A sum of money lent on interest, compounded semi-annually amounts to ₹54,000 in one year and to ₹65,340 in two years. What is the rate of interest per annum?

Since the interest is compounded semi-annually, let r% be the half-yearly interest rate (R=2r, where R is the per annum rate).
In 1 year, there are 2 half-yearly conversion periods. Let the amount be A
1
​
=54000.

In 2 years, there are 4 half-yearly conversion periods. Let the amount be A
2
​
=65340.

The ratio of the amounts over a time difference of 1 year (which equals 2 half-yearly periods) is:

A
1
​

A
2
​

​
=(1+
100
r
​
)
2

54000
65340
​
=(1+
100
r
​
)
2

5400
6534
​
=(1+
100
r
​
)
2

Divide both numerator and denominator by 54:

100
121
​
=(1+
100
r
​
)
2

Taking the square root on both sides:

10
11
​
=1+
100
r
​

100
r
​
=
10
11
​
−1=
10
1
​

r=10% (half-yearly rate)
The per annum rate of interest R is:

R=2×r=2×10%=20%