Chapter 2, Problem 18CT

i.

To determine

To calculate: A state imposes a sales tax on items of clothing that cost more than $175. The tax applies only to the difference of the price of the item and $175.Use the receipt shown to find the tax rate(as a percent).

  

Answer to Problem 18CT

The tax rate is 6.25%.

Explanation of Solution

Given information: According to the receipt price of an item without tax is $295 and on that tax imposed is $7.5 and tax applies only to the difference of the price of the item and $175.

Calculation:

Now, let $r$ be the rate of tax on receipt shown. so,

  $r*\left(productprice-175\right)=7.5$

  $r*\left(295-175\right)=7.5$

  $r*120=7.5$

  $r=\frac{7.5}{120}=6.25$

The tax rate on above product is 6.25.

ii.

To determine

To calculate: A state imposes a sales tax on items of clothing that cost more than $175. The tax applies only to the difference of the price of the item and $175. A shopper has $430 to spend on a winter coat. write and solve the inequality to find the prices $p$ of coats that the shopper can afford. Assume that $p\ge 175$ .

Answer to Problem 18CT

The shopper can afford price less than or equal to $415.

Explanation of Solution

Given information: $p$ is the price of coat, and shopper has $430 to spend on winter coat.

Calculation:

Using above information we can write the inequality

  $p+r*\left(p-175\right)\le 430$

Now we solve the above inequality

  $p*\left(1+r\right)-175r\le 430$

  $p*\left(1+\frac{6.25}{100}\right)-175*\frac{6.25}{100}\le 430$

  $1.0652p-10.9375\le 430$

  $1.0652*p\le 440.9375$

  $p\le \frac{440.9375}{1.0652}$

  $p\le 415$

iii.

To determine

To calculate: A state imposes a sales tax on items of clothing that cost more than $175. The tax applies only to the difference of the price of the item and $175. Another state imposes a 5% sales tax on the entire price of an item of clothing. For which prices would paying the 5% tax be cheaper than paying the tax described above? Write and solve an inequality to find your answer and list three prices that are solutions.

Answer to Problem 18CT

$875 is cheaper

Explanation of Solution

Given information: state imposes a 5% sales tax on the entire price of an item of clothing. Now we find the prices $t$ for which paying the 5% tax on the entire price of the object is cheaper than paying the 6.25% on the difference between the object's price and $175.

Calculation:

  $\left(1+0.05\right)*t\le x+0.0625\left(x-175\right)$

  $1.05*t\le 1.0625*t-10.9375$

  $0\le \left(1.0625-1.05\right)*t-10.9375$

  $10.9375\le 0.0125*t$

  $\frac{10.9375}{0.0125}\le t$

  $t\ge 875$