Chapter 5.5, Problem 13IP

a.

To determine

To find: The proportional relationship between the number of hot dog packages and the number of hot dogs. Also, write the constant of proportionality.

Answer to Problem 13IP

The number of hot dog packages is proportional to the number of hot dogs and the constant of proportionality is $8$ .

Explanation of Solution

Given information:The number of hot dogs per package is $8$ .

Calculation:

Two quantities in which the ratio or rate is constant, are to be said proportional and if the rates are not constant, then the quantities are said to be non-proportional. The constant ratio which is defining whether the two quantities are said to be proportional or not is known as constant of proportionality.

The number of hot dogs per package is:

  $1\text{package}=8\text{hot}\text{dogs}$

To find number of hot dogs for $2$ packages, multiply the both sides of the above expression of hot dogs per package by $2$ .

  $\begin{array}{c}2\times 1\text{package}=2\times 8\text{hot}\text{dogs}\\ =16\text{hot}\text{dogs}\end{array}$

To find number of hot dogs for $3$ packages, multiply the both sides of the above expression of hot dogs per package by $3$ .

  $\begin{array}{c}3\times 1\text{package}=3\times 8\text{hot}\text{dogs}\\ =24\text{hot}\text{dogs}\end{array}$

The table showing number of hot dogs packages to number of hot dogs is shown below:

Number of hot dogs packages	$1$	$2$	$3$
Hot Dogs	$8$	$16$	$24$

The set is said to be proportional, if number of hot dogs to number of hot dogs packages have equivalent rates. So, the expression for equivalent rates is:

  $\frac{8}{1}=\frac{16}{2}=\frac{24}{3}=8$

Therefore, from the above expression it is clear that the number of hot dogs to number of hot dogs packages and constant of proportionality is $8$ .

b.

To determine

To find: The proportional relationship between the number of hot dogs and the number of hot dog buns and the constant of proportionality.

Answer to Problem 13IP

The number of hot dogs is proportional to the number of hot dog buns and the constant of proportionality is $\frac{5}{4}$ .

Explanation of Solution

Given information:The number of hot dogs per package is $8$ and the number of hot dog buns per package is $10$ .

Calculation:

The number of hot dogs buns and hot dogs per package is:

  $10\text{hot}\text{dog}\text{buns}=8\text{hot}\text{dogs}$

To find number of hot dog buns and hot dogs for $2$ packages, multiply the both sides of the above expression of hot dogs and hot dog buns per package by $2$ .

  $\begin{array}{c}2\times 10\text{hot}\text{dog}\text{buns}=2\times 8\text{hot}\text{dogs}\\ 20\text{hot}\text{dog}\text{buns}=16\text{hot}\text{dogs}\end{array}$

To find number of hot dogs buns and hot dogs for $3$ packages, multiply the both sides of the above expression of hot dogs and hot dog buns per package by $3$ .

  $\begin{array}{c}3\times 10\text{hot}\text{dog}\text{buns}=3\times 8\text{hot}\text{dogs}\\ 30\text{hot}\text{dog}\text{buns}=24\text{hot}\text{dogs}\end{array}$

The table showing number of hot dog buns to number of hot dogs is shown below:

Hot dogs	$8$	$16$	$24$
Hot Dog buns	$10$	$20$	$30$

The set to be said proportional, if number of hot dog buns to number of hot dogs have equivalent rates. So, the expression for equivalent rates is:

  $\frac{10}{8}=\frac{20}{16}=\frac{30}{24}=\frac{5}{4}$

Therefore, from the above expression it is clear that the number of hot dog buns to number of hot dogs and constant of proportionality is $\frac{5}{4}$ .