Chapter 5.5, Problem 11IP

To determine

To find:The proportional relationship between the cost and the number of rides in an amusement park and the constant of proportionality by completing the table (1).

Answer to Problem 11IP

The cost of the ride is notproportional to the number of rides.

Explanation of Solution

Given information: The cost per ride to an amusement park is $\$4$ plus $\$1.50$ .

Calculation:

Two quantities in which the ratio or rate is constant, are to be said proportional and if the rates are not constant, then the quantities are said to be non-proportional. The constant ratio which is defining whether the two quantities are said to be proportional or not is known as constant of proportionality.

Let, number of rides be $x$ and cost of ride be $y$ .

The expression for proportional relationship for number of rides to cost of ride is:

  $x=\$4+\$1.50\times y$

To find the cost for $1$ ride, substitute $1$ for $x$ and $1$ for $y$ in the above expression.

  $\begin{array}{c}1\text{ride}=\$4+\$1.50\times 1\\ 1\text{ride}=\$4+\$1.5\\ =\$5.5\end{array}$

To find cost for $2$ rides, substitute $2$ for $x$ and $2$ for $y$ in the above expression.

  $\begin{array}{c}2\text{rides}=\$4+\$1.50\times 2\\ =\$4x+\$3\\ =\$7\end{array}$

To find cost for $3$ rides, substitute $3$ for $x$ and $3$ for $y$ in the above expression.

  $\begin{array}{c}3\text{rides}=\$4+\$1.50\times 3\\ =\$4+\$5\\ =\$9\end{array}$

To find cost for $4$ rides, substitute $4$ for $x$ and $4$ for $y$ in the above expression.

  $\begin{array}{c}4\text{rides}=\$4+\$1.50\times 4\\ =\$4+\$6\\ =\$10\end{array}$

To find cost for $5$ rides, substitute $5$ for $x$ and $5$ for $y$ in the above expression.

  $\begin{array}{c}5\text{rides}=\$4+\$1.50\times 5\\ =\$4+\$7.5\\ =\$11.5\end{array}$

The table showing rate of cost to number of rides is shown below:

Number of Rides	$1$	$2$	$3$	$4$	$5$
Cost ( $\$$ )	$\$5.5$	$\$7$	$\$9$	$\$10$	$\$11.5$

Table-( $1$ )

The set to be said proportional, if cost to number of rides have equivalent rates. So, the expression for equivalent rates is:

  $\begin{array}{l}\frac{\$5.5}{1}=5.5\\ \frac{\$7}{2}=3.5\\ \frac{\$9}{3}=3\\ \frac{\$10}{4}=2.5\\ \frac{\$11.5}{5}=2.3\end{array}$

The above expression shows that cost to number of rides is not proportional as they do not have a constant of proportionality.

  $\frac{\$5.5}{1}\ne \frac{\$7}{2}\ne \frac{\$9}{3}\ne \frac{\$10}{4}\ne \frac{\$11.5}{5}\ne 5.5$

Therefore, from the above expression it is clear that the cost of the ride is not proportional.