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Standard IX

Mathematics

Rational Numbers

Question 90 a...

Question

Question 90 (a)

Given that, $\frac{p}{q}$ and $\frac{r}{s}$ are two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers, we say that
$\frac{☐}{☐} < \frac{☐}{☐}$ , if $p \times s < r \times q$

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Solution

Given, $p \times s < r \times q$
$\Rightarrow \frac{p}{q} < \frac{r}{s}$ [by transferring sides]

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Question 90 (c)

Given that, $\frac{p}{q}$ and $\frac{r}{s}$ are two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers, we say that
$\frac{☐}{☐} < \frac{☐}{☐}$ , if $p \times s > r \times q$

In each of the following, fill in the blanks so as to make the statement true:

(i) A number which can be expressed in the form $p q$ , where $p$ and $q$ are integers and $q$ is not equal to zero, is called a .....

(ii) If the integers $p$ and $q$ have no common divisor other than 1 and $q$ is positive, then the rational number $p q$ is said to be in the ....

(iii) Two rational numbers are said to be equal, if they have the same .... form.

(vi) $\frac{a}{b} = \frac{a \div m}{b \div}$

(v) If $p$ and $q$ are positive integers, then $p q$ is a ..... rational number and $p - q$ is a ..... rational number.

(vi) The standard form of −1 is ...

(vii) If $p q$ is a rational number, then $q$ cannot be ....

(viii) Two rational numbers with different numerators are equal, if their numerators are in the same .... as their denominators.

Q. Which of the following options hold?
(1) Every integer is a rational number and every fraction is a rational number.

(2) A rational number

\frac{p}{q}

is positive if p and q are either both positive or both negative.

(3) A rational number

\frac{p}{q}

is negative if one of p and q is positive and other is negative.

(4) If there are two rational numbers with common denominator then the one with the larger numerator is larger than the other.

Q. Which of the following options is correct?
(1) Every integer and fraction is a rational number.
(2) A rational number

\frac{p}{q}

is positive if p and q are either both positive or both negative.

(3) A rational number

\frac{p}{q}

is negative if one of p and q is positive and other is negative.
(4) If there are two rational numbers with same denominator, then the one with the larger numerator is larger than the other.

Q. Question 3
State whether the following statements are true or false? Justify your answer.
(i)

\frac{\sqrt{2}}{3}

is a rational number.
(ii) There are infinitely many integers between any two integers.
(iii) Number of rational numbers between 15 and 18 is finite.
(iv) There are numbers which cannot be written in the form

\frac{p}{q}, q \neq 0

p and q both are integers.
(v) The square of an irrational number is always rational.
(vi)

\frac{\sqrt{12}}{\sqrt{3}}

is not a rational number as

\sqrt{1} 2

and

\sqrt{3}

are not integers.
(vii)

\frac{\sqrt{15}}{\sqrt{3}}

is written in the form

\frac{p}{q}

, so it is a rational number.

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Rational Numbers

Standard IX Mathematics

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Question 90 (a) Given that, pq and rs are two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers, we say that ☐☐<☐☐, if p×s<r×q

Given, p×s<r×q ⇒pq<rs [by transferring sides]

Question 90 (a)

Given that, $\frac{p}{q}$ and $\frac{r}{s}$ are two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers, we say that
$\frac{☐}{☐} < \frac{☐}{☐}$ , if $p \times s < r \times q$

Given, $p \times s < r \times q$
$\Rightarrow \frac{p}{q} < \frac{r}{s}$ [by transferring sides]