Question
The Chewy Candy Company is test marketing a new candy bar in Seoul and Manila.
In Seoul, the company sold
\[40\] candy bars by the end of the first month, and each month the cumulative total number of candy bars sold increases by
\[60\].
In Manila, the company sold
\[20\] candy bars by the end of the first month, and the cumulative total number of candy bars sold increases by approximately
\[80\%\] each month.
The company's board assembles at the end of each month to review the cumulative total number of candies sold in each city.
What is the first month in which the board will see the cumulative total number of candy bars sold in Manila exceeded the cumulative total in Seoul?
In Seoul, the company sold
\[40\] candy bars by the end of the first month, and each month the cumulative total number of candy bars sold increases by
\[60\].
In Manila, the company sold
\[20\] candy bars by the end of the first month, and the cumulative total number of candy bars sold increases by approximately
\[80\%\] each month.
The company's board assembles at the end of each month to review the cumulative total number of candies sold in each city.
What is the first month in which the board will see the cumulative total number of candy bars sold in Manila exceeded the cumulative total in Seoul?
Answers
GPT-4o mini
To solve the problem, we need to understand how the cumulative total of candy bars sold increases over the months in both Seoul and Manila.
1. **Seoul Candy Sales:**
- Cumulative sales after month 1: \( 40 \)
- Increase per month: \( 60 \)
The cumulative sales in Seoul after \( t \) months can be represented by the formula:
\[
S(t) = 40 + 60(t - 1)
\]
Here, \( t \) is the month number, and subtracting 1 accounts for the initial sales in the first month.
Simplifying this gives:
\[
S(t) = 40 + 60t - 60 = 60t - 20
\]
2. **Manila Candy Sales:**
- Cumulative sales after month 1: \( 20 \)
- Each month, the total increases by \( 80\% \). Thus, the total sold by the end of month \( t \) can be calculated using the formula for exponential growth.
Since each month the amount sold is multiplied by \( 1.8 \) (which is \( 1 + 0.8 \)):
\[
M(t) = 20 \cdot (1.8)^{(t - 1)}
\]
3. **Finding the first month where sales in Manila exceed Seoul:**
We need to find the smallest \( t \) such that:
\[
M(t) > S(t)
\]
This gives us the inequality:
\[
20 \cdot (1.8)^{(t - 1)} > 60t - 20
\]
Now we will calculate \( M(t) \) and \( S(t) \) month by month until we find the first \( t \) that satisfies this inequality.
**Calculating for each month:**
- **Month 1:**
\[
S(1) = 40 \quad \text{and} \quad M(1) = 20
\]
(not exceeding)
- **Month 2:**
\[
S(2) = 60 \cdot 2 - 20 = 100 \quad \text{and} \quad M(2) = 20 \cdot 1.8 = 36
\]
(not exceeding)
- **Month 3:**
\[
S(3) = 60 \cdot 3 - 20 = 160 \quad \text{and} \quad M(3) = 20 \cdot 1.8^2 \approx 64.8
\]
(not exceeding)
- **Month 4:**
\[
S(4) = 60 \cdot 4 - 20 = 220 \quad \text{and} \quad M(4) = 20 \cdot 1.8^3 \approx 116.64
\]
(not exceeding)
- **Month 5:**
\[
S(5) = 60 \cdot 5 - 20 = 280 \quad \text{and} \quad M(5) = 20 \cdot 1.8^4 \approx 209.952
\]
(not exceeding)
- **Month 6:**
\[
S(6) = 60 \cdot 6 - 20 = 340 \quad \text{and} \quad M(6) = 20 \cdot 1.8^5 \approx 377.92
\]
(exceeding)
**Conclusion:**
The first month when the cumulative total number of candy bars sold in Manila exceeds that in Seoul is:
\[
\boxed{6}
\]
1. **Seoul Candy Sales:**
- Cumulative sales after month 1: \( 40 \)
- Increase per month: \( 60 \)
The cumulative sales in Seoul after \( t \) months can be represented by the formula:
\[
S(t) = 40 + 60(t - 1)
\]
Here, \( t \) is the month number, and subtracting 1 accounts for the initial sales in the first month.
Simplifying this gives:
\[
S(t) = 40 + 60t - 60 = 60t - 20
\]
2. **Manila Candy Sales:**
- Cumulative sales after month 1: \( 20 \)
- Each month, the total increases by \( 80\% \). Thus, the total sold by the end of month \( t \) can be calculated using the formula for exponential growth.
Since each month the amount sold is multiplied by \( 1.8 \) (which is \( 1 + 0.8 \)):
\[
M(t) = 20 \cdot (1.8)^{(t - 1)}
\]
3. **Finding the first month where sales in Manila exceed Seoul:**
We need to find the smallest \( t \) such that:
\[
M(t) > S(t)
\]
This gives us the inequality:
\[
20 \cdot (1.8)^{(t - 1)} > 60t - 20
\]
Now we will calculate \( M(t) \) and \( S(t) \) month by month until we find the first \( t \) that satisfies this inequality.
**Calculating for each month:**
- **Month 1:**
\[
S(1) = 40 \quad \text{and} \quad M(1) = 20
\]
(not exceeding)
- **Month 2:**
\[
S(2) = 60 \cdot 2 - 20 = 100 \quad \text{and} \quad M(2) = 20 \cdot 1.8 = 36
\]
(not exceeding)
- **Month 3:**
\[
S(3) = 60 \cdot 3 - 20 = 160 \quad \text{and} \quad M(3) = 20 \cdot 1.8^2 \approx 64.8
\]
(not exceeding)
- **Month 4:**
\[
S(4) = 60 \cdot 4 - 20 = 220 \quad \text{and} \quad M(4) = 20 \cdot 1.8^3 \approx 116.64
\]
(not exceeding)
- **Month 5:**
\[
S(5) = 60 \cdot 5 - 20 = 280 \quad \text{and} \quad M(5) = 20 \cdot 1.8^4 \approx 209.952
\]
(not exceeding)
- **Month 6:**
\[
S(6) = 60 \cdot 6 - 20 = 340 \quad \text{and} \quad M(6) = 20 \cdot 1.8^5 \approx 377.92
\]
(exceeding)
**Conclusion:**
The first month when the cumulative total number of candy bars sold in Manila exceeds that in Seoul is:
\[
\boxed{6}
\]