How Much Money Does Precious Receive From Steven With A Simple Discount Rate Of 9.75% Per Year If She Must Pay Back R35,000 In 27 Months?

In the world of finance, understanding different methods of calculating interest and loan proceeds is crucial. One such method is the simple discount interest, where interest is deducted upfront from the loan amount. This article delves into the concept of simple discount, illustrating how to calculate the actual amount received by the borrower when a simple discount rate is applied. We will explore a practical scenario where Precious borrows money from Steven at a simple discount rate, and we aim to determine the precise amount Precious receives upfront.

When delving into the realm of finance, understanding the nuances of various interest calculation methods is paramount, and the simple discount approach stands as a notable example. In essence, simple discount is a method where the interest is calculated on the future value (or maturity value) of the loan and then deducted upfront from the principal amount. This contrasts with the more common simple interest method, where interest is calculated on the principal and added at the end of the loan term. The key difference lies in the timing of the interest deduction: upfront in simple discount versus at the end in simple interest. This seemingly small difference can have significant implications on the effective interest rate and the actual amount the borrower receives.

At its core, the simple discount method operates by calculating the interest amount based on the face value of the loan—the total amount due at the end of the term. This interest, often termed the 'discount,' is then subtracted from the face value. The remaining amount, known as the proceeds, is what the borrower actually receives. This upfront deduction of interest means the borrower has access to a smaller sum than the face value of the loan, which is a crucial consideration when evaluating the true cost of borrowing.

To illustrate, imagine a scenario where an individual borrows $1,000 with a simple discount rate of 10% for one year. The discount would be 10% of $1,000, which is $100. This $100 is deducted upfront, so the borrower receives $900. However, they are still obligated to repay $1,000 at the end of the year. This highlights an essential aspect of simple discount: the borrower is paying interest on an amount they never actually receive. The effective interest rate, therefore, is higher than the stated discount rate.

One of the critical advantages of understanding simple discount lies in its application in various short-term financial instruments, such as treasury bills and commercial paper. These instruments are often issued at a discount, meaning investors purchase them for less than their face value and receive the full face value at maturity. The difference between the purchase price and the face value represents the investor's return, effectively acting as the interest earned. Recognizing the simple discount principle allows investors to accurately assess the yield and make informed decisions about these investments.

Moreover, comprehending simple discount is vital for comparing different borrowing options. While a simple discount loan might appear attractive due to its upfront interest deduction, it's essential to calculate the effective interest rate to compare it accurately with other loan types, such as simple interest loans. The effective interest rate reflects the true cost of borrowing, considering the interest paid relative to the funds actually received. This comparison is crucial for borrowers seeking the most cost-effective financing solution.

In conclusion, the simple discount method presents a unique approach to interest calculation, where interest is deducted upfront. While it offers certain advantages in specific financial instruments, it's imperative to understand its implications on the actual amount received and the effective interest rate. By mastering the principles of simple discount, individuals and businesses can make sound financial decisions, whether they are borrowing funds or investing in short-term instruments.

In this specific scenario, Precious seeks a loan from Steven, with the agreement structured around a simple discount rate. The agreed-upon discount rate is 9.75% per year, and Precious is obligated to repay R35,000 in 27 months. This sets the stage for a classic simple discount problem where we need to determine the actual amount of money Precious will receive upfront. To solve this, we need to apply the simple discount formula, carefully considering the time period and the discount rate.

The critical elements of this problem are the discount rate, the repayment amount, and the loan term. The discount rate of 9.75% per year serves as the basis for calculating the interest that will be deducted upfront. The repayment amount of R35,000 represents the face value of the loan, which is the total amount Precious must repay at the end of the 27-month term. The loan term of 27 months needs to be converted into years to align with the annual discount rate. This conversion is crucial for accurate calculation, as the discount rate is given on an annual basis.

To effectively tackle this problem, we must first convert the loan term of 27 months into years. Since there are 12 months in a year, we divide 27 months by 12, resulting in 2.25 years. This conversion is essential because the discount rate is expressed as a percentage per year, and we need to ensure that the time period is consistent with the rate. Using the correct time period is paramount for accurate financial calculations, as even a slight discrepancy can lead to significant errors in the final result.

With the loan term correctly expressed in years, we can now proceed to calculate the discount amount. The discount is the interest that is deducted upfront from the face value of the loan. It is calculated by multiplying the face value (R35,000) by the discount rate (9.75% per year) and the loan term (2.25 years). This calculation will give us the total amount of interest that will be deducted upfront. Understanding this step is crucial as it directly impacts the amount Precious will actually receive.

Once we have calculated the discount amount, we can determine the proceeds, which is the actual amount Precious receives. The proceeds are calculated by subtracting the discount from the face value of the loan. This subtraction reveals the amount Precious has at her disposal after the upfront interest deduction. The difference between the face value and the proceeds represents the cost of borrowing under the simple discount method. This understanding is vital for Precious to assess the true cost of the loan and compare it with other financing options.

By carefully setting up the problem and identifying the key variables, we are well-positioned to apply the simple discount formula and calculate the proceeds. This step-by-step approach ensures accuracy and clarity in the solution process. The correct identification and conversion of the loan term, combined with the precise application of the discount rate, will lead us to the final answer, revealing the actual amount Precious receives from Steven.

To determine the amount Precious receives, we utilize the formula for simple discount proceeds: Proceeds = Face Value * (1 - Discount Rate * Time). Applying this formula to the scenario, we have a face value of R35,000, a discount rate of 9.75% (or 0.0975 as a decimal), and a time period of 2.25 years (27 months converted to years). Plugging these values into the formula allows us to calculate the proceeds, which represent the actual amount Precious receives upfront.

Now, let's apply the simple discount formula step by step to ensure clarity and accuracy in our calculation. The formula, Proceeds = Face Value * (1 - Discount Rate * Time), is the cornerstone of determining the actual amount received in a simple discount loan. We begin by substituting the known values into the formula. The face value, which is the amount Precious is obligated to repay, is R35,000. The discount rate, given as 9.75% per year, is converted to its decimal form, 0.0975. The time period, 27 months, has been converted to 2.25 years to align with the annual discount rate. These values are now ready to be inserted into the equation.

Substituting these values, the formula becomes: Proceeds = R35,000 * (1 - 0.0975 * 2.25). The next step is to perform the calculation within the parentheses. We start by multiplying the discount rate (0.0975) by the time period (2.25 years). This multiplication yields the total discount applied over the loan term, expressed as a decimal. This value represents the proportion of the face value that will be deducted as interest upfront. The accurate calculation of this product is crucial for determining the final proceeds.

Continuing with the calculation, we find that 0.0975 multiplied by 2.25 equals 0.219375. This value is then subtracted from 1, representing the portion of the face value that Precious will actually receive after the discount. This subtraction is a key step in the simple discount calculation, as it directly reflects the impact of the upfront interest deduction. The result of this subtraction is a decimal that, when multiplied by the face value, will give us the proceeds.

Subtracting 0.219375 from 1, we get 0.780625. This number represents the fraction of the face value that Precious will receive. To find the actual proceeds, we multiply this fraction by the face value of R35,000. This final multiplication will give us the amount Precious receives upfront, which is the solution to our problem. The accuracy of this multiplication is paramount, as it provides the definitive answer to how much money Precious has at her disposal.

Performing the multiplication, we find that R35,000 multiplied by 0.780625 equals R27,321.875. Rounding this to two decimal places, as is standard practice in financial calculations, we get R27,321.88. This is the amount Precious receives from Steven after the simple discount is applied. This final result provides a clear and concise answer to the problem, demonstrating the practical application of the simple discount formula.

Therefore, by meticulously applying the formula and performing each step with precision, we have successfully calculated the proceeds of the loan. The result, R27,321.88, represents the actual amount Precious receives, highlighting the impact of the simple discount method on the loan's initial disbursement. This calculation underscores the importance of understanding simple discount and its effects on borrowing and lending transactions.

Based on the calculation, the amount of money Precious receives from Steven now is R27,321.88. This corresponds to option a in the provided choices. This amount reflects the face value of the loan (R35,000) less the interest deducted upfront due to the simple discount rate.

To arrive at the correct solution, we systematically applied the simple discount formula, ensuring each step was executed with precision. The process began with identifying the key variables: the face value of the loan (R35,000), the simple discount rate (9.75% per year), and the loan term (27 months). A crucial initial step was converting the loan term from months to years, resulting in 2.25 years. This conversion is essential to align the time period with the annual discount rate, thereby ensuring the accuracy of the subsequent calculations.

Next, we substituted these values into the simple discount formula: Proceeds = Face Value * (1 - Discount Rate * Time). Plugging in the values, we had: Proceeds = R35,000 * (1 - 0.0975 * 2.25). The calculation within the parentheses was performed first. Multiplying the discount rate (0.0975) by the time (2.25 years) gave us 0.219375. This value represents the total discount applied over the loan term as a proportion of the face value.

Subtracting this discount factor from 1, we obtained 0.780625. This figure represents the portion of the face value that Precious will actually receive after the upfront interest deduction. This step is critical in understanding the essence of simple discount, where interest is deducted at the beginning of the loan term rather than being added at the end.

Finally, multiplying the face value (R35,000) by this fraction (0.780625) yielded the proceeds: R27,321.875. Rounding this amount to two decimal places, which is standard practice in financial calculations, we arrived at R27,321.88. This is the precise amount Precious receives upfront from Steven after the application of the simple discount.

This methodical approach not only provides the correct answer but also underscores the importance of understanding the simple discount method. The upfront deduction of interest means that Precious receives less than the face value of the loan, and this difference represents the cost of borrowing. This understanding is crucial for making informed financial decisions and comparing the true cost of different borrowing options.

Therefore, the solution R27,321.88 accurately reflects the proceeds of the loan under the simple discount arrangement. This outcome aligns with option a in the provided choices, confirming that our calculations and application of the simple discount formula were correct. The clarity and precision of this solution highlight the significance of mastering the principles of simple discount in financial transactions.

In conclusion, calculating loan proceeds under a simple discount arrangement requires careful application of the formula and attention to detail. The correct answer in this scenario is R27,321.88, which highlights the impact of discounting interest upfront. Understanding simple discount is essential for anyone involved in financial transactions, as it provides a clear picture of the actual funds received by the borrower.

To summarize the key takeaways, this exercise in calculating loan proceeds under a simple discount arrangement underscores several critical principles. Firstly, the simple discount method involves deducting interest upfront from the face value of the loan, which means the borrower receives less than the total amount they are obligated to repay. This contrasts with simple interest, where interest is added to the principal at the end of the loan term. Understanding this fundamental difference is crucial for comparing borrowing options and assessing the true cost of financing.

Secondly, the precise application of the simple discount formula is paramount for accurate calculations. The formula, Proceeds = Face Value * (1 - Discount Rate * Time), requires careful substitution of the correct values and meticulous execution of the mathematical operations. Errors in any step of the calculation can lead to significant discrepancies in the final result. Therefore, a step-by-step approach, as demonstrated in this article, is highly recommended to ensure accuracy.

Thirdly, the conversion of the loan term to match the time period of the discount rate is a critical step. In this scenario, the loan term was given in months (27 months), while the discount rate was expressed as an annual rate (9.75% per year). To align these, we converted the loan term to years (2.25 years). This conversion is essential because using inconsistent time periods can result in a miscalculation of the discount and, consequently, the proceeds.

Furthermore, the proceeds calculated under simple discount represent the actual amount the borrower has at their disposal. This amount is less than the face value of the loan, with the difference being the interest deducted upfront. This upfront deduction has implications for the effective interest rate, which is the true cost of borrowing when considering the funds actually received. Borrowers should be aware of this difference and consider the effective interest rate when comparing loan options.

In summary, understanding simple discount and its calculations is a valuable skill in financial literacy. It allows borrowers and lenders alike to accurately assess the terms of a loan and make informed decisions. The key takeaways from this discussion emphasize the importance of precise formula application, time period alignment, and awareness of the upfront interest deduction's impact on the actual proceeds. By mastering these principles, individuals and businesses can navigate financial transactions with greater confidence and clarity.