Chapter 5.5, Problem 14IP

(a)

To determine

To find:. The proportional situation between Isabel savings and her sister savings by showing the first six weeks saving for each girl in the tables.

Answer to Problem 14IP

The money saved by Isabel is not proportional to the number of weeks and the money saved by her sister is proportional to the number of weeks.

Explanation of Solution

Given information: The cost saved by Isabel per week is $\$20$ . The cost Isabel sister has is $\$10$ and an additional money saved by her each week is $\$20$ .

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 cost saved by Isabel per week is:

  $1\text{week}=\$20$

To find cost for $2$ weeks, multiply the both sides of the above expression of $1$ week cost by $2$ .

  $\begin{array}{c}2\times 1\text{week}=2\times \$20\\ =\$40\end{array}$

To find cost for $3$ weeks, multiply the both sides of the above expression of $1$ week cost by $3$ .

  $\begin{array}{c}3\times 1\text{week}=3\times \$20\\ =\$60\end{array}$

To find cost for $4$ weeks, multiply the both sides of the above expression of $1$ week cost by $4$ .

  $\begin{array}{c}4\times 1\text{week}=4\times \$20\\ =\$80\end{array}$

To find cost for $5$ weeks, multiply the both sides of the above expression of $1$ week cost by $5$ .

  $\begin{array}{c}5\times 1\text{week}=5\times \$20\\ =\$100\end{array}$

To find cost for $6$ weeks, multiply the both sides of the above expression of $1$ week cost by $6$ .

  $\begin{array}{c}6\times 1\text{week}=6\times \$20\\ =\$120\end{array}$

The table showing rate of cost to number of weeks is shown below:

Number of weeks	$1$	$2$	$3$	$4$	$5$	$6$
Cost ( $\$$ )	$\$20$	$\$40$	$\$60$	$\$80$	$\$100$	$\$120$

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

  $\frac{\$20}{1}=\frac{\$40}{2}=\frac{\$60}{3}=\frac{\$80}{4}=\frac{\$100}{5}=\frac{\$120}{6}=20$

Therefore, from the above expression it is clear that the cost of the savings is proportional to the number of the weeks and constant of proportionality is $20$ .

The cost saved by Isabel sister per week is:

  $1\text{week}=\$20+10$

To find cost for $2$ weeks, multiply the both sides of the above expression of $1$ week cost by $2$ .

  $\begin{array}{c}2\times 1\text{week}=2\times \$20+10\\ =\$50\end{array}$

To find cost for $3$ weeks, multiply the both sides of the above expression of $1$ week cost by $3$ .

  $\begin{array}{c}3\times 1\text{week}=3\times \$20+10\\ =\$70\end{array}$

To find cost for $4$ weeks, multiply the both sides of the above expression of $1$ week cost by $4$ .

  $\begin{array}{c}4\times 1\text{week}=4\times \$20+10\\ =\$90\end{array}$

To find cost for $5$ weeks, multiply the both sides of the above expression of $1$ week cost by $5$ .

  $\begin{array}{c}5\times 1\text{week}=5\times \$20+10\\ =\$110\end{array}$

To find cost for $6$ weeks, multiply the both sides of the above expression of $1$ week cost by $6$ .

  $\begin{array}{c}6\times 1\text{week}=6\times \$20+10\\ =\$130\end{array}$

The table showing rate of cost to number of weeks is shown below:

Number of weeks	$1$	$2$	$3$	$4$	$5$	$6$
Cost ( $\$$ )	$\$30$	$\$50$	$\$70$	$\$90$	$\$110$	$\$130$

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

  $\frac{\$30}{1}\ne \frac{\$50}{2}\ne \frac{\$70}{3}\ne \frac{\$90}{4}\ne \frac{\$110}{5}\ne \frac{\$130}{6}$

Therefore, from the above expression it is clear that the cost of the savings is proportional to the number of the weeks and thus no constant of proportionality.

(b)

To determine

To find: The equation showing the number of weeks and saving cost proportional relationship.

Answer to Problem 14IP

The equation showing the number of weeks and saving cost of Isabel proportional relationship is $t=20p$ and the equation showing the number of weeks and saving cost of Isabel’s sister proportional relationshipis $t=30p$ .

Explanation of Solution

Given information:The cost saved by Isabel per week is $\$20$ . The cost Isabel sister has is $\$10$ and an additional money saved by her each week is $\$20$ .

Calculation:

The constant of proportionality of Isabel is:

  $\frac{\text{savings}\text{cost}}{\text{number}\text{of}\text{weeks}}=\frac{\$20}{1}=20$

The expression of Proportional relationships in equation form is:

  $y=kx$

Here, the constant of proportionality is $k$ .

The amount of savings to the number of weeks, in which the number of weeks is $p$ and the savings cost is $t$ .

Substitute $t$ for $y$ , $p$ for $x$ and $20$ for $k$ in the expression of proportional relationship.

  $t=20p$

Therefore, the equation showing the number of weeks and saving cost of Isabel proportional relationshipis $t=20p$ .

The constant of proportionality for Isabel’s sister is:

  $\frac{\text{savings}\text{cost}}{\text{number}\text{of}\text{weeks}}=\frac{\$20}{1}=20$

The expression of Proportional relationships for Isabel’s sister in equation form is:

  $y=kx+\$\text{}10$

Here, the constant of proportionality is $k$ and $\$\text{}10$ is the additional saving of her.

Substitute $t$ for $y$ , $p$ for $x$ and $20$ for $k$ in the expression of proportional relationship.

  $t=20p+\$10$

Therefore, the equation showing the number of weeks and saving cost of Isabel sister’s proportional relationshipis $t=20p+\$10$ .

(c)

To determine

To explain: The comparison of Graph A and Graph B.

Explanation of Solution

Given information: The savings per week of Isabel and Isabel’s sister is shown in the following graph

Graph:

  

Interpretation:

To determine whether the relationship is proportional or not, the graph must be passing through straight line through the origin.

In the above graph A, $x-\text{axis}$ shows the number of weeks and $y-\text{axis}$ determine the money saved. Here, the line is not passing through the origin but is a straight line. This concluded that cost is not proportional to number of weeks.This implies that Graph A is for Isabel’s sister savings.

In the above graph B, $x-\text{axis}$ shows the number of weeks and $y-\text{axis}$ determine the money saved. Here, the line is passing through the origin and is a straight line. This concluded that cost is proportional to number of weeks. This implies that Graph B is for Isabel’s savings.