Question
A car dealer sold 100 cars in his first year and plans to increase his sales by 10 percent each year for the next 5 years. How many cars will he have sold at the end of his fifth year?
1245
725
610
920
1245
725
610
920
Answers
GPT-4o mini
To calculate the number of cars sold at the end of the fifth year, we can use the formula for exponential growth. The formula for the number of cars sold in year \( n \) is:
\[
S_n = S_0 \times (1 + r)^n
\]
where:
- \( S_0 \) = initial sales (100 cars)
- \( r \) = growth rate (10% or 0.10)
- \( n \) = number of years
We calculate it for each of the 5 years:
1. **Year 1**:
\[
S_1 = 100 \times (1 + 0.10)^1 = 100 \times 1.10 = 110
\]
2. **Year 2**:
\[
S_2 = 100 \times (1 + 0.10)^2 = 100 \times 1.10^2 = 100 \times 1.21 = 121
\]
3. **Year 3**:
\[
S_3 = 100 \times (1 + 0.10)^3 = 100 \times 1.10^3 = 100 \times 1.331 = 133.1
\]
4. **Year 4**:
\[
S_4 = 100 \times (1 + 0.10)^4 = 100 \times 1.10^4 = 100 \times 1.4641 = 146.41
\]
5. **Year 5**:
\[
S_5 = 100 \times (1 + 0.10)^5 = 100 \times 1.10^5 = 100 \times 1.61051 \approx 161.051
\]
Now, let’s find the total number of cars sold over the 5 years:
\[
\text{Total cars sold} = S_1 + S_2 + S_3 + S_4 + S_5
\]
Calculating this gives us:
\[
\text{Total cars sold} = 110 + 121 + 133.1 + 146.41 + 161.051
\]
Let's sum these numbers:
\[
110 + 121 = 231
\]
\[
231 + 133.1 = 364.1
\]
\[
364.1 + 146.41 = 510.51
\]
\[
510.51 + 161.051 \approx 671.561
\]
At the end of the fifth year, the total cars sold is approximately **671.561**.
The calculated amounts and the options provided (1245, 725, 610, 920) do not match exactly. It appears there is some discrepancy between the calculation approach I provided and the expected outcome.
For a quick summation:
1. Year 1 = 110
2. Year 2 = 121
3. Year 3 = 133.1
4. Year 4 = 146.41
5. Year 5 = 161.051
And the given options state:
- 610❌
- 725❌
- 920❌
- 1245❌
None of these match the computed car sales.
Thus, the previous logic of compounding growth was correctly cited for estimates beyond 100.
Let me reflect again over the entire growth model:
The dealer's sales in the 5th year would be \( 100 * (1.10)^5 = 100 * 1.61051 \approx 161.051 \), then add iterative previous years.
Summing looks closer till yielding near:
Total after year's sales at a glance for accurate figures will revise herald the closest logical sales estimates into concise end displays.
However, if totality gives otherwise needing function of list options could infer accordingly.
\[
S_n = S_0 \times (1 + r)^n
\]
where:
- \( S_0 \) = initial sales (100 cars)
- \( r \) = growth rate (10% or 0.10)
- \( n \) = number of years
We calculate it for each of the 5 years:
1. **Year 1**:
\[
S_1 = 100 \times (1 + 0.10)^1 = 100 \times 1.10 = 110
\]
2. **Year 2**:
\[
S_2 = 100 \times (1 + 0.10)^2 = 100 \times 1.10^2 = 100 \times 1.21 = 121
\]
3. **Year 3**:
\[
S_3 = 100 \times (1 + 0.10)^3 = 100 \times 1.10^3 = 100 \times 1.331 = 133.1
\]
4. **Year 4**:
\[
S_4 = 100 \times (1 + 0.10)^4 = 100 \times 1.10^4 = 100 \times 1.4641 = 146.41
\]
5. **Year 5**:
\[
S_5 = 100 \times (1 + 0.10)^5 = 100 \times 1.10^5 = 100 \times 1.61051 \approx 161.051
\]
Now, let’s find the total number of cars sold over the 5 years:
\[
\text{Total cars sold} = S_1 + S_2 + S_3 + S_4 + S_5
\]
Calculating this gives us:
\[
\text{Total cars sold} = 110 + 121 + 133.1 + 146.41 + 161.051
\]
Let's sum these numbers:
\[
110 + 121 = 231
\]
\[
231 + 133.1 = 364.1
\]
\[
364.1 + 146.41 = 510.51
\]
\[
510.51 + 161.051 \approx 671.561
\]
At the end of the fifth year, the total cars sold is approximately **671.561**.
The calculated amounts and the options provided (1245, 725, 610, 920) do not match exactly. It appears there is some discrepancy between the calculation approach I provided and the expected outcome.
For a quick summation:
1. Year 1 = 110
2. Year 2 = 121
3. Year 3 = 133.1
4. Year 4 = 146.41
5. Year 5 = 161.051
And the given options state:
- 610❌
- 725❌
- 920❌
- 1245❌
None of these match the computed car sales.
Thus, the previous logic of compounding growth was correctly cited for estimates beyond 100.
Let me reflect again over the entire growth model:
The dealer's sales in the 5th year would be \( 100 * (1.10)^5 = 100 * 1.61051 \approx 161.051 \), then add iterative previous years.
Summing looks closer till yielding near:
Total after year's sales at a glance for accurate figures will revise herald the closest logical sales estimates into concise end displays.
However, if totality gives otherwise needing function of list options could infer accordingly.