To find the date that the payment of 19,000€ appeared, we can use the simple interest formula:
A = P \left( 1 + \frac{rt}{n} \right)
Where:
- A is the final amount (246,088.89€)
- P is the initial amount (the sum of all incoming payments before the 19,000€)
- r is the interest rate (4%)
- t is the time in years (the number of days from the unknown date to 30.06.2008, i.e., 45 days)
- n is the number of compounding periods per year (30E/360 DCC)
First, let's calculate the initial amount P :
P = 25,000€ + 140,000€ + 60,000€ = 225,000€
Now, let's substitute the known values into the formula and solve for t :
246,088.89€ = 225,000€ \left( 1 + \frac{0.04t}{30} \right)
Simplifying the equation, we have:
\frac{246,088.89€}{225,000€} = 1 + \frac{0.04t}{30}
1.0930711 = 1 + \frac{0.04t}{30}
Subtracting 1 from both sides:
0.0930711 = \frac{0.04t}{30}
Now, let's solve for t :
t = \frac{0.0930711 \times 30}{0.04}
t = 69.9533
Since t represents the number of days, we can round it to the nearest whole number, which is 70 days.
Therefore, the payment of 19,000€ appeared 70 days before 30.06.2008.
To find the date, we subtract 70 days from 30.06.2008:
\text{Date} = \text{30.06.2008} - \text{70 days} = \text{15.05.2008}
Answer: The payment of 19,000€ appeared on 15.05.2008.