8th Class Mathematics Rational Numbers Properties of Division on the Set Q

         Properties of Division on the Set Q  

 

            Closure Property for Division on Q

For any two rational numbers\[\frac{a}{b}\]and\[\frac{c}{d}\],

\[\left[ \left( \frac{a}{b} \right)\div \left( \frac{c}{d} \right) \right]\]is also a rational number. This is called the Closure Property of Division. Look at the following example:  

 

 

\[-\frac{1}{8}\div -\frac{4}{3}=-\frac{1}{8}\times -\frac{3}{4}=\frac{3}{32}\in Q;\]

It is true for number Q. So, we can say that division of two rational number is also a rational number.  

 

Division is not Commutative

For any two rational numbers\[\frac{a}{b}\] and \[\frac{c}{d}\],

we have \[\frac{a}{b}\div \frac{c}{d}\ne \frac{c}{d}\div \frac{a}{b}\]

Look at the following example:

 

               

 

\[\frac{11}{4}\div -\frac{1}{9}\ne -\frac{1}{9}\div \frac{11}{4}\]

Therefore, we can say that division is not commutative.  

 

Division is not associative

For any three rational numbers \[\frac{a}{b},\frac{c}{d}\] and \[\frac{e}{f}\in Q\],

we have \[\left( \frac{a}{b}\div \frac{c}{d} \right)\div \frac{e}{f}\ne \frac{a}{b}\div \left( \frac{c}{d}\div \frac{e}{f} \right)\]  

Look at the following example:

 

 

 

\[\left( \frac{5}{4}\div -\frac{1}{6} \right)\div \frac{1}{3}\ne \frac{5}{4}\div \left( -\frac{1}{6}\div \frac{1}{3} \right)\]

From the above example, we can say that division is not associative.  

 

            Decimal Representation of a Rational Number  

There are two forms of decimal representation of the rational numbers. They are terminating or no terminating.

Thus, we can represent every rational number as either terminating or no terminating decimals. The non-terminating decimals may be repeating or non-repeating. For example:  

 

 

0.25, 0.625, etc. are terminating decimals and 0.3333...... is non-terminating but repeating decimal whereas 0.01001000100001.........is a non-terminating and non-repeating decimal.    

 

 

 

 

• The numbers 'e' and pie are the two mostly used irrational numbers.
• The set of rational number is finite and the set of irrational number is infinite.
• The decimal expansion of  \[\frac{1}{7}\] never ends.
• One of the interesting fact about the irrational number is that when it is expressed in decimal form, the digits following the decimal points do not ever get repeated to terminated.    

 

 

 

 

• Rational number is closed under addition, multiplication, subtraction, and division.
• Rational number is commutative and associative over addition and multiplication.
• 0 is the additive identity and 0 is the multiplicative identity.
• Rational numbers can be represented on the number lines.
• Between any two rational numbers there are infinite number of rational numbers.  

 

 

 

 

  Thomas's monthly salary is Rs. 78,000. He spends 20% on fooding and 10% on house rent. From the remaining he spends 30% on his only son's education and donates 10% of the rest to charity. His monthly saving is:

(a) Rs. 35498                                      

(b) Rs. 34398

(c) Rs. 34389                                      

(d) Rs. 34498

(e) None of these  

 

Answer: (b)

Explanation:

Amount spent on fooding = 20% of Rs. 78,000 = Rs. 15,600

Amount spent on house rent = 10% of Rs. 78,000 = Rs. 7,800

Amount left with him = 78,000 - (15,600 + 7,800) = Rs. 54,600

Amount spent on son's education = 30% of 54,600 = Rs. 16,380

Amount left with him = Rs. 54,600 - Rs. 16,380 = Rs. 38,220

Amount donated to charity = 10 % of Rs 38,220 = Rs. 3,822

Amount left with him = Rs. 38,220 - Rs. 3,822 = Rs. 34,398  

 

 

  Find the sum \[\left( \frac{4}{5}+\frac{25}{15}-10 \right)+\frac{21}{45}\].

(a) \[-\frac{1}{9}\]                                                          

(b) \[1\frac{1}{9}\]

(c) \[-7\frac{1}{15}\]                                                      

(d) \[4\frac{22}{27}\]

(e) None of these  

 

Answer: (c)        

Explanation:

The L. C. M. of the denominators

L. C. M.(5,15, 45) = 45

Now 45 is the common denominator

\[=\left( \frac{12+25-150}{15} \right)+\frac{21}{45}\]

\[=-\frac{113}{15}+\frac{21}{45}=\frac{-113+7}{15}=\frac{106}{15}\]

\[=-7\frac{1}{15}\]  

 

 

  The rational number lying between 85 and 90 is_________.

(a) \[\frac{355}{4}\]                                       

(b) \[\frac{355}{2}\]

(c) \[\frac{355}{3}\]                                                        

(d) \[\frac{355}{5}\]

(e) None of these                                        

 

Answer: (a)  

 

 

  Find the value of \[\text{4}.\overline{\text{12}}\].                 

(a) \[4\frac{11}{90}\]                                     

(b) \[4\frac{5}{85}\]

(c) \[4\frac{11}{80}\]                                     

(d) \[4\frac{11}{60}\]

(e) None of these                                       

 

Answer: (a)  

 

 

  Mary went to the market to purchase some vegetables. The total weight of her vegetable was 25 kg. Her vegetable weight includes 15% of cabbage, 35% potato, 15% garlic, 25% onion and rest is chilli. The weight of chilli in her bag is:

(a) 5 kg                                                

(b) 3.5 kg

(c) 2.5 kg                                             

(d) 0.5 kg                            

(e) None of these

 

Answer: (c)

Explanation:

\[\sqrt{2}=\text{414213562373}0\text{95}0\text{488}0\text{16887242}0\text{9698}0\text{78569}...)\]  

 

 

  There are infinite number of rational numbers between any two rational number. Which one of the following rational number lies between \[\frac{20}{30}\] and \[\frac{40}{50}\]?

(a) \[\frac{11}{15}\]                                                       

(b) \[\frac{21}{30}\]           

(c) \[\frac{41}{30}\]                                                        

(d) \[\frac{2}{3}\]

(e) None of these  

 

Answer: (a)

Explanation:

We can find the rational number lying between any two rational numbers by equalizing the denominator. We can equate the denominator by multiplying both the rational numbers with the denominator each other and vice versa and then find the rational number between them.

\[\frac{2}{3}\times \frac{5}{5}=\frac{10}{15} and \frac{4}{5}\times \frac{3}{3}=\frac{12}{15}\]

Thus the rational number lying between \[\frac{12}{15}\] and \[\frac{10}{15}\] is \[\frac{11}{15}\].  

 

 

  The real numbers are either rational or irrational. The irrational numbers are the numbers which are non-terminating and non-repeating. Identify the numbers given below as non-terminating and non- repeating.

(a) \[\frac{1}{7}\]                                            

(b) \[\frac{4}{5}\]            

(c) \[\frac{5}{2}\]                                            

(d) \[\frac{\sqrt{216}}{3}\]

(e) None of these  

 

Answer: (a)

Explanation:

Options (b), (c) and (d) all are rational numbers, whereas option (a) is an irrational number.  

 

 

  The value of the given expression \[\left[ \frac{156}{24}+\left\{ \frac{-24}{56}+\frac{26}{112} \right\}\times \frac{112}{44} \right]\] is given by:

(a) \[\frac{1}{6}\]                                            

(b) \[\frac{2}{5}\]             

(c) \[\frac{36}{6}\]                                          

(d) 216

(e) None of these  

 

Answer: (c)    

 

 

  Which one of the following rational numbers has no reciprocal?

(a) \[\frac{4}{7}\]                                            

(b) \[\frac{9}{3}\]            

(c) 0                                                      

(d) \[\frac{5}{9}\]

(e) None of these  

 

Answer: (c)  

 

 

  The multiplicative identity, of the rational number \[\frac{455}{1024}\] is:

(a) \[\frac{1024}{455}\]                                                

(b) \[\frac{1}{455}\]          

(c) \[\frac{1}{1024}\]                                                     

(d) 1

(e) None of these  

 

Answer: (d)  

 

 

  Which one of the following rational numbers lies between \[\frac{45}{78}\] and \[\frac{26}{52}\]?

(a) \[\frac{75}{156}\]                                                     

(b) \[\frac{85}{156}\]          

(c) \[\frac{95}{156}\]                                                     

(d) \[\frac{105}{156}\]

(e) None of these  

 

Answer: (b)

Explanation On equating the denominator, the given rational number reduces to \[\frac{90}{156}\] and \[\frac{78}{156}\] and the rational number lying between these two is \[\frac{85}{156}\].  

 

 

  The largest rational number among the following rational numbers is:

(a) \[\frac{44}{34}\]                                                      

(b) \[\frac{55}{85}\]           

(c) \[\frac{76}{68}\]                                                        

(d) \[\frac{98}{102}\]

(e) None of these  

 

Answer: (a)

Explanation On equating the denominators we find that numerator of option (a) is largest and is equal to \[\frac{132}{102}\].  

 

 

  Which one of the following is a natural number?

(a) \[\frac{14}{56}\]                                                       

(b) \[\frac{19}{57}\]

(c) \[\frac{91}{13}\]                                                        

(d) \[\frac{45}{135}\]

(e) None of these  

 

Answer: (c)  

 

 

  The multiplicative inverse of \[\frac{97}{89}\] is:

(a) \[\frac{97}{89}\]                                                       

(b) \[\frac{89}{97}\]           

(c) \[\frac{1}{97}\]                                          

(d) \[\frac{1}{89}\]

(e) None of these  

 

Answer: (b)  

 

 

  The additive identity of the given number \[\frac{576}{890}\] is:

(a) 1                                                      

(b) \[\frac{576}{890}\]          

(c) \[\frac{890}{576}\]                                                   

(d) 0

(e) None of these  

 

Answer: (d)