Question

Two toy rental companies charge a fixed amount of money for booking a toy and some extra money for each day the toy is booked.
Company 1: Booking charge of $\$17.5$ and an additional $\$0.25$ for each day the customers keep the toys.

Company 2:



If Joy wants to keep the toys for $30$days, then which company will he choose for less rental pay?

• A. Both the companies will charge the same rent
  
• B. Company $1$
• C. Company $2$
• D. Cannot be determined

Answer 1

The correct option is B Company $1$

We will find out the function rules for both the toy companies.
The fixed rate is always considered as the $y$-intercept,
and the variable rate is considered as the rate of change or slope.

Company 1: Booking charge of $\$17.5$ and an additional $\$0.25$ for each day the customers keep the toys.
Here, the y-intercept $=b=17.5$
And slope or rate of change $=m=0.25$
The standard form of a linear function is $y=mx+b$.
So, Function $1$ can be written as
$y=0.25x+17.5$.
$x=$ Number of hours for which the toy will be rented
$y=$ Total rental pay

If Joy wants to keep the toys for $30$ days, then he’ll have to pay $=y=mx+b$

$=0.2530+17.5$
$==7.5+17.5$
$=25$

Joy needs to pay $\$25$ for $30$ days.

Company 2:


Slope or rate of change $=\frac{Changeinyvalues}{Changeinxvalues}=\frac{1}{2}or\frac{1.5}{3}or\frac{2}{4}=0.5$

We know the standard form of the linear function is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.
Let’s consider an input and output order pair of the function.
$(1,15)$ is an input and output order pair, where $x=1$ and $y=15$.
Replacing the above two values of $x$ and $y$ in $y=mx+b$,

$\Rightarrow 15=0.5\times 1+b$
$\Rightarrow 15=0.5+b$
$\Rightarrow 15-0.5=b$
$\Rightarrow 14.5=b$

So, the y-intercept $=14.5$ and it means for $x=0,y=14.5$.
So, Function $2$ can be written as $y=mx+b$.

$\Rightarrow y=0.5x+14.5$

$x=$ Number of hours for which the toy will be rented
$y=$Total rental pay
If Joy wants to keep the toys for $30$ days, then he’ll have to pay $=y=mx+b$

$=0.5\times 30+14.5$

$=15+14.5$

$=\$29.5$

Joy needs to pay $\$29.5$ for $30$ days.

So, comparing the price for $30$ days of both the companies, it can be concluded that Company $1$ is charging less as compared to Company $2$ (as $\$25<\$29.5$).

So, option B is the correct answer.