Question

A lends a sum of money for 10 years at \(5 \%\) SI. \(B\) lends double the amount for 5 years at the same rate of interest. Which of the following statements is true in this regard? (a) \(A\) will get twice the amount of interest that \(B\) we"ld get (b) B will get twice the amount of interest that \(A\) wo thet (c) \(A\) will get the same amount of interest that \(B\) wou' ! get (d) \(B\) will get four times the amount of interest that \(A\) wou. 1 get

Short answer

Answer: A will get the same amount of interest that B would get.

Step-by-step solution

Define the problem and variables

Let x be the amount A lends. In this case, A lends x dollars for 10 years at a 5% simple interest rate. B lends double the amount, or 2x dollars, for 5 years at the same 5% simple interest rate. We will determine the relationship between the interest amounts A and B receive.

Calculate the interest for A

We will use the formula for simple interest, which is: Simple Interest = Principal × Rate × Time In A's case, the principal is x dollars, the rate is 5% or \(\dfrac{5}{100}\), and the time is 10 years. Therefore, the simple interest for A is: Simple Interest A = x × \(\dfrac{5}{100}\) × 10

Calculate the interest for B

For B, the principal is double A's, which is 2x dollars, the rate is 5% or \(\dfrac{5}{100}\), and the time is 5 years. Therefore, the simple interest for B is: Simple Interest B = 2x × \(\dfrac{5}{100}\) × 5

Compare the interest amounts

Now, we need to compare the interest that A and B receive, i.e., Simple Interest A and Simple Interest B. Simple Interest A = x × \(\dfrac{5}{100}\) × 10 Simple Interest B = 2x × \(\dfrac{5}{100}\) × 5 Dividing Simple Interest A by Simple Interest B, we get: \(\dfrac{\text{Simple Interest A}}{\text{Simple Interest B}} = \dfrac{x × \dfrac{5}{100} × 10}{2x × \dfrac{5}{100} × 5}\) Canceling out common factors, we get: \(\dfrac{\text{Simple Interest A}}{\text{Simple Interest B}} = \dfrac{10}{10}\) Therefore, Simple Interest A = Simple Interest B

Choose the correct answer

Now that we have determined Simple Interest A = Simple Interest B, we can choose the correct answer, which is: (c) A will get the same amount of interest that B would get.

Key concepts

Simple Interest Calculation

Understanding the concept of simple interest is fundamental to many financial transactions and problems involving loaning or investing money over time. Simple interest is the additional amount earned or paid on the principal amount without taking compounding into account - that is, interest is not earned on interest.

Let's delve into the actual calculation. The formula used for computing simple interest is as follows:
Simple Interest = Principal × Rate × Time.

In these calculations:

• Principal is the initial amount of money before interest.
• Rate is the percentage of the principal charged as interest for each time period.
• Time is the duration for which the money is borrowed or invested, typically expressed in years.

When we talk about rate, it is crucial to convert the percentage to a decimal by dividing by 100 before using it in our formula. Similarly, it is important to ensure that the time period is consistent with the rate's time frame. For example, if the rate is annual, then time should also be in years.

To apply this to our problem, we defined the principal for A and B, determined the rate as 5% (or 0.05 when converted), and applied the respective times of 10 years for A and 5 years for B. This straightforward approach helps students easily grasp how simple interest accumulates over time.

Comparing Interest Amounts

In financial literacy and many applied math situations, it's important to not just calculate interest, but also to be able to compare different interest amounts. This skill can help in evaluating various investment or loan options to choose the most beneficial one.

When comparing interest amounts, as we've done in our exercise, we calculated simple interest for two scenarios and then evaluated the relationship between them. Here's how you can approach comparisons step by step:

• Ensure you express all variables in consistent units (dollars with dollars, years with years, etc.).
• Calculate the simple interest for each scenario individually using the formula.
• Arrange the interest calculations such that you can directly compare them, either by using a ratio or by subtracting one from the other.

By dividing Simple Interest A by Simple Interest B and simplifying, we discovered that both A and B receive the same amount of interest. This kind of comparison is very common in evaluating loans or investment opportunities, and mastering these calculations can significantly impact personal and business finance decisions.

Quantitative Aptitude

Quantitative aptitude refers to our ability to handle numbers and execute mathematical operations. It is a key skill for students and professionals alike, especially for those in fields requiring analytical and problem-solving abilities, such as finance, engineering, and computer science.

In our discussion about simple interest and comparing interest amounts, we've engaged in quantitative reasoning—processing numerical information to reach a logical conclusion. Here are some aspects essential to developing this aptitude:

• Understanding basic mathematical concepts and formulas.
• Applying mathematical concepts to solve real-world problems.
• Displaying accuracy and speed in numerical calculations.
• Recognizing patterns and analyzing data efficiently.

Practicing problems like the one in our exercise regularly can boost quantitative aptitude significantly. This, in turn, can help in numerous competitive exams and in making informed decisions where quantitative analysis is vital.