Chapter 1.2, Problem 33E

a.

To determine

To find: $v\left(p\right)$

Answer to Problem 33E

  $\frac{7}{47}p$

Explanation of Solution

Given information:

  $\text{VAT paid, }v\left(p\right)=17.5\%\text{ of raw material}$

Calculation:

Let x be the raw price.

The price including VAT (p) is as follows:

  $p=x+v\left(p\right)$

Substitute the required values in the above equation to evaluated raw price (x) as follows:

  $\begin{array}{c}p=x+v\left(p\right)\\ =x+17.5\%x\\ =x+0.175x\\ p=1.175x\\ x=\frac{p}{1.175}\end{array}$

Substitute x in $v\left(p\right)$ and solve as follows:

  $\begin{array}{c}v\left(p\right)=17.5\%x\\ =17.5\%\left(\frac{p}{1.175}\right)\\ =0.175\left(\frac{p}{1.175}\right)\\ =\frac{7}{47}p\end{array}$

Thus, the value of v(p) is $\frac{7}{47}p$ .

b.

To determine

To find: $r\left(v\right)$

Answer to Problem 33E

  $0.84v$

Explanation of Solution

Given information:

Function for refund amount = r(v)

  $\begin{array}{c}r\left(v\right)=84\%v\\ v=\text{ VAT or }v\left(p\right)\end{array}$

Calculation:

Convert $r\left(v\right)$ from percentage into decimal.

  $\begin{array}{c}r\left(v\right)=84\%v\\ =0.84v\end{array}$

Thus, the value is 0.84v.

c.

To determine

To find: $r\left(p\right)$

Answer to Problem 33E

  $\frac{147}{1175}p$

Explanation of Solution

Given information:

  $\begin{array}{c}r\left(p\right)=\text{ refund}\\ v\left(p\right)=\frac{7}{47}p\\ r\left(v\right)=0.84v\\ p=\text{ price including VAT}\\ v\left(p\right)=\text{ VAT paid}\end{array}$

Calculation:

Since $v\approx v\left(p\right)$ , the refund function can be rewritten as follows:

  $r\left(v\left(p\right)\right)=0.84v\left(p\right)$

Substitute the values of $v\left(p\right)$ to calculate the refund price function as follows:

  $\begin{array}{c}r\left(v\left(p\right)\right)=0.84\left(\frac{7}{47}p\right)\\ r\left(p\right)=0.84\left(\frac{7}{47}p\right)\\ =\left(\frac{84}{100}\right)\left(\frac{7}{47}p\right)\\ =\left(\frac{21}{25}\right)\left(\frac{7}{47}p\right)\\ =\frac{147}{1175}p\end{array}$

Thus, the function of refund amount is $\frac{147}{1175}p$ .

d.

To determine

To find: refund due on given automobile price.

Answer to Problem 33E

  $\begin{array}{c}r\left(p_1\right)=\$52.94\\ r\left(p_2\right)=\$28.23\\ r\left(p_3\right)=\$99.72\end{array}$

Explanation of Solution

Given information:

  $r\left(p\right)=\frac{147}{1175}p$

The three prices are as follows:

  $\begin{array}{l}{p}_1=\$423.18\\ p_2=\$225.64\\ p_3=\$797.05\end{array}$

Calculation:

Substitute the value of $p_1$ in the given equation, to evaluate refund due as shown below.

  $\begin{array}{c}r\left(p_1\right)=\frac{147}{1175}{p}_1\\ =\frac{147}{1175}\left(\$423.18\right)\\ =\$52.94\end{array}$

Substitute the value of $p_2$ in the given equation, to evaluate refund due as shown below.

  $\begin{array}{c}r\left(p_2\right)=\frac{147}{1175}{p}_2\\ =\frac{147}{1175}\left(\$225.64\right)\\ =\$28.23\end{array}$

Substitute the value of $p_3$ in the given equation, to evaluate refund due as shown below.

  $\begin{array}{c}r\left(p_3\right)=\frac{147}{1175}{p}_3\\ =\frac{147}{1175}\left(\$797.05\right)\\ =\$99.72\end{array}$

Thus, the refund due on $p_1,p_2,\text{ and }{p}_3$ is $\$52.94,\$28.23,$ and $\$99.72$respectively.