Simple Interest: Calculating Time to Triple an Amount

This solution explains how to find the time needed for an amount to triple under simple interest, given the time it takes to double.

Understanding Simple Interest and the Problem

Simple Interest is a method of calculating the interest charge on a loan. It is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments.

The formula for Simple Interest (SI) is:

$ SI = \frac{P \times R \times T}{100} $

Where:

• $P$ = Principal amount (the initial amount of money)
• $R$ = Rate of interest per year (%)
• $T$ = Time period (in years)

The total Amount ($A$) after time $T$ is the sum of the Principal and the Simple Interest:

$ A = P + SI $

The problem states:

1. An amount ($P$) doubles ($A = 2P$) in 10 years ($T = 10$).
2. We need to find the time ($T_2$) for the same amount ($P$) to triple ($A = 3P$) at the same fixed rate ($R$).

Step-by-Step Solution

Step 1: Analyze the condition for the amount doubling

When the amount doubles, the final amount $A = 2P$. This means the interest earned ($SI$) is equal to the principal amount:

$ SI = A - P = 2P - P = P $

We know this takes 10 years ($T = 10$). So, earning interest equal to the principal takes 10 years.

Step 2: Calculate the rate of interest (R)

Using the simple interest formula with $SI = P$ and $T = 10$ years:

$ P = \frac{P \times R \times 10}{100} $

We can cancel $P$ from both sides (assuming $P \neq 0$):

$ 1 = \frac{R \times 10}{100} $

Now, solve for $R$:

$ 1 = \frac{R}{10} $

$ R = 1 \times 10 = 10\% $

So, the fixed rate of simple interest is 10% per year.

Step 3: Analyze the condition for the amount tripling

We want to find the time ($T_2$) when the amount triples. If the initial amount is $P$, the final amount $A = 3P$. The interest earned ($SI$) required for this is:

$ SI = A - P = 3P - P = 2P $

So, we need to earn interest equal to twice the principal amount.

Step 4: Calculate the time required for the amount to triple

Using the simple interest formula with $SI = 2P$ and the calculated rate $R = 10\%$:

$ 2P = \frac{P \times 10 \times T_2}{100} $

Cancel $P$ from both sides:

$ 2 = \frac{10 \times T_2}{100} $

Simplify the right side:

$ 2 = \frac{T_2}{10} $

Solve for $T_2$:

$ T_2 = 2 \times 10 = 20 $

Therefore, it will take 20 years for the amount to triple.

Alternative Conceptual Method

Simple Interest implies that the interest earned is the same every year.

• To double the amount ($P$ to $2P$), the interest earned must be $P$. This took 10 years.
• To triple the amount ($P$ to $3P$), the interest earned must be $2P$.
• Since earning $P$ interest takes 10 years, earning $2P$ interest (which is twice as much) will take twice the time.
• Time = $2 \times 10$ years = 20 years.

This method confirms the result obtained using the formulas.