Chapter 1.7, Problem 10E

Solution

To write: A mathematical expression for the quantity described verbally.

The required mathematical expression is $x\cdot \frac{100}{108.75}$ dollars.

Given data:

The verbal expression states: ‘Actual cost of an item selling for x dollars, if the sales tax rate is 8.75%.’

Method/Formula used:

Convert verbal expression into mathematical expression by assume base value x or any other variable.

Calculations:

The given statement has two parts, (1) the item is sold for $ x , (2) with an additional tax of 8.75 %. Let y be the actual price of the time. The sale price ‘ y’ dollar is the base number.

Denote the mathematical expression for sale price x after a levy of tax of 8.75% on the actual price y of the item. Then the expression x has the value,

  $\begin{array}{c}x=\stackrel{\text{Statement}\text{}\text{(1)}}{\overbracket{\text{actual saleprice}\text{of}\text{}\text{}\text{item}\$y}}\text{then}\stackrel{\text{Statement 2}}{\overbracket{8.75\%\text{of}x\text{more}}}\\ =y+\frac{8.75}{100}y\because a\%\text{means}\frac{1}{100}\text{of}a\\ =y\left(1+\frac{8.75}{100}\right)\\ =y\left(\frac{100+8.75}{100}\right)\\ =y\left(\frac{108.75}{100}\right)\end{array}$

Solve, further for y :

  $x=\left(\frac{108.75}{100}\right)y$

Multiply both sides by $\frac{100}{108.75}$ ,

  $\begin{array}{c}x\times \frac{100}{108.75}=\left(\frac{\cancel{108.75}}{\cancel{100}}\times \frac{\bcancel{100}}{\cancel{108.75}}\right)y\\ x\times \frac{100}{108.75}=y\end{array}$

Thus, the actual price y of the item is $x\cdot \frac{100}{108.75}$ .

Therefore, the mathematical expression $y\left(x\right)$ for the quantity described verbally( actual price of the item) is $y\left(x\right)=x\cdot \frac{100}{108.75}$ dollars.