Rational Numbers

{- 4/3 x 5/4 x - 3/5 x 6/7 x (14/12 x 13/11 x 33/39) } / 2

= {- 4/3 x 5/4 x - 3/5 x 6/7 x 14/12 x 13/11 x 33/39} / 2

= {1 x 6/7 x 7/6 x 13/11 x 11/13} / 2

= {1}/2

= 1/2

Let the number be X.
 
 1/12 - X = 1/10
 
 1/12 - 1/10 = X
 
(5 - 6) / 60 = X

• 1/60    = X 

Therefore, the number to be subtracted is - 1/ 60.

Option (1) is true, as for any three integers a,b and c, a + (b + c) = (a + b) + c
Therefore, addition is associative.
Option (2) is true as 5 - (8 - 3) is not equal to ( 5 - 8) - 3.
Or 5 - 5 is not equal to - 3 - 3.
Option (3) is true, as for three integers a, b and c, a  x ( b x c ) = ( a x b ) x c
Therefore, all the three options are correct. 

Value of 2/3 = 0.66
Value of 4/5 = 0.8
Value of - 7/10 = - 0.7
Value of 16/20 = 0.8
Value of 3/4 = 0.75
Valvue of - 3/4 = - 0.75
Value of 3/10 = 0.3
Therefore, 3/4 lies between the given two rational numbers.

Statement (1) is correct, as additive inverse of - 121/10 is 121/10, which is greater than 10.
Statement (2) is incorrect, as multiplicative inverse of 10/18 is 18/10, which is more than 1.
Statement (3) is correct, as sum of a number and its additive inverse is 0.
Statement (4) is incorrect, as product of a number and its multiplicative inverse is 1.
Therefore, both (1) and (3) are correct.

Correct, as this option is less than the given number.

• 17771/ 23333 > - 17772/ 23333

Let the number be X.
(X + ¼ ) (4/8 - 8/16 + ½ + 1/3) = 1/10 
(X + ¼ ) (1/2 - 1/2 + ½ + 1/3) = 1/10
(X + ¼ ) ( ½ + 1/3) = 1/10
(X + ¼ ) (5/6) = 1/10
X + ¼ = 6/50
X = 6/50 - ¼
X = (12 - 25)/100
X = - 13/100

Given expression is (abc/de).
Additive inverse of the given expression is (- abc/de).
Thus, abc/de - abc/de = 0
Multiplicative inverse of the given expression is (de/abc).
Thus, abc/de x de/abc = 1
So, the required sum of additive inverse and multiplicative inverse = - abc/de + de/abc
= {(de)² - (abc)²}/ abcde

Option (1) is incorrect, as there are infinite rational numbers between 1/20 and 11/20.
Option (2) is incorrect, as there are infinite rational numbers between 1/41 and 31/41.
Option (3) is correct, as there are 11 whole numbers between 20 and 32, i.e. 21 to 31.
Option (4) is correct, as there can be infinite rational numbers between any two rational numbers.
<span style="font-family:" arial;="" font-size:="" small;="">Hence, 5<sup style="font-family:" arial;="">th<span style="font-family:" arial;="" font-size:="" small;=""> option is the answer.

Statement (1) is incorrect, as there are only 10 natural numbers in the range from 1 to 10.
Statement (2) is incorrect, as there are 11 whole numbers in the given range of numbers.
Statement (3) is correct, as there are 11 whole numbers in the range from 0 to 10.
Statement (4) is correct, as there can be infinite rational numbers in between any 2 or more integers or rational numbers.
Hence, both (3) and (4) options are correct.

The given expression is {( 25/36 x 216/125) + (2401/343  x 1296/216) } / {(- 25/36 x 6/5) + (- 4/9 x 81/64)}
Solve the expression part by part:
( 25/36 x 216/125) = 25/36 x 216/125 = 6/5

(2401/343 x 1296/216) = 7 x 6 = 42

• 25/36 x 6/5 = - 5/6
• 4/9 x 81/64 = - 9/16
  Therefore, the required value of the expression is
     6/5 + 42 / - 5/6 - 9/16 
  = {(216/5) / {(- 40 - 27)/48}
  = - 216/5 x 48/67

<span style="font-family:" arial;="" font-size:="" small;="">(3/7) + (- 6/11) + (- 16/42) + (10/44)
For finding the additive inverse of the above expression, we need to find the value of the above expression.
Additive inverse of a number is negative value of the same number. So, addition of these two numbers will be zero. For example, additive inverse of 1 is - 1.
Now, the value of the given expression:
3/7 - 6/11 - 16/42 + 10/42
= 3/7 - 6/11 - 8/21 + 5/22
= 198/462 - 252/462 - 176/462 + 105/462
= (198 - 252 - 176 + 105) / 462
= - 125/462

Therefore, additive inverse of the given expression is (125/462) so that 125/ 462 - 125/462 = 0.

<span style="font-family:" arial;="" font-size:="" small;="">(1/2 + 1/3 + ¼ ) / (½ - 1/3)
For finding the additive inverse of the above expression, we need to find the value of the above expression.

Additive inverse of a number is negative value of the same number. So, addition of these two numbers will be zero. For example, additive inverse of 1 is - 1 (As 1 - 1 = 0).

Now, the value of the given expression is
(1/2 + 1/3 + ¼ ) / ( ½ - 1/3)

= {(6 + 4 + 3) / 12} / {( 3-2)/6}

= {13/12} / {1/6}

= 13/2

Therefore, additive inverse of the given expression is - 13/2 so that 13/2 - 13/2 = 0.

Multiplicative inverse of a number is equal to the reciprocal of the given number such that product of the given number and its reciprocal is equal to 1.
The given expression is
{ ½ - 2/4 + 1/3 - ¼ } / { ½ - 2/4 + 1/3 + ¼ }
= { ½ - ½ + 1/3 - ¼ } / { ½ - ½ + 1/3 + ¼ }
= { 1/3 - ¼ } / {1/3 + ¼ }
= {1/12} / {7/12}
= 1/7

Therefore, multiplicative inverse of the given number is 7.