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Let’s break down this problem step-by-step.

### Problem Analysis:
You want to buy a bicycle, and the problem has two parts:
1. You are saving money each week with some initial savings.
2. There’s an additional target for a more expensive bicycle, which requires extra saving weeks.

---

#### Given Information:
- Cost of the first bicycle = $155.75
- Initial savings = $30
- Weekly savings from you = $5
- Additional weekly contribution from your aunt = $10
- Effective weekly saving total = $5 + $10 = $15

#### Part (a): Find the Number of Weeks to Save $155.75
Let's set up an equation:
1. **Starting amount**: $30
2. **Weekly addition**: $15
3. **Goal**: $155.75

The formula for total savings after $ w $ weeks is:
\[
\text{Total Savings} = 30 + 15w
\]

To find $ w $, set this equal to the bicycle cost:
\[
30 + 15w = 155.75
\]

Solving for $ w $:
1. Subtract 30 from both sides:
   \[
   15w = 125.75
   \]
2. Divide by 15:
   \[
   w = \frac{125.75}{15} = 8.3833
   \]

Since you can't save for a fraction of a week, round up to the next whole week. Therefore, **it will take 9 weeks** to have enough money to buy the bicycle.

---

#### Part (b): Additional Weeks to Save for a More Expensive Bicycle
Now, let's find the number of weeks needed to save for a bicycle that costs $203.89.

1. New goal: $203.89
2. The equation becomes:
   \[
   30 + 15w = 203.89
   \]
3. Solving for $ w $:
   - Subtract 30 from both sides:
     \[
     15w = 173.89
     \]
   - Divide by 15:
     \[
     w = \frac{173.89}{15} = 11.5927
     \]

Rounding up, **it will take 12 weeks** to save enough for this more expensive bicycle.

---

#### Summary:
- **Answer to (a)**: 9 weeks to save $155.75.
- **Answer to (b)**: 12 weeks for $203.89, which is **3 more weeks than for the first bicycle**.

Would you like further clarification or additional help on any part?

---

### Further Questions:
1. How would the number of weeks change if the weekly savings increased?
2. What if the bicycle’s price decreased by $10? How many weeks would that save?
3. How much would you save in total if you continued saving for 15 weeks?
4. What if your aunt stopped contributing? How would that affect the saving time?
5. How much would you need to save weekly to buy the bicycle in 6 weeks?

### Tip:
For budgeting, setting small milestones helps track your progress and keeps you motivated!

You are saving money to buy a new bicycle that costs $155.75. You have $30 and plan to save $5 each week. Your aunt decides to give you an additional $10 each week.
a. How many weeks will you have to save until you have enough money to buy the bicycle?
b. How many more weeks would you have to save to buy a new bicycle that costs $203.89? Explain how you found your answer.

Learn how to calculate the number of weeks needed to save for a bicycle costing $155.75 with an initial savings of $30, weekly savings of $5, and an additional weekly contribution of $10. This problem covers linear equations and savings calculations. Also find out how many more weeks are required if the bicycle costs $203.89.

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