7th Class Mathematics Rational Numbers Rational Number

       Rational Number    

 

            Important Point Related to Rational Numbers

• Zero is a rational number because 0 can be written as \[\frac{0}{a}\] where a \[a\ne 0\]
• Every natural number is a rational number but the rational number may or may not be a natural number, for example \[\frac{2}{3}\] is a rational number which is not a natural number.
• Every whole number is a rational number but the rational number may or may not be a whole number, for example \[\frac{-2}{3}\] is a rational number which is not a whole number.
• Every integers is a rational number but the rational number may or may not be an integer, for example \[\frac{3}{5}\] is a rational number which is not an integer.  

 

        Types of Rational Number                

Rational numbers are of two types.

•    Positive rational numbers
•    Negative rational numbers  

 

Positive Rational Numbers                

If the numerator and denominator of a rational number having same sign then it is said to be positive rational number, for example \[\frac{3}{7},\frac{-4}{-3},\frac{15}{25},\frac{-13}{-15}\] are positive rational numbers.                                

 

Negative Rational Number                

If the numerator and denominator of a rational number having different sign then it is said to be negative rational number, for example \[\frac{3}{7},\frac{-4}{-3},\frac{15}{25},\frac{-13}{-15}\] are the negative rational numbers.                

 

Note: 0 is non - negative and non- positive rational number, in other words we can say that it is neither negative nor positive rational number.  

 

           Properties of Rational Number

• A rational number remains unaltered if we multiply numerator or denominator by the same non-zero numbers .i.e\[\frac{x}{y}\] remains same if we multiply the numerator and denominator by the same non-zero number ?m? i.e. \[\frac{x}{y}=\frac{m\times x}{m\times y}\]
• A rational number remains same if we divide numerator or denominator by the same non - zero numbers .i.e.\[\frac{x}{y}\] remains same if we divide numerator and denominator by the same non zero number "n" \[\frac{x}{y}=\frac{x\div n}{y\div n}.\]  

 

           Equivalent Rational Numbers                

If we multiply or divide the numerator or denominator of a rational by the same none zero integers then we get equivalent rational number.  

 

               

 

Find the equivalent rational numbers of \[\frac{p}{q}\].

Solution:                

\[\frac{2\times P}{2\times q},\frac{3\times p}{3\times q},\frac{4\times p}{4\times q},\frac{0.235\times p}{0.235\times q}.\]                

We can write infinite equivalent rational numbers of a rational number.                

 

          Lowest Form                

Divide the numerator and denominator by the HCF of (Numerator, Denominator) by ignoring the sign of it, so that we get the new numerator and denominator which are co-prime.  

 

               

Find the lowest form of \[\frac{-60}{96}.\]                

Solution:                

The HCF of 60 and 96 is 12. Therefore, divide the numerator and denominator                

By 12. We get \[\frac{-60}{96}=\frac{-60\div 12}{96\div 12}=\frac{-5}{8}\]                

 

            Standard Form of a Rational Number                

In the standard form we always write the denominator as positive. For this if the denominator of the rational number is negative then we multiply the numerator and denominator of the rational number by (-1)  

 

               

 

Write the following rational numbers in standard form:                

Solution:                

In the given problem \[\frac{-5}{7},\frac{7}{8}\] are in the standard form because its denominator is positive.

But in the rational numbers \[\frac{12}{-5},\frac{-3}{-4}\] are not in the standard form so we multiply the numerator and denominator by \[\left( -1 \right).\]                

So the standard form of the given rational numbers are \[\frac{-12}{5},\frac{3}{4},\frac{-5}{7},\frac{7}{8}\]                  

 

           Comparison of Rational Numbers

Write the rational numbers in the standard form.                

Every positive rational number is greater than the negative rational number.  

We can compare two rational numbers in the following way:

•    By making the denominators same
•    By short-cut method                                

 

Comparing Rational Number by Making the Denominator Same                

Step 1:   Write the rational number in the standard form.                

Step 2:   Find the LCM of all the denominators.                

Step 3:   Make the denominator same for all the rational numbers.                

Step 4:   Write all the rational number on the same denominator.                

Step 5:   Compare the numerators so obtained.  

 

 

               

 

Arrange the following rational numbers in ascending order                

\[\frac{3}{-5},\frac{5}{7},\frac{-4}{15},\frac{-7}{-15}\]                

Solution:                

Step 1: Standard form of rational numbers is\[\frac{-3}{5},\frac{5}{7},\frac{-4}{15},\frac{7}{15}\]                

Step 2: The LCM of denominator 5, 7, 15, 15 is 105.                                    

Step 3: Now all the rational number with denominator 105                

\[\frac{-3}{5}=\frac{\left( -3 \right)\times 21}{5\times 21}=\frac{-63}{105};\frac{5}{7}=\frac{5\times 15}{7\times 15}=\frac{75}{105}\]                

\[\frac{-4}{15}=\frac{\left( -4 \right)\times 7}{15\times 7}=\frac{-28}{105};\frac{7}{15}=\frac{7\times 7}{15\times 7}=\frac{49}{105}\]                

Step 4: The numerators are \[\left( -63 \right),75,\left( -28 \right)\] and 49.

The ascending order of numerators are\[-63<-28<49<75.\]                

Therefore \[\frac{-63}{105}<\frac{-28}{105}<\frac{49}{105}<\frac{75}{105}\]                

The ascending order of the given rational numbers is \[\frac{-3}{5}<\frac{-4}{15}<\frac{7}{15}<\frac{5}{7}.\]  

 

Short cut Method (for comparing two Rational Numbers)                

Step 1:  Write the rational number in the standard form.                

Step 2:  Multiply the numerator of first rational number with the denominator of the second and vice - versa.                

For two rational numbers which are in the standard form \[\frac{A}{B}\]and \[\frac{C}{D}\]                

We find the product of A and D, similarly B and C and compare AD and BC

•    If AD> BC then \[\frac{A}{B}>\frac{C}{D}\]
•    If AD> BC then \[\frac{A}{B}<\frac{C}{D}\]    

 

               

Which one of the two fractions is greater \[\frac{7}{-8}\] or \[\frac{6}{-7}?\]                

Solution:                

Step 1: The standard form of the given rational numbers are \[\frac{-7}{8}\] and \[\frac{-6}{7}.\]                

Step 2: The product of the numerator of first and denominator of second is \[\left( -7 \right)\times 7=-49\]

The product of numerator of second and denominator of first is \[\left( -6 \right)\times 8=\left( -48 \right)\]                

Here\[\left( -49 \right)\text{ }<\left( -48 \right)\]. Therefore, \[\frac{7}{-8}<\frac{6}{-7}\]                                                                                                                                                          

 

 

 

  Arrange the following rational numbers in descending order                

\[\frac{17}{30},\frac{-3}{-5},\frac{4}{-15},\frac{-7}{15}\]                  

(a) \[\frac{-3}{-5}<\frac{17}{30}<\frac{4}{-15}\]                                

(b) \[\frac{-3}{-5}>\frac{17}{30}>\frac{4}{-15}>\frac{-7}{15}\]                

(c)  \[\frac{17}{30}>\frac{-3}{-5}>\frac{4}{-15}>\frac{-7}{15}\]    

(d) \[\frac{4}{-15}<\frac{-7}{15}<\frac{-3}{-5}<\frac{17}{30}\]                

(e) None of these                                

 

Answer: (b)                

Explanation                

The given rational numbers are:                

\[\frac{17}{30},\frac{-3}{-5},\frac{4}{-15},\frac{-7}{15}\]                

Write the above rational numbers in standard form.                

\[\frac{17}{30},\frac{-3}{-5},\frac{4}{-15},\frac{-7}{15}\]                

The LCM of denominators 30, 5, 15, 15 is 30.                

Now, \[\frac{17}{30},\frac{-3}{-5},\frac{4}{-15},\frac{-7}{15}\]                

\[\frac{-4}{15}=\frac{-4\times 2}{15\times 2}=\frac{-8}{30};\frac{-7}{15}=\frac{-7\times 2}{15\times 2}=\frac{-14}{30}\]                

Now denominators are equal. The descending order is                

\[\frac{18}{30}>\frac{17}{30}>\frac{-8}{30}>\frac{14}{30}\]                

Or \[\frac{3}{5}>\frac{17}{30}>\frac{-4}{15}>\frac{-7}{15}\]                                           

 

 

  For three rational number x, y, z such that x > y and y < z. Which one of the following is true?                

(a) x < z

(b) x > z               

(c) y is the smallest rational number       

(d) Both A and Bare correct                

(e) None of these                  

 

Answer: (c)                

Explanation                

x > y and y < z => y is the smallest rational number among x, y, z.                

 

 

  Peter defines a rational number in the following ways. "It is or the form \[\frac{p}{q}\] where q is the smallest whole number.? This definition is:                

(a) Always true                                                                

(b) Represents some rational number only                

(c) Some as the definition of rational number    

(d) Always false                

(e) None of these                  

 

Answer: (d)                                                                        

 

 

  Which one of the following statement is true?                  

(a) The equivalent of \[\left( \frac{p}{q} \right)\] is different from \[\left( \frac{p}{q} \right)\]                

(b) \[\frac{35p}{33q}\] is the equivalent of \[\frac{p}{q}\]                

(c) There are only 5 equivalent rational numbers of the form \[\frac{p}{q}\]                

(d) We can write infinite equivalent rational numbers of the form \[\frac{p}{q}\]                

(e) None of these                                

 

Answer: (d)                

 

 

Which one of the following statement is false?                

(a) "0" is a positive rational number                

(b) A positive rational number is in the form \[\frac{p}{q},\] where p and q are integers of same sign and \[q\ne 0\]                

(c) A negative rational number is of the form of \[\frac{p}{q},\] where p and q are integer of opposite sign and \[q\ne 0\]                

(d) A positive rational number is always greater than the negative rational number                

(e) None of these                                

 

Answer: (a)