Question

The overall width in $cm$ of several wide-screen television are $97.28cm$, $98\frac{4}{9}cm$, $98\frac{1}{25}cm$ and $97.94cm$. Express these numbers as rational numbers in the form $\frac{p}{q}$ and arrange the widths in ascending order.

Answer 1

Step 1: Express $97.28cm$ in $\frac{p}{q}$ form

The given widths of the wide screens are $97.28cm$, $98\frac{4}{9}cm$, $98\frac{1}{25}cm$ and $97.94cm$.

Consider the rational number $97.28$.

$$\begin{gathered} \Rightarrow 97.28=97+0.28 \\ \Rightarrow 97.28=97+\frac{28}{100} \\ \Rightarrow 97.28=97+\frac{7}{25} \\ \Rightarrow 97.28=\frac{\left(97\times 25\right)+7}{25} \\ \Rightarrow 97.28=\frac{2425+7}{25} \\ \Rightarrow 97.28=\frac{2432}{25} \end{gathered}$$

Step 2: Express $98\frac{4}{9}cm$ in $\frac{p}{q}$ form

Consider the rational number $98\frac{4}{9}$.

$$\begin{gathered} \Rightarrow 98\frac{4}{9}=\frac{\left(98\times 9\right)+4}{9} \\ \Rightarrow 98\frac{4}{9}=\frac{882+4}{9} \\ \Rightarrow 98\frac{4}{9}=\frac{886}{9} \end{gathered}$$

Step 3: Express $98\frac{1}{25}cm$ in $\frac{p}{q}$ form

Consider the rational number $98\frac{1}{25}$.

$$\begin{gathered} \Rightarrow 98\frac{1}{25}=\frac{\left(98\times 25\right)+1}{25} \\ \Rightarrow 98\frac{1}{25}=\frac{2450+1}{25} \\ \Rightarrow 98\frac{1}{25}=\frac{2451}{25} \end{gathered}$$

Step 4: Express $97.94cm$ in $\frac{p}{q}$ form

Consider the rational number $97.94$.

$$\begin{gathered} \Rightarrow 97.94=97+0.94 \\ \Rightarrow 97.94=97+\frac{94}{100} \\ \Rightarrow 97.94=97+\frac{47}{50} \\ \Rightarrow 97.94=\frac{\left(97\times 50\right)+47}{50} \\ \Rightarrow 97.94=\frac{4850+47}{50} \\ \Rightarrow 97.94=\frac{4897}{50} \end{gathered}$$

Step 5: Arrange the rational numbers in ascending order

The given rational numbers in $\frac{p}{q}$ form are $\frac{2432}{25}$, $\frac{886}{9}$, $\frac{2451}{25}$ and $\frac{4897}{50}$.

The least common multiple of the denominators $25,9,25,50$ can be given by, $\left(50\times 9\right)=450$.

Thus the rational numbers can also be expressed as,

$97.28=\frac{2432}{25}=\frac{2432\times 18}{25\times 18}=\frac{43776}{450}$.

$98\frac{4}{9}=\frac{886}{9}=\frac{886\times 50}{9\times 50}=\frac{44300}{450}$.

$98\frac{1}{25}=\frac{2451}{25}=\frac{2451\times 18}{18\times 25}=\frac{44118}{450}$.

$97.94=\frac{4897}{50}=\frac{4897\times 9}{50\times 9}=\frac{44073}{450}$.

As $43776<44073<44118<44300$, thus $97.28<97.94<98\frac{1}{25}<98\frac{4}{9}$.

Hence, the rational numbers can be expressed in $\frac{p}{q}$ form as $97.28=\frac{2432}{25}$, $98\frac{4}{9}=\frac{886}{9}$, $98\frac{1}{25}=\frac{44118}{450}$ and $97.94=\frac{44073}{450}$. The rational numbers can be arranged in ascending order as $97.28,97.94,98\frac{1}{25},98\frac{4}{9}$.