Question

State, true or false :
$\left(i\right)\frac{7}{9}=\frac{7+5}{9+5}\left(ii\right)\frac{7}{9}=\frac{7-5}{9-5}\left(iii\right)\frac{7}{9}=\frac{7\times 5}{9\times 5}\left(iv\right)\frac{7}{9}=\frac{7÷5}{9÷5}\left(v\right)\frac{-5}{-12}\text{is a negative rational number}\left(vi\right)\frac{-13}{25}\text{is smaller than}\frac{-25}{13}.$

Answer 1

$\left(i\right)\frac{7}{9}=\frac{7+5}{9+5}$

Lets simplify, $\frac{7+5}{9+5}$

$\frac{7+5}{9+5}=\frac{12}{14}=\frac{6}{7}$

$\Rightarrow$ The Simplest form of $\frac{7+5}{9+5}$ is $\frac{6}{7}$

$\Rightarrow$ $\frac{7}{9}\ne \frac{7+5}{9+5}$

So, $\frac{7}{9}=\frac{7+5}{9+5}$ is False.

$\left(ii\right)\frac{7}{9}=\frac{7-5}{9-5}$

Lets simplify, $\frac{7-5}{9-5}$

$\frac{7-5}{9-5}=\frac{2}{4}=\frac{1}{2}$

$\Rightarrow$ The Simplest form of $\frac{7-5}{9-5}$ is $\frac{1}{2}$

$\Rightarrow$ $\frac{7}{9}\ne \frac{7-5}{9-5}$

So, $\frac{7}{9}=\frac{7-5}{9-5}$ is False.

$\left(iii\right)\frac{7}{9}=\frac{7\times 5}{9\times 5}$

Lets simplify, $\frac{7\times 5}{9\times 5}$

$\frac{7\times 5}{9\times 5}=\frac{7}{9}$

$\Rightarrow$ The Simplest form of $\frac{7\times 5}{9\times 5}$ is $\frac{7}{9}$

$\Rightarrow$ $\frac{7}{9}=\frac{7\times 5}{9\times 5}$

So, $\frac{7}{9}=\frac{7\times 5}{9\times 5}$ is True.

$\left(iv\right)\frac{7}{9}=\frac{7÷5}{9÷5}$

Lets simplify, $\frac{7÷5}{9÷5}$

$\frac{7÷5}{9÷5}=\frac{7}{5}\times \frac{5}{9}=\frac{7}{9}$

$\Rightarrow$ The Simplest form of $\frac{7÷5}{9÷5}$ is $\frac{7}{9}$

$\Rightarrow$ $\frac{7}{9}=\frac{7÷5}{9÷5}$

So, $\frac{7}{9}=\frac{7÷5}{9÷5}$ is True.

$\left(v\right)\frac{-5}{-12}\text{is a negative rational number}$

Lets simplify $\frac{-5}{-12}$

$\frac{-5}{-12}=\frac{5}{12}$ is a positive rational number.

So, given statement is False.

$\left(vi\right)\frac{-13}{25}\text{is smaller than}\frac{-25}{13}.$

Lets convert the fractions $\frac{-13}{25}$, $\frac{-25}{13}$ into like fractions.

$\frac{-13}{25}=\frac{-13}{25}\times \frac{13}{13}=\frac{-169}{325}$

$\frac{-25}{13}=\frac{-25}{13}\times \frac{25}{25}=\frac{-625}{325}$

In like fractions, the fraction which has smaller numerator is smaller.

$\frac{-625}{325}<\frac{-169}{325}$

$\Rightarrow \frac{-25}{13}<\frac{-13}{25}$

Hence, the given statement is False.