Jonathan and Oscar began saving money the same week. The table shows the models for the amount of money Jonathan and Oscar have saved after x weeks.

Jonathan's savings: f(x) =18x +5
Oscars savings: g(X)=8x+25

1 answer

To analyze the savings of Jonathan and Oscar, we can use the given equations to determine their respective savings over time.

Jonathan's Savings

The model for Jonathan's savings is: \[ f(x) = 18x + 5 \] This means that Jonathan saves $18 each week and started with $5.

Oscar's Savings

The model for Oscar's savings is: \[ g(x) = 8x + 25 \] This means that Oscar saves $8 each week and started with $25.

Calculating Savings Over Time

Let's calculate their savings over a few weeks to understand how each person's savings grow:

  1. After 0 weeks (initial savings):

    • \( f(0) = 18(0) + 5 = 5 \) (Jonathan)
    • \( g(0) = 8(0) + 25 = 25 \) (Oscar)
  2. After 1 week:

    • \( f(1) = 18(1) + 5 = 23 \)
    • \( g(1) = 8(1) + 25 = 33 \)
  3. After 2 weeks:

    • \( f(2) = 18(2) + 5 = 41 \)
    • \( g(2) = 8(2) + 25 = 41 \)
  4. After 3 weeks:

    • \( f(3) = 18(3) + 5 = 59 \)
    • \( g(3) = 8(3) + 25 = 49 \)
  5. After 4 weeks:

    • \( f(4) = 18(4) + 5 = 77 \)
    • \( g(4) = 8(4) + 25 = 57 \)

Summary of Savings

  • After 0 weeks: Jonathan - $5, Oscar - $25
  • After 1 week: Jonathan - $23, Oscar - $33
  • After 2 weeks: Jonathan - $41, Oscar - $41
  • After 3 weeks: Jonathan - $59, Oscar - $49
  • After 4 weeks: Jonathan - $77, Oscar - $57

Comparison

From this, we can see that:

  • In the first week, Oscar is ahead, but they tie after 2 weeks.
  • After 2 weeks, Jonathan begins to pull ahead, showing that his savings rate allows him to surpass Oscar's total savings eventually.

Finding the Week They Are Equal

To find out when they have the same amount saved, we can set the equations equal to each other: \[ 18x + 5 = 8x + 25 \]

  1. Subtract \(8x\) from both sides: \[ 10x + 5 = 25 \]

  2. Subtract 5 from both sides: \[ 10x = 20 \]

  3. Divide both sides by 10: \[ x = 2 \]

Thus, Jonathan and Oscar will have the same amount of savings after 2 weeks.