how much does he still owe after 3 years? (8) A car costs $$ \$ 64000 $$. Ir depreciates by $$ 25 \% $$ in the first year, by $$ 20 \% $$ in the second year, and by $$ 15 \% $$ in each of the following years. Find the value of the car to the nearest $$ \$ 50 $$ after 4 years. (9) A person saves $$ \mathrm{z} \ddagger 30000 $$ at $$ 4 \frac{1}{2} \% $$ compound interest. She adds 000 to her amount at the end of each year. Find her total savings after 2 years. (10) A man borrows 185000 at $$ 6 \% $$ compound interest. He pays back $$
ot
ot \pm 45000 $$ at the end of each year. How much does he still owe after he has made his second repayment?

Question

how much does he still owe after 3 years? (8) A car costs $$ \$ 64000 $$. Ir depreciates by $$ 25 \% $$ in the first year, by $$ 20 \% $$ in the second year, and by $$ 15 \% $$ in each of the following years. Find the value of the car to the nearest $$ \$ 50 $$ after 4 years. (9) A person saves $$ \mathrm{z} \ddagger 30000 $$ at $$ 4 \frac{1}{2} \% $$ compound interest. She adds 000 to her amount at the end of each year. Find her total savings after 2 years. (10) A man borrows 185000 at $$ 6 \% $$ compound interest. He pays back $$ 
ot 
ot \pm 45000 $$ at the end of each year. How much does he still owe after he has made his second repayment?

UpStudy AI · Accepted Answer

Let's solve each problem step by step.

### Problem 8: Car Depreciation

1. **Initial Cost**: The car costs $$ C = 64000 $$.
2. **Depreciation in the first year**: The car depreciates by $$ 25\% $$.
   $$
   \text{Value after Year 1} = C \times (1 - 0.25) = 64000 \times 0.75
   $$
3. **Depreciation in the second year**: The car depreciates by $$ 20\% $$.
   $$
   \text{Value after Year 2} = \text{Value after Year 1} \times (1 - 0.20)
   $$
4. **Depreciation in the third year**: The car depreciates by $$ 15\% $$.
   $$
   \text{Value after Year 3} = \text{Value after Year 2} \times (1 - 0.15)
   $$
5. **Depreciation in the fourth year**: The car depreciates by $$ 15\% $$.
   $$
   \text{Value after Year 4} = \text{Value after Year 3} \times (1 - 0.15)
   $$

Now, let's calculate these values step by step.

### Problem 9: Savings with Compound Interest

1. **Initial Savings**: The person saves $$ S = 30000 $$.
2. **Interest Rate**: The interest rate is $$ r = 4.5\% = 0.045 $$.
3. **Annual Addition**: The person adds $$ A = 1000 $$ at the end of each year.
4. **Total Savings after 2 years**:
   - After Year 1:
   $$
   \text{Total after Year 1} = S \times (1 + r) + A
   $$
   - After Year 2:
   $$
   \text{Total after Year 2} = \text{Total after Year 1} \times (1 + r) + A
   $$

Now, let's calculate these values step by step.

### Problem 10: Loan Repayment

1. **Initial Loan Amount**: The man borrows $$ L = 185000 $$.
2. **Interest Rate**: The interest rate is $$ r = 6\% = 0.06 $$.
3. **Annual Repayment**: He pays back $$ P = 45000 $$ at the end of each year.
4. **Amount owed after the first year**:
   $$
   \text{Amount after Year 1} = L \times (1 + r) - P
   $$
5. **Amount owed after the second year**:
   $$
   \text{Amount after Year 2} = \text{Amount after Year 1} \times (1 + r) - P
   $$

Now, let's calculate the values for all three problems.
Calculate the value by following steps:
- step0: Calculate:
  $$30000\left(1+0.045\right)+1000$$
- step1: Add the numbers:
  $$30000\times 1.045+1000$$
- step2: Multiply the numbers:
  $$31350+1000$$
- step3: Add the numbers:
  $$32350$$
Calculate or simplify the expression $$ 64000 * 0.75 * (1 - 0.20) * (1 - 0.15) * (1 - 0.15) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$64000\times 0.75\left(1-0.2\right)\left(1-0.15\right)\left(1-0.15\right)$$
- step1: Subtract the numbers:
  $$64000\times 0.75\times 0.8\left(1-0.15\right)\left(1-0.15\right)$$
- step2: Subtract the numbers:
  $$64000\times 0.75\times 0.8\times 0.85\left(1-0.15\right)$$
- step3: Subtract the numbers:
  $$64000\times 0.75\times 0.8\times 0.85\times 0.85$$
- step4: Multiply the terms:
  $$48000\times 0.8\times 0.85\times 0.85$$
- step5: Multiply the terms:
  $$38400\times 0.85\times 0.85$$
- step6: Multiply the terms:
  $$32640\times 0.85$$
- step7: Multiply the numbers:
  $$27744$$
Calculate or simplify the expression $$ (185000 * (1 + 0.06) - 45000) * (1 + 0.06) - 45000 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(185000\left(1+0.06\right)-45000\right)\left(1+0.06\right)-45000$$
- step1: Add the numbers:
  $$\left(185000\times 1.06-45000\right)\left(1+0.06\right)-45000$$
- step2: Multiply the numbers:
  $$\left(196100-45000\right)\left(1+0.06\right)-45000$$
- step3: Subtract the numbers:
  $$151100\left(1+0.06\right)-45000$$
- step4: Add the numbers:
  $$151100\times 1.06-45000$$
- step5: Multiply the numbers:
  $$160166-45000$$
- step6: Subtract the numbers:
  $$115166$$
Here are the detailed solutions for each problem:

### Problem 8: Car Depreciation
After calculating the depreciation over 4 years, the value of the car is approximately **$27,744**.

### Problem 9: Savings with Compound Interest
After 2 years, the total savings amount to **$32,350**.

### Problem 10: Loan Repayment
After making his second repayment, the man still owes **$115,166**.

If you have any further questions or need additional assistance, feel free to ask!
After 3 years, he still owes **$115,166**.

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