Abu's Simple Interest Investment Analysis

Overview of the Investment Scenario

The central focus of the transcript is a financial problem involving an individual named Abu. Abu has engaged in a dual-investment strategy by placing his capital into two separate accounts. These accounts operate under simple interest terms rather than compound interest. The primary objective is to determine the exact amount of currency (denominated in GH¢) that Abu invested in the account paying the higher interest rate, based on the total interest accumulated over a specific duration and the proportional relationship between the two initial investment amounts.

Theoretical Foundations: The Simple Interest Framework

To solve the problem described in the transcript, one must apply the fundamental principles of simple interest. Simple interest is a method of calculating the interest charge on a loan or the growth of an investment where the interest is calculated only on the original principal amount. The standard mathematical formula for simple interest is expressed as:

$I = P \times R \times T$

In this equation, $I$ represents the Interest earned, $P$ represents the Principal amount (the initial sum of money invested), $R$ represents the annual Interest Rate (expressed as a decimal), and $T$ represents the Time the money is invested in years. Because Abu has two separate investments, the total interest earned is the horizontal sum of the interest from each individual account, which can be modeled as:

$I_{Total} = (P_1 \times R_1 \times T) + (P_2 \times R_2 \times T)$

Identification of Quantities and Variables

The transcript provides several specific quantitative values that are essential for constructing the algebraic model. The interest rates for the two accounts are given as $11\%$ and $10\%$ . In decimal form, these are represented as $0.11$ and $0.10$ respectively. The total interest earned by Abu across both accounts is stated as $GH¢\,22,580.60$ .

Regarding the time period ( $T$ ), there is a discrepancy between the two provided transcript segments. Page 1 contains a fragmented reference to "35 years," which may be interpreted as a formatting error for $3.5$ years, whereas Page 2 explicitly states "3 years." For the purpose of these notes, the primary calculation will follow the explicit duration of $3$ years as provided in the clear text of Page 2, while acknowledging the potential variation.

Establishing Mathematical Relationships Between Accounts

A critical detail in the transcript is the proportional relationship between the two principals. It states that the amount invested at $11\%$ is $140\%$ of the amount invested at $10\%$ . To represent this algebraically, let $x$ be the amount invested at the $10\%$ rate ( $P_{10\%}$ ). Therefore, the amount invested at the $11\%$ rate ( $P_{11\%}$ ) is defined as:

$P_{11\%} = 1.40 \times x$

This relationship allows us to reduce the problem to a single-variable linear equation, making it possible to solve for both investment amounts using the total interest figure provided.

Formulation of the Total Interest Equation

By substituting the known variables and the defined relationship into the total interest formula, we construct the following equation based on the $3$ -year duration provided on Page 2:

$22,580.60 = (1.4x \times 0.11 \times 3) + (x \times 0.10 \times 3)$

Here, the first term $(1.4x \times 0.11 \times 3)$ represents the interest from the first account, and the second term $(x \times 0.10 \times 3)$ represents the interest from the second account. We can simplify the components of the equation as follows:

Calculate interest for the first account: $(1.4 \times 0.11 \times 3) = 0.462$
Calculate interest for the second account: $(0.10 \times 3) = 0.30$
Combine the terms: $0.462x + 0.30x = 22,580.60$

Step-by-Step Calculation: Solving for Principal Amounts

To find the value of $x$ (the principal at $10\%$ ), we continue the algebraic resolution of the simplified equation:

$0.762x = 22,580.60$

Dividing both sides by $0.762$ results in:

$x = \frac{22,580.60}{0.762}$

$x \approx 29,633.33$

This value represents the amount invested at the $10\%$ interest rate. To find the amount invested at the $11\%$ rate, which was the specific question asked in the transcript, we apply the $140\%$ multiplier:

$P_{11\%} = 1.40 \times 29,633.33$

$P_{11\%} \approx 41,486.67$

If the time period were interpreted as $3.5$ years (based on the "35 years" formatting on Page 1), the coefficients would change to $0.539$ and $0.35$ . This would result in a total coefficient of $0.889$ . Solving $0.889x = 22,580.60$ would yield an exact principal of $x = 25,400$ , making the investment at $11\%$ exactly $35,560$ . While the text on Page 2 explicitly says "3 years," the cleaner numerical result of the $3.5$ -year calculation suggests it may have been the intended duration.

Conclusion and Final Determinations

Based on the clear instruction of the Page 2 transcript, Abu invested approximately $GH¢\,41,486.67$ in the account paying $11\%$ interest. The calculation demonstrates how the proportional relationship between investment buckets and the fixed rates of simple interest interact over time to reach a specific total yield. This problem emphasizes the importance of correctly identifying the principal multipliers and meticulously tracking the duration (T) in interest formulas.