Math CentralQuandaries & Queries


Question from Kenneth:


Is compound interest on $\$1.00$ for one year the same as simple interest on $1.00 for one year?

Compound interest would be $\$1.01,$ and I believe the same amount would be the same at simple interest.


Hi Kenneth,

You haven't said what the interest rate is but from your calculations I assume it's 1% per year. You also didn't say how many times per year the interest is compounded but again from your calculations I assume you are compounding once per year. If both my assumptions are correct then yes, the return at the end of a year would be the same as with simple interest. If however you compound twice per year, every six months, the calculation is different.

What is the value of $\$1.00$ invested for one year at a rate of 1% per year compounded every six months? For this question, $\$1.00$ invested at 1$ per year for six months would yield

\[\$1.00 + \$1.00 \times \frac{0.01}{2} = \$1.00\times \left(1+ 0.005\right) = \$1.005.\]

This amount is then invested at 1% per year for 6 months and would yield

\[\$1.005 + \$1.005 \times \frac{0.01}{2} = \$1.005\times \left(1+ 0.005\right) = \$1.010025.\]

For a general development of the value at the end of $m$ years of $\$p,$ invested at a rate of $100 \times r\% = r$ per year, compounded $k$ times per year I suggest you look at my response to Kayla.


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