Rational Numbers Class 8th Mathematics AP Board Solution

Class 8th Mathematics AP Board Solution
Exercise 1.1
  1. Name the property involved in the following examples
  2. 2 (3/5 + 1/2) = 2 (3/5) Name the property involved in the following examples…
  3. 3/7 x 1 = 3/7 = 1 x 3/7 Name the property involved in the following examples…
  4. (-2/5) x 1 = -2/5 Name the property involved in the following examples…
  5. 2/5 + 1/3 = 1/3 + 2/5 Name the property involved in the following examples…
  6. 5/2 x 3/7 = 15/14 Name the property involved in the following examples…
  7. 7a + (-7a) = 0 Name the property involved in the following examples…
  8. x x 1/x = 1 (x not equal 0) Name the property involved in the following…
  9. (2 x x) + (2 x 6) = 2 x (x+6) Name the property involved in the following…
  10. -3/5 Write the additive and the multiplicative inverses of the following.…
  11. 1 Write the additive and the multiplicative inverses of the following.…
  12. 0 Write the additive and the multiplicative inverses of the following.…
  13. 7/9 Write the additive and the multiplicative inverses of the following.…
  14. −1 Write the additive and the multiplicative inverses of the following.…
  15. Fill in the blanks.
  16. Fill in the blanks.
  17. 1 x 9/11 Fill in the blanks.
  18. Fill in the blanks.
  19. Multiply by the reciprocal of -5/14
  20. Which properties can be used in computing 2/5 x (5 x 7/6) + 1/3 x (3 x 4/11)…
  21. Verify the following (5/4 + -1/2) + -3/2 = -5/4 + (-1/2 + -3/2)
  22. Evaluate after rearrangement.
  23. 3/4 from 1/3 Subtract
  24. -32/13 from 2 Subtract
  25. −7 from -4/7 Subtract
  26. What numbers should be added to -5/8 so as to get -3/2 .
  27. The sum of two rational numbers is 8 If one of the numbers is -5/6 find the…
  28. Is subtraction associative in rational numbers? Explain with an example.…
  29. x = 2/15 Verify that - (-x) = x for
  30. x = -13/17 Verify that - (-x) = x for
  31. Write- (i) The set of numbers which do not have any additive identity (ii) The…
Exercise 1.2
  1. Represent these numbers on the number line. (i) 9/7 (ii) -7/5
  2. Write five rational numbers which are smaller than 5/6 .
  3. Find 12 rational numbers between -1 and 2.
  4. Find a rational number between 2/3 and 3/4 . [Hint: First write the rational…
  5. Find ten rational numbers between - 3/4 and 5/6 .
Exercise 1.3
  1. Express each of the following decimal in the p/q form. (i) 0.57 (ii) 0.176 (iii)…
  2. Express each of the following decimals in the rational form (p/q) .…
  3. Express each of the following decimals in the rational form (p/q) . 0. bar 57…
  4. Find (x + y) (x y) if
  5. Find (x + y) ÷ (x − y) if x = 1/4 , y = 3/2
  6. Divide the sum of - 13/5 and 12/7 by the product of - 13/7 and - 1/2 .…
  7. If of a number exceeds 1/7 of the same number by 36. Find the number.…
  8. Two pieces of lengths 2 3/5 m and 3 3/10 m are cut off from a rope 11 m long.…
  9. The cost of meters of cloth is 12 3/4 . Find the cost per metre.
  10. Find the area of a rectangular park which is 18 3/5 m long and 8 2/3 m broad.…
  11. What number should be divided by to get - 11/4 ?
  12. If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is…
  13. When the repeating decimal 0.363636 .... is written in simplest fractional form…

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 




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 9th marking is.



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





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


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