Q.Which yields more interest: 8% effective or 7.8% compounded semi-annually?

Concept: Effective Interest Rate – the true annual rate that accounts for compounding within the year.

Step 1: The 8% effective rate is already the true annual rate: $r_{\text{eff}} = 8\%$.

Step 2: Convert 7.8% compounded semi-annually to an effective rate.

Nominal rate $j = 7.8\% = 0.078$, compounding frequency $m = 2$.

Effective rate formula:

Step 3: Compute: …

The core of this problem is understanding what "effective interest rate" actually means. An effective rate tells you the actual percentage increase in your money after one year, accounting for compounding. A nominal rate like "7.8% compounded semi-annually" is a quoted rate that doesn't directly tell you the real growth — you have to compute the effective rate from it.

So the question reduces to: What is the effective annual rate of 7.8% compounded semi-annually? Then we simply compare that number to 8%.

1. Recall the formula for effective annual rate (EAR). If a nominal rate $r$ is compounded $n$ times per year, the effective annual rate is:

This formula comes from the idea that after one year, a principal $P$ grows to $P\left(1 + \frac{r}{n}\right)^n$, so the interest earned is $P\left[\left(1 + \frac{r}{n}\right)^n - 1\right]$, and the effective rate is that interest divided by $P$.

2. Plug in the values. Here $r = 7.8\% = 0.078$ and $n = 2$ (semi-annual means twice a year).

3. Compute step by step.

   First, $1 + 0.039 = 1.039$.

   Then square it: $1.039^2 = 1.039 \times 1.039$.

   Let's do it carefully: $1.039 \times 1.039 = (1.04 - 0.001)^2 = 1.0816 - 2(1.04)(0.001) + 0.000001 = 1.0816 - 0.00208 + 0.000001 = 1.079521$.

   (You can also just multiply: $1.039 \times 1.039 = 1.079521$.)

   So $\text{EAR} = 1.079521 - 1 = 0.079521$.

4. Convert to percentage.

   $0.079521 \times 100\% = 7.9521\%$. …