Rational Numbers Solution of TS & AP Board Class 8 Mathematics

Rational Numbers Solution of  TS & AP Board Class 8 Mathematics

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Exercise 1.1

Question 1.

Name the property involved in the following examples

Answer: Role of zero/additive identity:
0 is the additive identity for rational numbers, i.e. if 'a' is any rational number then
a + 0 = a = 0 + a
Question 2.

Name the property involved in the following examples

Answer:

Distributive law

This property is know as the distributive law of multiplication over addition.

For all rational numbers a, b and c

a(b + c) = ab + ac

⇒

Question 3.

Name the property involved in the following examples

Answer:

Multiplicative identity

The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.

Question 4.

Name the property involved in the following examples

Answer:

Multiplicative identity

The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.

Question 5.

Name the property involved in the following examples

Answer:

Commutative law of addition

In commutative law of addition, a, b is rational number where

a + b = b + a

Question 6.

Name the property involved in the following examples

Answer:

Closure law in multiplication

We find that rational numbers are closed under multiplication. For any two rational numbers a and b, ab is also rational number.

Question 7.

Name the property involved in the following examples

7a + (-7a) = 0

Answer:

(vii) Additive inverse law

Any two numbers whose sum is 0 are called the additive inverses of each other. In general if ‘a’ represents any rational number then a + (-a) = 0 and (-a ) + a = 0

Then a, (-a) are additive inverse of each other.

Question 8.

Name the property involved in the following examples

Answer:

(viii) Multiplicative inverse

We say that a rational number is called the reciprocal or the multiplicative inverse of another rational number  if 

Question 9.

Name the property involved in the following examples

Answer:

Distributive property

_{This property is known as distribution law of multiplication over addition. For all rational numbers a, b and c}

_{a + b = b + a}

Question 10.

Write the additive and the multiplicative inverses of the following.

Answer:

Additive inverse, 

Multiplicative inverse, 

Explanation:- When a number is added to its additive inverse, the result zero. When a number is multiplied to its multiplicative inverse, the result is 1.

Solving for the additive inverse:-

(add  to both sides)

Solving for the multiplicative inverse:-

(dividing both sides by )

Question 11.

Write the additive and the multiplicative inverses of the following.

1

Answer:

Additive inverse, -1

Multiplication inverse, 1

Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.

Solving for additive inverse:-

1 + x = 0

(add (-1) to both sides)

X = -1

Solving for the multiplicative inverse:-

1X = 1

(dividing both sides by 1)

X = 1

Question 12.

Write the additive and the multiplicative inverses of the following.

0

Answer:

Does not exist as the answer will be 0 itself.

Question 13.

Write the additive and the multiplicative inverses of the following.

Answer:

Additive inverse, 

Multiplication inverse, 

Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.

Solving for additive inverse:-

(add () to both sides)

Solving for the multiplicative inverse:-

(dividing both sides by )

Question 14.

Write the additive and the multiplicative inverses of the following.

−1

Answer:

Additive inverse, -1

Multiplication inverse, 1

Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.

Solving for additive inverse:-

(-1) + x = 0

(add 1 to both sides)

X = 1

Solving for the multiplicative inverse:-

(-1)X = 1

(dividing both sides by(-1))

X = 1

Question 15.

Fill in the blanks.

Answer:

let the blank be x

⇒ 

⇒ 

⇒ 

Question 16.

Fill in the blanks.

Answer:

let the blank be x

(add  on both sides)

X = 0

Question 17.

Fill in the blanks.

Answer:

let the blank be x

Question 18.

Fill in the blanks.

Answer: We know, that rational numbers are associative over addition, i.e. if are three rational number then,

Therefore, on comparing we get

Question 19.

Multiply by the reciprocal of

Answer:

Reciprocal of  is 

According to the given question

Question 20.

Which properties can be used in computing

Answer:

 Multiplicative associative, multiplicative inverse, multiplicative identity, closure with addition are the properties used in computing.

Question 21.

Verify the following

Answer:

LHS:- 

RHS:- 

LHS = RHS

Hence verified

Question 22.

Evaluate
after rearrangement.

Answer:

Given, 

(rearranging the like fractions at one place)

(Since + - = -)

(L.C.M. of 53 = 15)

Question 23.

Subtract

from

Answer:

Given, 

(By taking L.C.M.)

Question 24.

Subtract

from 2

Answer:

Given, 

(Since - - = + )

(By taking L.C.M.)

Question 25.

Subtract

−7 from

Answer:

Given, 

(Since - - = + )

(By taking L.C.M.)

Question 26.

What numbers should be added toso as to get.

Answer:

let the unknown number be x

According to the given question,

(Add  on both the sides)

(By taking L.C.M.)

Question 27.

The sum of two rational numbers is 8 If one of the numbers is find the other.

Answer:

let the unknown number be x

According to the given question,

(Add  on both the sides)

(By taking L.C.M.)

Question 28.

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

Answer:

Subtraction is not associative for rational numbers because when the numbers (say a,b,c) are subtracted by grouping any two at first and the other two at second [(a-b)-c and then a-(b-c)] the answer is not same. Thus subtraction is not associative in rational numbers.

Example:- let the 3 numbers be 5,8,9

Then, at first --(9-5)-8 = -4

And in second - 9-(5-8) = 9-(-3) = 12

Since, first case is not equal to second case’s answer

Question 29.

Verify that – (–x) = x for

x = 

Answer:

According to the question,

-(-x) = x

LHS:- -(-X) RHS:- X

(Since - - = + )

LHS = RHS

Hence verified

Question 30.

Verify that – (–x) = x for

x = 

Answer:

According to the question,

-(-x) = x

LHS:- -(-X) RHS:- X

(Since - - = + )

LHS = RHS

Hence verified

Question 31.

Write-

(i) The set of numbers which do not have any additive identity

(ii) The rational number that does not have any reciprocal

(iii) The reciprocal of a negative rational number.

Answer:

(i) Natural numbers

(ii) 0(zero) is the rational number which does not have a reciprocal.

(iii) Is a negative rational number

The reciprocal of a negative number must itself be a negative number so that the number and its reciprocal multiply to 1.

Example:- 

Reciprocal 

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Exercise 1.2

Question 1.

Represent these numbers on the number line.

(i) 9/7 (ii) -7/5

Answer:

In a rational number, the number below the bar i.e. the denominator tells the number of equal parts into which the first unit has been divided. The numerator tells ‘how many’ of these parts are considered.

(i) Here  means 9 markings of  each on the right of zero and starting from 0. The 9^th marking is.

(ii) Here  means 7 markings of  each on the left of zero and starting from 0. The 7^th marking is.

Question 2.

Write five rational numbers which are smaller than.

Answer:

Now we have to write 5 numbers which are less than 

It is very simple.

Therefore

 are 5 numbers which are less than 

Question 3.

Find 12 rational numbers between -1 and 2.

Answer:

let us multiply and divide (-1) and 2 by 12

 12 rational numbers are,

Question 4.

Find a rational number betweenand.

[Hint: First write the rational numbers with equal denominators.]

Answer:

make denominators same

Therefore lies between  and 

Question 5.

Find ten rational numbers betweenand .

Answer:

make denominators same

 10 rational numbers are,

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Exercise 1.3

Question 1.

Express each of the following decimal in theform.

(i) 0.57

(ii) 0.176

(iii) 1.00001

(iv) 25.125

Answer:

(i) 

(ii) 

(iii) 

(iv) 

Question 2.

Express each of the following decimals in the rational form .

Answer:

let x = 

x = 0.99999….. -→(i)

Here the periodicity of the decimal is one

So, we multiply both sides of (i) by 10 and we get

10x = 9.999…-→(ii)

Subtract (i) from (ii)

10x = 9.999….

x = 0.999…

10x - x = 9.999... - 0.999...

9x = 9.0

x = 1

Hence _{= 1}

Question 3.

Express each of the following decimals in the rational form .

Answer:

let x = 

X = 0.575757….. -→(i)

Here the periodicity of the decimal is two

So, we multiply both sides of (i) by 100 and we get

100x = 57.575757…-→(ii)

Subtract (i) from (ii)

100x = 57.5757….

X = 0.5757…

99x = 57.0

(divide by 99 on both sides)

Hence ₌

Question 4.

Find (x + y) ÷ (x − y) if

Answer:

According to the question we have,

(x + y) ÷ (x − y)

Put the values of x and y

Question 5.

Find (x + y) ÷ (x − y) if

x =  , y = 

Answer:

According to the question we have,

(x + y) ÷ (x − y)

Put the values of x and y

Question 6.

Divide the sum of and by the product ofand.

Answer:

To find the sum,

To find the sum,

According to the given question,

Question 7.

If of a number exceeds of the same number by 36. Find the number.

Answer:

let the number is x

Then of x 

According to the question,

9x = 3635

(multiply 35 on both the sides)

9X = 1260

(divide by 9 on both the sides)

X = 140

Question 8.

Two pieces of lengths m and m are cut off from a rope 11 m long. What is the length of the remaining rope?

Answer:

total length of the rope = 11m

Let the third part of rope be x

Length of first piece = m = m

Length of second piece = m = m

According to the question,

(By taking L.C.M.)

Question 9.

The cost of  meters of cloth is. Find the cost per metre.

Answer:

cost of cloth =  = 

Length of cloth =  = 

Cost per meter = 

(divide cost of cloth by length of cloth)

= 1.66 is the cost per meter

Question 10.

Find the area of a rectangular park which ism long and m broad.

Answer:

length = m = 

Breadth = m = 

Area = length breadth

= 

Question 11.

What number shouldbe divided by to get?

Answer:

let the number which be divided by be x

According to the question

(Multiply by x on both the sides)

Question 12.

If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is the length of the cloth required for each trouser?

Answer:

number of trousers = 36

Length of cloth = 64m

Length required for each trouser = m

Question 13.

When the repeating decimal 0.363636 .... is written in simplest fractional form, find the sum p + q.

Answer:

x = 0.363636…(i)

Periodicity = 2

So,

100x = 36.363636….(ii)

From (i) and (ii)

100x = 36.363636….

X = 0.3636363…

_____________________

99x = 36

X = 

Then p = 4 and q = 11

p + q = 4 + 11 = 15