Question
If you had invested only 1 month’s worth of the emergency fund in the saving account at a 3.15% APR and the remainder in the 45-day CD at a 4.65% APR, what is the difference in the interest earned in 45 days when compared with question #3? Round to the nearest penny.
question #3~Your fixed expenses are $1,500.45/month. Your emergency fund has 4 month’s worth of coverage. You invest half in a savings account with an interest rate of 3.15% APR and the other half in a 45-day CD with an interest rate of 4.65% APR. How much is your total interest in 45 days? Round to the nearest penny.
question #3~Your fixed expenses are $1,500.45/month. Your emergency fund has 4 month’s worth of coverage. You invest half in a savings account with an interest rate of 3.15% APR and the other half in a 45-day CD with an interest rate of 4.65% APR. How much is your total interest in 45 days? Round to the nearest penny.
Answers
GPT-4o mini
To solve this problem, we first need to determine a few key figures such as how much is being invested in the savings account versus the CD in both scenarios.
### Step 1: Calculate Initial Amount
Based on question #3, your monthly expenses are $1,500.45, and your emergency fund is 4 months’ worth of those expenses.
\[
\text{Total Emergency Fund} = 4 \times 1,500.45 = 6,001.80
\]
### Step 2: Distributions in Question #3
In question #3, you're investing half of the emergency fund in a savings account and half in a CD.
\[
\text{Amount in Savings} = \frac{6,001.80}{2} = 3,000.90
\]
\[
\text{Amount in CD} = 3,000.90
\]
### Step 3: Calculate Interest in Question #3
For the savings account (3.15% APR over 45 days):
- Convert 3.15% APR to a daily interest rate (since a year typically has 365 days):
\[
\text{Daily interest rate} = \frac{3.15\%}{365} = \frac{0.0315}{365} \approx 0.0000865753425
\]
- Calculate the interest earned in the savings account over 45 days:
\[
\text{Interest from Savings} = 3,000.90 \times 0.0000865753425 \times 45 \approx 11.48
\]
For the CD (4.65% APR over 45 days):
- Convert 4.65% APR to a daily interest rate:
\[
\text{Daily interest rate} = \frac{4.65\%}{365} = \frac{0.0465}{365} \approx 0.0001273972603
\]
- Calculate the interest earned in the CD over 45 days:
\[
\text{Interest from CD} = 3,000.90 \times 0.0001273972603 \times 45 \approx 17.36
\]
So the total interest from question #3 is:
\[
\text{Total Interest (Question 3)} = 11.48 + 17.36 = 28.84
\]
### Step 4: New Scenario
Now, in the new scenario, only one month's worth of the emergency fund is being considered to invest in the savings account, while the remainder (3 months' worth) is invested in the CD, which means:
\[
\text{Amount in Savings} = 1,500.45
\]
\[
\text{Amount in CD} = 6,001.80 - 1,500.45 = 4,501.35
\]
**Interest earned in the savings account (for the new scenario)**:
\[
\text{Interest from Savings} = 1,500.45 \times 0.0000865753425 \times 45 \approx 5.84
\]
**Interest earned in the CD (for the new scenario)**:
\[
\text{Interest from CD} = 4,501.35 \times 0.0001273972603 \times 45 \approx 25.94
\]
**Total interest for the new scenario**:
\[
\text{Total Interest (New Scenario)} = 5.84 + 25.94 = 31.78
\]
### Step 5: Calculate the Difference in Interest
Now, we calculate the difference in interest between the two scenarios:
\[
\text{Difference} = \text{Total Interest (New Scenario)} - \text{Total Interest (Question 3)}
= 31.78 - 28.84 = 2.94
\]
### Final Answer
So, the difference in the interest earned in 45 days is:
**$2.94** (rounded to the nearest penny).
### Step 1: Calculate Initial Amount
Based on question #3, your monthly expenses are $1,500.45, and your emergency fund is 4 months’ worth of those expenses.
\[
\text{Total Emergency Fund} = 4 \times 1,500.45 = 6,001.80
\]
### Step 2: Distributions in Question #3
In question #3, you're investing half of the emergency fund in a savings account and half in a CD.
\[
\text{Amount in Savings} = \frac{6,001.80}{2} = 3,000.90
\]
\[
\text{Amount in CD} = 3,000.90
\]
### Step 3: Calculate Interest in Question #3
For the savings account (3.15% APR over 45 days):
- Convert 3.15% APR to a daily interest rate (since a year typically has 365 days):
\[
\text{Daily interest rate} = \frac{3.15\%}{365} = \frac{0.0315}{365} \approx 0.0000865753425
\]
- Calculate the interest earned in the savings account over 45 days:
\[
\text{Interest from Savings} = 3,000.90 \times 0.0000865753425 \times 45 \approx 11.48
\]
For the CD (4.65% APR over 45 days):
- Convert 4.65% APR to a daily interest rate:
\[
\text{Daily interest rate} = \frac{4.65\%}{365} = \frac{0.0465}{365} \approx 0.0001273972603
\]
- Calculate the interest earned in the CD over 45 days:
\[
\text{Interest from CD} = 3,000.90 \times 0.0001273972603 \times 45 \approx 17.36
\]
So the total interest from question #3 is:
\[
\text{Total Interest (Question 3)} = 11.48 + 17.36 = 28.84
\]
### Step 4: New Scenario
Now, in the new scenario, only one month's worth of the emergency fund is being considered to invest in the savings account, while the remainder (3 months' worth) is invested in the CD, which means:
\[
\text{Amount in Savings} = 1,500.45
\]
\[
\text{Amount in CD} = 6,001.80 - 1,500.45 = 4,501.35
\]
**Interest earned in the savings account (for the new scenario)**:
\[
\text{Interest from Savings} = 1,500.45 \times 0.0000865753425 \times 45 \approx 5.84
\]
**Interest earned in the CD (for the new scenario)**:
\[
\text{Interest from CD} = 4,501.35 \times 0.0001273972603 \times 45 \approx 25.94
\]
**Total interest for the new scenario**:
\[
\text{Total Interest (New Scenario)} = 5.84 + 25.94 = 31.78
\]
### Step 5: Calculate the Difference in Interest
Now, we calculate the difference in interest between the two scenarios:
\[
\text{Difference} = \text{Total Interest (New Scenario)} - \text{Total Interest (Question 3)}
= 31.78 - 28.84 = 2.94
\]
### Final Answer
So, the difference in the interest earned in 45 days is:
**$2.94** (rounded to the nearest penny).