Q17. - Algebra 1

Question

Record your answers on a sheet of paper. Show your work.

Theresa is saving money for a vacation. She needs to save at least $\backslash (640$ to pay for her expenses. Each week, she puts 19 $\backslash )35$ towards her vacation savings.

a. Let $\mathit{w}$ represent the number of weeks Theresa saves money. Write an inequality to model the situation.

b. Solve the inequality from part a. What is the minimum number of weeks Theresa must save money to reach her goal?

c. If Theresa were to save $\$45$ each week instead, by how many weeks would the minimum savings time be decreased?

Step-by-Step Solution

Answer

a. The inequality to model the situation is $\mathbf{35}\mathit{w}\mathbf{\ge }\mathbf{640}$.

b. The minimum number of weeks that Theresa must save money to reach her goal is 19.

c. If Theresa were to save $\$45$ each week instead, the number of weeks by which the minimum savings time would be decreased is 4.

Step 1. Write the given information.

It is given that each week the money saved by Theresa is $\$35$ and she needs to save at least $\$640$.

Step 2. Write the inequality for the given situation.

Let $\mathit{w}$ be the number of weeks Theresa saves money.

It is given that each week the money saved by Theresa is $\$35$ and she needs to save at least $\$640$. 

Step3. Calculation

a.

Therefore, the inequality for the given situation is:

$35w\ge 640$

The inequality to model the situation is $35w\ge 640$.

b.

The solution of the inequality $35w\ge 640$ is given by:

$\begin{array}{c}35w\ge 640\\ w\ge \frac{640}{35}\\ w\ge 18.3\end{array}$

The solution to the inequality $35w\ge 640$ is $w\ge 18.3$.

Therefore, the minimum value of $w$ an integer is 19.

Therefore, the minimum number of weeks that Theresa must save money to reach her goal is 19 .

c.

Let $w$ be the number of weeks Theresa saves money.

It is given that each week the money saved by Theresa is $\$45$ and she needs to save at least $\$640$.

Therefore, the inequality for the given situation is:

$45w\ge 640$

The inequality to model the situation is $45w\ge 640$.

d. 

The solution of the inequality $45w\ge 640$ is given by:

$\begin{array}{c}45w\ge 640\\ w\ge \frac{640}{45}\\ w\ge 14.2\end{array}$

 

The solution to inequality $45w\ge 640$ is $w\ge 14.2$.

Therefore, the minimum value of $w$ an integer is 15.

Therefore, If Theresa were to save $\$45$ each week instead, the number of weeks by which the minimum savings time would be decreased is given by: $19-15=4$

Therefore, If Theresa were to save $\$45$ each week instead, the number of weeks by which the minimum savings time would be decreased is 4.