 3 is .

12 = , 13 =

Now Add them: 36 + 26 =

Now, let's simplify the right side: 34 × 12 =

To solve for the missing value, subtract on both sides:

56 - 38 = 38 + (34 × ?) - 38

To substract 56 - 38, we need a common . The least common denominator of 6 and 8 is

56 = , 38 =

2024 - 924 =

So the equation becomes: 1124 = 34 × ?

To isolate the missing value, divide both sides of the equation by

Dividing by a fraction is the same as multiplying by its reciprocal:

× 43 = ​ {.reveal(when="blank-22")}Simplify 4472 by dividing both the numerator and the denominator by their greatest common divisor (which is 4):

= (444)/(724) =

The value that fills the blank: 1118

(v) ? × (12 + 13) = (34 × 12) + (34 × ?)

Solution:

This equation demonstrates the distributive property. Let's break it down:

The left side of the equation is: (?) × (12 + 13)

The right side of the equation is: 34×12 + (34 × ?)

The distributive property states that a × (b + c) = (a × b) + (a × c)

In this case, 'a' is the missing value. 'b' is , and 'c' is .

Therefore, the missing value must be .

The values that fill the blanks: 34 , 13

(vi) −167 + ___ = −167

Solution: The missing value is .

4. Multiply 211 by the reciprocal of −514

The reciprocal of a fraction is found by flipping it. So, the reciprocal of −514 is .

Multiply the fractions: (211) × (−145) = 2×−1411×5 =

So, the answer is −2855.

5. Which properties can be used in computing 25 × (5 × 76) + 13 × (3 × 411).

Solution:

Multiplicative AssociativeDistributive, Multiplicative InverseCommutative and Multiplicative .

6. Verify the following and write the property used: (54 + 14) + −34 = 54 + (−12 + −32).

Solution:

Let's break down both sides of the equation to verify if they are equal:

Left side:

(54 + 14) + −34 = + −34 = 64 + −34 =

Right side:

54 + (−12 + −32) = 54 + = 54 + = 54 + =

Verification:

34 ≠= −34

The left side of the equation equals 34, while the right side equals −34. Therefore, the equation is not verified.

7. Evaluate 35 + 73 + (−25) + (−23) after rearrangement.

Solution:

Rearrange the terms: Group the fractions with the same denominator together to make the addition easier:

(35 + −25) + (73 + −23)

Add the fractions within each group: 3/5+-2/5(3 - 2)/5` =

(73 + −23) = 7−23 =

Add the results:

15 + 53

Find a common denominator: The least common denominator for 5 and 3 is .

Convert the fractions to equivalent fractions with the common denominator:

15 =

53 =

Add the equivalent fractions:

315 + 2515 =

Therefore, 35 + 73 + (−25) + (−23) = 2815

8. Subtract

i. 34 from 13

Solution:

Find a common denominator: The least common multiple of 3 and 4 is .

Convert the fractions:

13 =

34 =

Subtract the fractions:

412 - 912 = 4−912 =

Therefore, subtracting 3/4 from 1/3 gives us −512.

ii. −3213 from 2

Solution:

To subtract -32/13 from 2, we can rewrite the expression as:

2 - (−3213)

Subtracting a negative number is the same as its positive counterpart:

2 + 3213

To add these together, we need a common .

We can express 2 as 21. The common denominator for 1 and 13 is .

So we convert 21 to an equivalent fraction with a denominator of 13:

(21) × (1313) =

Now we can add the fractions:

2613 + 3213 = 26+3213 =

So, the result is 5813.

iii. -7 from −47

Solution:

To subtract -7 from −47, we set up the subtraction like this:

−47 - (-7)

Subtracting a negative number is the same as its positive counterpart:

−47 +- 7

To add these together, we need a common denominator. We can express 7 as 71. The common denominator for 7 and 1 is .

So we convert 71 to an equivalent fraction with a denominator of 7:

(71) × (77) =

Now we can add the fractions:

−47 + 497 = 49−47 =

So, the result is 457.

9. What number should be added to −58 so as to get −32?

Solution:

Let "x" be the number you need to add. The problem can be written as an equation:

−58 + x = −32

To find x, you need to get it by itself on one side of the equation. Add 58 to both sides:

x = −32 +

The fractions have different denominators (2 and 8). To add them, you need a common denominator. The least common denominator for 2 and 8 is .

Convert −32 to a fraction with a denominator of 8:

−32 × 44 =

x = −128 + 58

x = −12+58

x =

The number you need to add to −58 to get −32 is −78.

10. The sum of two rational numbers is 8 if one of the numbers is −56 find the other.

Solution:

Let 'x' be the other rational number. We know that the sum of the two numbers is 8, so we can write the equation:

x + (−56) = 8

To find the value of x, we need to get it by itself on one side of the equation. We can do this by adding 5/6 to both sides:

x = 8 +

To add the numbers, we need a common denominator. We can write 8 as .

To get a denominator of 6, we multiply both the numerator and denominator by 6:

81 × 66 =

Add the fractions: Now we can add the fractions:

x = + 56

x = 48+56

x =

Therefore, the other rational number is 536.

11. Is subtraction associative in rational numbers? Explain with an example

Solution:

NoYes, subtraction is notis associative in rational numbers.

Example:Let's take three rational numbers: a = 1, b = 2, and c = 3.

(a - b) - c: (1 - 2) - 3 = - =

a - (b - c): 1 - (2 - 3) = 1 - (-1) = + =

Since -4 ≠ 2, we've shown that (a - b) - c ≠ a - (b - c).

12. Verify that – (–x) = x for:

i. x = 215

Solution:

If x = 215, then:

-x =

-(-x) = -(−215) =

Since 215 is indeed equal to x, we have verified that – (–x) = for x = 215.

ii. x = −1317

Solution:

The given value of x is −1317.

We need to find -(-x).

-(-x) = -(-(−1317))

Since a negative times a negative is a , we have:

-(-(−1317)) = -(1317) =

Since -(-x) = −1317 and x = −1317, we have -(-x) = x.

Therefore, it is verified that -(-x) = x for x = −1317.

13. Write:

i. The set of numbers which do not have an additive identity.

Solution: numbers

ii. The rational number that does not have any reciprocal.

Solution:

iii. The reciprocal of a negative rational number.

Solution: The reciprocal of a negative rational number −ab (where a and b are positive) is .