Question

If , $\mathbf{5}\mathbf{<}\mathbf{a}\mathbf{<}\mathbf{7}\mathbf{<}\mathbf{b}\mathbf{<}\mathbf{14}$then which of the following best defines $\frac{a}{b}$.?

a.$\frac{5}{7}\mathbf{<}\frac{a}{b}\mathbf{<}\frac{1}{2}$ B. $\frac{5}{14}\mathbf{<}\frac{a}{b}\mathbf{<}\frac{1}{2}$ C. $\frac{5}{7}\mathbf{<}\frac{a}{b}\mathbf{<}\mathbf{1}$ D. $\frac{5}{14}\mathbf{<}\frac{a}{b}\mathbf{<}\mathbf{1}$

Short answer

The statement that best defines $\frac{a}{b}$is $\frac{5}{14}\mathbf{<}\frac{a}{b}\mathbf{<}\mathbf{1}$.Therefore the option D is correct.

Step-by-step solution

­- Description of step.

It is given that $5<a<7<b<14$.

Therefore, it can be obtained that:

$\frac{1}{5}>\frac{1}{a}>\frac{1}{7}>\frac{1}{b}>\frac{1}{14}$

As $5<a$, therefore it can be obtained that:

$\frac{5}{14}<\frac{a}{14}$

As, $\frac{1}{14}<\frac{1}{b}$, therefore it can be obtained that:

$\begin{array}{l}\frac{1}{14}<\frac{1}{b}\\ \frac{a}{14}<\frac{a}{b}\end{array}$

As, $\frac{5}{14}<\frac{a}{14}$and $\frac{a}{14}<\frac{a}{b}$, therefore it can be obtained that:

$\begin{array}{c}\frac{5}{14}<\frac{a}{14}<\frac{a}{b}\\ \Rightarrow \frac{5}{14}<\frac{a}{b}\end{array}$

Therefore, it is obtained that $\frac{5}{14}<\frac{a}{b}$.

­- Description of step.

As $a<7$, therefore it can be obtained that:

$\frac{a}{b}<\frac{7}{b}$

As, $\frac{1}{b}<\frac{1}{7}$, therefore it can be obtained that:

$\begin{array}{l}\frac{7}{b}<\frac{7}{7}\\ \frac{7}{b}<1\end{array}$

As, $\frac{a}{b}<\frac{7}{b}$and $\frac{7}{b}<1$, therefore it can be obtained that:

$\begin{array}{c}\frac{a}{b}<\frac{7}{b}<1\\ \Rightarrow \frac{a}{b}<1\end{array}$

Therefore, it is obtained that $\frac{a}{b}<1$.

­- Description of step.

As, $\frac{a}{b}<1$ and $\frac{5}{14}<\frac{a}{b}$, therefore it can be written as:

$\frac{5}{14}<\frac{a}{b}<1$

The statement that best defines ${\frac{a}{b}}^{}$ is $\frac{5}{14}<\frac{a}{b}<1$.

Therefore the option D is correct.