Question
You save $5000 and invest 60% of it in stocks while leaving the rest
in a savings account earning 4.9% APR. The stock increases 9% in the
first year and loses 4% of its value the second year. What is the total
gained during the 2 years? What would the gain have been if
all of the investment had been left in the savings account?
in a savings account earning 4.9% APR. The stock increases 9% in the
first year and loses 4% of its value the second year. What is the total
gained during the 2 years? What would the gain have been if
all of the investment had been left in the savings account?
Answers
GPT-4o mini
To calculate the total gained during the two years, we need to find out how much money is made from the stock investment and from the savings account separately.
### Step 1: Determine the amounts in stocks and savings account
You save $5,000 and invest 60% in stocks.
- Investment in stocks: \( 5000 \times 0.60 = 3000 \)
- Amount in savings account: \( 5000 - 3000 = 2000 \)
### Step 2: Calculate the growth of the stock investment
The stock investment increases by 9% in the first year and then loses 4% in the second year.
1. **First year:**
\[
\text{Value after first year} = 3000 \times (1 + 0.09) = 3000 \times 1.09 = 3270
\]
2. **Second year:**
\[
\text{Value after second year} = 3270 \times (1 - 0.04) = 3270 \times 0.96 = 3139.20
\]
### Step 3: Calculate the growth of the savings account
The savings account earns 4.9% APR.
1. **First year:**
\[
\text{Value after first year} = 2000 \times (1 + 0.049) = 2000 \times 1.049 = 2098
\]
2. **Second year:**
\[
\text{Value after second year} = 2098 \times (1 + 0.049) = 2098 \times 1.049 = 2208.902
\]
### Step 4: Total value after 2 years
Now, add the final values from both investments after two years.
- Total value = Value from stocks + Value from savings
\[
\text{Total value} = 3139.20 + 2208.902 = 5348.102
\]
### Step 5: Calculate total gain
Total gain is the final total value minus the initial investment:
\[
\text{Total gain} = 5348.102 - 5000 = 348.102
\]
### Step 6: Calculate gain if all money was in the savings account
If the entire $5,000 was invested in the savings account at 4.9%:
1. **First year:**
\[
\text{Value after first year} = 5000 \times (1 + 0.049) = 5000 \times 1.049 = 5245
\]
2. **Second year:**
\[
\text{Value after second year} = 5245 \times (1 + 0.049) = 5245 \times 1.049 = 5505.805
\]
### Gain with complete savings account investment
\[
\text{Total gain with savings only} = 5505.805 - 5000 = 505.805
\]
### Summary of results
- Total gained during 2 years with split investments: **$348.10**
- Total gained if all money was in the savings account: **$505.81**
### Step 1: Determine the amounts in stocks and savings account
You save $5,000 and invest 60% in stocks.
- Investment in stocks: \( 5000 \times 0.60 = 3000 \)
- Amount in savings account: \( 5000 - 3000 = 2000 \)
### Step 2: Calculate the growth of the stock investment
The stock investment increases by 9% in the first year and then loses 4% in the second year.
1. **First year:**
\[
\text{Value after first year} = 3000 \times (1 + 0.09) = 3000 \times 1.09 = 3270
\]
2. **Second year:**
\[
\text{Value after second year} = 3270 \times (1 - 0.04) = 3270 \times 0.96 = 3139.20
\]
### Step 3: Calculate the growth of the savings account
The savings account earns 4.9% APR.
1. **First year:**
\[
\text{Value after first year} = 2000 \times (1 + 0.049) = 2000 \times 1.049 = 2098
\]
2. **Second year:**
\[
\text{Value after second year} = 2098 \times (1 + 0.049) = 2098 \times 1.049 = 2208.902
\]
### Step 4: Total value after 2 years
Now, add the final values from both investments after two years.
- Total value = Value from stocks + Value from savings
\[
\text{Total value} = 3139.20 + 2208.902 = 5348.102
\]
### Step 5: Calculate total gain
Total gain is the final total value minus the initial investment:
\[
\text{Total gain} = 5348.102 - 5000 = 348.102
\]
### Step 6: Calculate gain if all money was in the savings account
If the entire $5,000 was invested in the savings account at 4.9%:
1. **First year:**
\[
\text{Value after first year} = 5000 \times (1 + 0.049) = 5000 \times 1.049 = 5245
\]
2. **Second year:**
\[
\text{Value after second year} = 5245 \times (1 + 0.049) = 5245 \times 1.049 = 5505.805
\]
### Gain with complete savings account investment
\[
\text{Total gain with savings only} = 5505.805 - 5000 = 505.805
\]
### Summary of results
- Total gained during 2 years with split investments: **$348.10**
- Total gained if all money was in the savings account: **$505.81**