Question

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

Answer 1

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**.