Chapter 5.5, Problem 18HP

To determine

To find:The proportional relationship between the Monica’s age and Patrice’s age by using a table and the value of constant of proportionality.

Answer to Problem 18HP

The age relationship of Monica’s to Patrice’s is proportional and constant of proportionalityis $2$ .

Explanation of Solution

Given information:

The age of Monica is $12\text{years}$ and Patrice is $6\text{years}$ .

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 expression of Proportional relationships having constant of proportionality $k$ in equation form is:

  $y=kx$

To obtain the constant rate, substitute $12\text{years}$ for $y$ and $6\text{years}$ for $x$ in the expression of proportional relationship.

  $\begin{array}{c}12\text{years}=k\times 6\text{years}\\ k=\frac{\text{12}\text{years}}{\text{6}\text{years}}\\ =2\end{array}$

The relationship between Monika’s age to Patrice’s age to make table is as follows:

  $\text{Patrice's}\text{age}=\frac{\text{Monika's}\text{age}}{k}$

To find Patrice’s age when Monika is $24\text{years}$ , substitute $24\text{years}$ for Monika’s age and $2$ for $k$ in the above expression.

  $\begin{array}{c}\text{Patrice's}\text{age}=\frac{24\text{years}}{2}\\ =12\text{years}\end{array}$

To find Patrice’s age when Monika is $36\text{years}$ , substitute $36\text{years}$ for Monika’s age and $2$ for $k$ in the above expression.

  $\begin{array}{c}\text{Patrice's}\text{age}=\frac{36\text{years}}{2}\\ =18\text{years}\end{array}$

The table showing Monica’s age to Patrice’s age is shown below:

Monika’s Age ( $\text{years}$ )	$12$	$24$	$36$
Patrice’s Age ( $\text{years}$ )	$6$	$12$	$18$

The set to be said proportional, if Monica’s age to Patrice’s age have equivalent rates. So, the expression for equivalent rates is:

  $\frac{12}{6}=\frac{24}{12}=\frac{36}{18}=2$

Therefore, from the above expression it is clear that age relationship of Monica’s to Patrice’s is proportional and constant of proportionality is $2$ .