Chapter 9, Problem 20CLT

To determine

To find: The dollars worth of gym shoes, jogging shoes and walking shoes, sold that year.

Answer to Problem 20CLT

Gym shoes worth $97.55\text{ dollars}$ , jogging shoes worth $73.42\text{ dollars}$ and walking shoes worth $250.35\text{ dollars}$ were sold that year..

Explanation of Solution

Given:

The percents (in decimals) of the total amounts spent on three types of footwear in the recent year is:

  $\begin{array}{c}\text{Age}\\ \text{Group}\end{array}\{\begin{array}{c}14-17\\ 18-24\\ 25-34\end{array}\stackrel{\begin{array}{l}\text{Gym Jogging Walking}\\ \text{Shoes Shoes Shoes}\end{array}}{\overbrace{\left[\begin{array}{ccc}0.079& 0.064& 0.029\\ 0.050& 0.060& 0.022\\ 0.103& 0.259& 0.085\end{array}\right]}}$

The total amount (in millions of dollars) spent by the age groups on the three types of footwear are:

  $\begin{array}{c}14-17\text{ age group}\to 479.88\text{ dollars}\\ 18-24\text{ age group}\to \text{ 365}.88\text{ dollars}\\ 25-44\text{ age group}\to \text{ 1248}.89\text{ dollars}\end{array}$

Calculation:

In order to determine the individual prices of the gym shoes, jogging shoes and walking shoes, multiply the percent matrix with the amount that the three age groups spent on the three types of shoes, as shown below:

  $\begin{array}{c}\left[\begin{array}{ccc}0.079& 0.064& 0.029\\ 0.050& 0.060& 0.022\\ 0.103& 0.259& 0.085\end{array}\right]\times \left[\begin{array}{c}479.88\\ 365.88\\ 1248.89\end{array}\right]\\ =\left[\begin{array}{c}97.54465\\ 73.42238\\ 250.34621\end{array}\right]\end{array}$

So, gym shoes worth $97.55\text{ dollars}$ , jogging shoes worth $73.42\text{ dollars}$ and walking shoes worth $250.35\text{ dollars}$ were sold that year.