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:
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After 0 weeks (initial savings):
- \( f(0) = 18(0) + 5 = 5 \) (Jonathan)
- \( g(0) = 8(0) + 25 = 25 \) (Oscar)
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After 1 week:
- \( f(1) = 18(1) + 5 = 23 \)
- \( g(1) = 8(1) + 25 = 33 \)
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After 2 weeks:
- \( f(2) = 18(2) + 5 = 41 \)
- \( g(2) = 8(2) + 25 = 41 \)
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After 3 weeks:
- \( f(3) = 18(3) + 5 = 59 \)
- \( g(3) = 8(3) + 25 = 49 \)
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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: \[ 18x + 5 = 8x + 25 \]
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Subtract \(8x\) from both sides: \[ 10x + 5 = 25 \]
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Subtract 5 from both sides: \[ 10x = 20 \]
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Divide both sides by 10: \[ x = 2 \]
Thus, Jonathan and Oscar will have the same amount of savings after 2 weeks.