Understanding the Time Required for a Sum to Treble at Simple Interest

In the realm of financial management and the principles of simple interest, understanding how long it takes for a sum of money to treble itself at a specific interest rate is fundamental. This article delves into the mathematical computations involved and provides clear, illustrative examples to ensure comprehension.

Formula and Basic Understanding

When dealing with simple interest, the formula that stands out is:

A P PRT/100

Where:

A final amount P principal amount (initial sum) R rate of interest per year (as a percentage) T time period in years

Principal Sum of 100 at 16% per annum

Let us consider a principal sum of 100.

A P (PTR/100)

300 100 (100 x 16 x T/100)

300 - 100 16T

16T 200

T 200/16 12.5 years

Derivation for Trebling a Principal Sum

Now, let's derive the time it takes for a sum to treble itself at a simple interest rate of 16% per annum. For clarity, let's use the formula again:

A P PRT/100

We know that if the principal trebles itself:

A 3P

3P P (PRT/100)

2P PRT/100

Dividing both sides by P (assuming P ≠ 0):

2 RT/100

Substituting the Given Rate

Now, substitute the rate R 16 (as a decimal, R 0.16):

2 0.16T

T 2/0.16

T 12.5 years

Thus, it will take 12.5 years for the sum of money to treble itself at a simple interest rate of 16% per annum.

Exploring Further

Why does the question sound "bassless" and what does it mean by the principal sum trebling? Let's break this down further:

The value of the principal P remains fixed every year in simple interest. The interest calculated annually is a percentage of P (16% in this case). If the principal trebles, it becomes 3P. The simple interest (SI) in this case is (3P - P) 2P. Using the formula for simple interest:

SI P × R × T / 100

2P P × 16 × T / 100

200 16T

T 200 / 16 12.5 years

So, it takes 12.5 years for the principal to treble.

Additional Insight

Let's consider another example to solidify the understanding:

SI Amount (A) - Principal (P)

A 5P (as the amount trebles)

SI 5P - P 4P

SI P × R × T / 100 4P

P × R × T 400P

R × T 400

R 16 (as the rate of interest is 16%)

T 400 / 16 25 years

Hence, it will take 25 years for the principal to treble at a simple interest rate of 16% per annum.