25 埃德温（Edwin）将等额的资金 $P$ 分别存入 A 银行和 B 银行。A 银行提供年利率为 4% 的单利，而 B 银行提供每半年 2% 的利率，每半年计息一次（复利）。7 年后，埃德温从两家银行收到的总金额之差为 $513$。求 $P$ 的值，将答案精确到百位。

1. Find the amount in Bank A after 7 years using simple interest.

For Bank A, simple interest at $4\%$ per annum for $7$ years:

$\text{Amount in A}=P\left(1+\frac{4}{100}\times 7\right)$

$=P(1+0.28)=1.28P$

2. Find the amount in Bank B after 7 years using half-yearly compound interest.

For Bank B, interest is $2\%$ per half-year, compounded every six months.

Number of half-years in $7$ years:

$7\times 2=14$

$\text{Amount in B}=P\left(1+\frac{2}{100}\right)^{14}=P(1.02)^{14}$

3. Use the given difference between the two amounts to form an equation.

Since Bank B gives the larger amount,

$P(1.02)^{14}-1.28P=513$

$P\left((1.02)^{14}-1.28\right)=513$

4. Solve for $P$ and round to the nearest hundred.

$P=\frac{513}{(1.02)^{14}-1.28}$

$P\approx 12993.3$

Correct to the nearest hundred,

$P=13000$