Question

Four cities lie along a straight highway in this order, City A, City B, City C, City D. The distance from City A to City B is 5 times the distance from City B to City C. Thedistance from City A to City Dis 2 times the distance from City A to City B. The distance from City C to City Dis 16 miles.

Answer 1

We are given that there are 4 cities that lie on a straight highway. We can visulaize our problem to be 1-D line. We will go ahead and place these cities on a line as follows:

The above placement of 4 cities is NOT to scale. We will go ahead and update our graphics with each step.

`\text{\textcolor{#FF7968}{Distance from City A to City B is 5 times the distance from City B to City C}}`

We will first go ahead and decrypt the above statement and write it down as a mathematical equation as follows:

`\textcolor{#FF7968}\text{\textcolor{#FF7968}{ = 5}}\textcolor{#FF7968}\ldots\text{\textcolor{#FF7968}{ Eq1}}`

Where,

`\begin{gathered} |AB|\colon\text{ Distance from City A to B} \\ |BC|\text{ : Distance from City B to C} \end{gathered}`

The next statement is as follows:

`\text{\textcolor{#FF7968}{Distance from City A to City D is 2 times the distance from City A to City B}}`

We will first go ahead and decrypt the above statement and write it down as a mathematical equation as follows:

`\textcolor{#FF7968}\text{\textcolor{#FF7968} = 2}\textcolor{#FF7968}AB\text{\textcolor{#FF7968}{ Eq2}}`

Where,

`|AD|\colon\text{ Distance from City A to City D}`

The last statement states as follows:

`\text{\textcolor{#FF7968}{The distance from City C to City D is 16 miles}}`

We will first go ahead and decrypt the above statement and write it down as a mathematical equation as follows:

`\textcolor{#FF7968}CD\text{\textcolor{#FF7968}{ = 16 }}\textcolor{#FF7968}{\ldots}\text{\textcolor{#FF7968}{ Eq3}}`

Where,

`|CD|\text{ : Distance from City C to City D}`

WE have the following three equations summarized as follows:

`\begin{gathered} |AB|\text{ = 5}\cdot|BC| \\ |AD|\text{ = 2}\cdot|AB| \\ |CD|\text{ = 16} \end{gathered}`

We will now update our graphical representation.

Substitute the (Eq1) and (Eq2) as follows:

`\begin{gathered} |\text AD \cdot(\text{ 5}\cdot|BC|\text{ ) } \\ \textcolor{#FF7968}\text{\textcolor{#FF7968}{ = 10}}\textcolor{#FF7968}\cdot \end{gathered}`

The distance ( AD ) can be split into three parts as follows:

`\begin{gathered} |\text BC \\ \text{Then use the above manipulation,} \\ \text + \cdot\text{ ( }|BC|\text{ )} \\ \textcolor{#FF7968}\text{\textcolor{#FF7968}{ = 9}}\textcolor{#FF7968}BC\text{\textcolor{#FF7968}CD} \end{gathered}`

Substitute Eq1 and Eq3 into the above result:

`\begin{gathered} 5\cdot|BC|\text{ = 9}\cdot|BC|\text{ - 16} \\ 4\cdot|BC|\text{ = 16} \\ \textcolor{#FF7968}BC\text{\textcolor{#FF7968}{ = 4 miles}} \end{gathered}`

Use Eq1 to evaluate ( AB ) as follows:

`\begin{gathered} |AB|\text{ = 5}\cdot|BC| \\ |AB|\text{ = 5}\cdot4 \\ \textcolor{#FF7968}\text{\textcolor{#FF7968}{ = 20 miles}} \end{gathered}`

The correspnding inter-city distances are as follows:

`\textcolor{#FF7968}\text{\textcolor{#FF7968} = 4 miles, }`