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Carly receives $275 every week for babysitting for her neighbors. She decides to save the money she receives to help pay for a camp she wants to attend next summer. She tracks how much money she saves from babysitting over the course of the year.

11 answers

To calculate how much money Carly will save over the course of the year from her babysitting job, we can follow these steps:

  1. Determine Weekly Earnings: Carly earns $275 each week.
  2. Determine Number of Weeks in a Year: There are 52 weeks in a year.
  3. Calculate Total Savings for the Year: Multiply her weekly earnings by the number of weeks in a year.

\[ \text{Total Savings} = \text{Weekly Earnings} \times \text{Number of Weeks} \]

\[ \text{Total Savings} = 275 \times 52 \]

\[ \text{Total Savings} = 14,300 \]

Thus, if Carly saves all her babysitting earnings, she will save $14,300 over the course of the year to help pay for the camp she wants to attend next summer.

Explain why Carly’s savings over time is an example of linear change. in red font

Carly’s savings over time is an example of linear change because:

  1. Constant Rate of Change: Carly saves a fixed amount of $275 every week. This consistent amount indicates that her savings increase at a constant rate without variation.

  2. Proportional Relationship: The amount of money she saves is directly proportional to the number of weeks she babysits. For every week, her savings increase by the same amount ($275), resulting in a straight-line graph when plotted.

  3. Straight-Line Graph: If you were to graph her savings over time, the x-axis would represent time (in weeks), and the y-axis would represent savings. The points plotted on the graph would form a straight line, reflecting the uniform increase in her savings.

  4. Predictable Future Savings: Because the savings increase in a uniform manner, it's easy to predict how much Carly will have saved after any given number of weeks. This predictability is a hallmark of linear change.

In summary, Carly’s weekly savings of $275 leads to a linear relationship, where her total savings grow at a steady and predictable rate over time.

(In Red Font): Carly's savings over time exemplifies linear change due to her constant weekly earnings and the direct proportionality of her savings to time, resulting in a straight-line relationship when graphed.

all together and less words

Carly's savings over time is an example of linear change because she saves a constant amount of $275 every week. This consistent rate results in a straight-line graph when her savings are plotted against time, demonstrating a predictable and uniform increase. Thus, her total savings grow steadily, reflecting a linear relationship.

Create a sequence that describes Carly’s growing savings each week over the course of 20 weeks. (Show all your work). (2 pts)
again less words and more just math

Carly's weekly savings sequence for 20 weeks can be described as follows:

  1. Initial Savings (Week 0): $0
  2. Weekly Increase: $275

Sequence Calculation:

  • Week 1: \( 275 \times 1 = 275 \)
  • Week 2: \( 275 \times 2 = 550 \)
  • Week 3: \( 275 \times 3 = 825 \)
  • Week 4: \( 275 \times 4 = 1,100 \)
  • Week 5: \( 275 \times 5 = 1,375 \)
  • Week 6: \( 275 \times 6 = 1,650 \)
  • Week 7: \( 275 \times 7 = 1,925 \)
  • Week 8: \( 275 \times 8 = 2,200 \)
  • Week 9: \( 275 \times 9 = 2,475 \)
  • Week 10: \( 275 \times 10 = 2,750 \)
  • Week 11: \( 275 \times 11 = 3,025 \)
  • Week 12: \( 275 \times 12 = 3,300 \)
  • Week 13: \( 275 \times 13 = 3,575 \)
  • Week 14: \( 275 \times 14 = 3,850 \)
  • Week 15: \( 275 \times 15 = 4,125 \)
  • Week 16: \( 275 \times 16 = 4,400 \)
  • Week 17: \( 275 \times 17 = 4,675 \)
  • Week 18: \( 275 \times 18 = 4,950 \)
  • Week 19: \( 275 \times 19 = 5,225 \)
  • Week 20: \( 275 \times 20 = 5,500 \)

Sequence of Savings:

0, 275, 550, 825, 1,100, 1,375, 1,650, 1,925, 2,200, 2,475, 2,750, 3,025, 3,300, 3,575, 3,850, 4,125, 4,400, 4,675, 4,950, 5,225, 5,500.

Create a formula for Carly’s savings over the course of time. (Show all your work) (2 pts)

To create a formula for Carly's savings over time, we define the variables and follow these steps:

Step 1: Define Variables

  • Let S be the total savings after t weeks.
  • Let t be the time in weeks.
  • Let R be the amount saved per week (which is $275).

Step 2: Establish the Relationship

Carly starts with $0 in savings, and she saves a constant amount of $275 each week. Therefore, her total savings can be expressed as:

\[ S(t) = R \times t \]

Step 3: Substitute the Value of R

Since Carly saves $275 each week:

\[ S(t) = 275 \times t \]

Final Formula

Thus, the formula for Carly's savings over time is:

\[ S(t) = 275t \]

Explanation

  • t represents the number of weeks.
  • S(t) gives the total savings after t weeks.

This formula allows us to calculate Carly's total savings at any point in time by simply substituting the number of weeks (t) into the equation.

less words and more math

Formula Derivation for Carly's Savings

  1. Variables:

    • \( S \) = total savings
    • \( t \) = time in weeks
    • \( R = 275 \) (savings per week)

  2. Relationship: \[ S(t) = R \times t \]

  3. Substitute \( R \): \[ S(t) = 275 \times t \]

Final Formula:

\[ S(t) = 275t \]