To solve this problem by hand using the formula for continuously compounded interest:

Given:

• The final amount in Robert's account: $A = ₱23,005.48$
• The interest rate: $r = 5\% = 0.05$
• The time period: $t = 6 \text{ years}$

The formula for continuously compounded interest is:

$A = P \times e^{r \times t}$

where:

• $A$ is the amount after interest
• $P$ is the principal (initial amount, which we need to find)
• $r$ is the interest rate
• $t$ is the time in years
• $e$ is Euler's number (approximately 2.71828)

Step 1: Rearrange the formula to solve for $P$:

$P = \frac{A}{e^{r \times t}}$

Step 2: Substitute the given values:

$P = \frac{23,005.48}{e^{0.05 \times 6}}$

First, calculate $e^{0.05 \times 6}$: $e^{0.30} \approx 1.34986$

Step 3: Compute $P$:

$P = \frac{23,005.48}{1.34986} \approx 17,046.84$

So, Robert received approximately ₱17,046.84 as a graduation gift from his parents.

If you have any further questions or need more details, feel free to ask!

Related questions:

1. How does continuous compounding differ from annual compounding?
2. What would the amount be if the interest rate were 6% instead of 5%?
3. How would the result change if the interest was compounded monthly instead of continuously?
4. What is the formula for compound interest when it is compounded annually?
5. How long would it take for Robert's money to double at 5% continuous interest?

Tip:

In continuously compounded interest, the amount grows at an exponential rate, which means even small changes in the rate or time can significantly impact the final amount!