Question

What is the conversion period and rate, if a sum is borrowed for \[n\] years at the rate of \[R%\] per annum.
a). Compounded half yearly
b). Compounded quarterly

Answer 1

Hint: In this question we have given the time that is \[n\] years at the rate of \[R%\] per annum and we have to calculate the rate and conversion period compounded half yearly and quarterly therefore apply the conditions for rate of interest compounded half yearly that is two times in a year and quarterly that is four times in a year, so that you can obtain the given result.

Complete step-by-step solution:
The main distinction between simple and compound interest is that simple interest is calculated on the principal amount, whereas compound interest is calculated on the principal amount plus the interest compounded for a period cycle.
Simple interest and compound interest are two essential concepts that are commonly employed in various financial services, particularly in banking. Simple interest is used in loans such as instalment loans, auto loans, student loans, and mortgages. Compound interest is employed by the majority of savings accounts to pay interest. It pays a lot more than just interest. Let's take a closer look at the difference between simple and compound interest in this post.
Compound interest is interest earned on both the principal and the interest over a set period of time. The principal is also used to account for the interest that has accrued on a principal over time. Furthermore, the cumulative principal value is used to calculate interest for the next time period. Compound interest is a novel method of calculating interest that is utilised in all financial and economic operations worldwide. When we look at the compound interest values accumulated over successive time periods, we can see how powerful compounding is.
Now, according to the question:
We have given time that is \[n\] years and rate \[R%\] per annum
Compounded half yearly
If the rate of interest is compounded half yearly that is six months or two times in a year then the number of years will be doubled and the rate of interest is halved.
Therefore the rate will be:
\[\Rightarrow rate=\dfrac{R%}{2}\]
Conversion time period will be:
\[\Rightarrow \text{conversion period}=n\times 2\]
\[\Rightarrow \text{conversion period}=2n\]
Compounded quarterly
If the rate of interest is compounded quarterly that is three months or four times in a year then the number of years will be four times and the rate of interest is one fourth.
Therefore the rate will be:
\[\Rightarrow rate=\dfrac{R%}{4}\]
Conversion time period will be:
\[\Rightarrow \text{conversion period}=n\times 4\]
\[\Rightarrow \text{conversion period}=4n\]

Note: Student should note that Compounding is the process of adding interest to the current principal amount on a daily, weekly, half-yearly, quarterly, or yearly basis. In the case of compound interest, interest is always higher than in the case of simple interest and the rate of formula is always given in fraction.