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Deconstructing the Question

Patty invests \(2000\) dollars for one year at \(p\%\) interest.

We compare

annual compounding

and

semiannual compounding.

The question asks how much extra interest she earns with semiannual compounding.

Step-by-step

Convert the rate

\(p\%=\frac{p}{100}\)

Interest with annual compounding

\(2000\times\frac{p}{100}=20p\)

Now consider semiannual compounding.

Each half-year rate is

\(\frac{p}{2}\%=\frac{p}{200}\)

Amount after one year

\(2000\left(1+\frac{p}{200}\right)^2\)

Expand

\((1+x)^2=1+2x+x^2\)

with

\(x=\frac{p}{200}\)

\(1+\frac{2p}{200}+\frac{p^2}{40000}\)

\(1+\frac{p}{100}+\frac{p^2}{40000}\)

Multiply by \(2000\)

Interest earned

\(2000\left(\frac{p}{100}+\frac{p^2}{40000}\right)\)

\(=20p+\frac{p^2}{20}\)

Difference from annual compounding

\((20p+\frac{p^2}{20})-20p\)

\(\frac{p^2}{20}\)

Answer D
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