Compound Interest Problem Solving Finding The Principal Sum

Hey guys! Let's dive into a fascinating problem involving compound interest. We're going to break down a scenario where a sum of money is lent at 8% per annum compound interest. The twist? The interest earned in the second year exceeds the interest earned in the first year by Rs. 96. Our mission, should we choose to accept it, is to find the original sum of money.

Understanding Compound Interest: The Foundation of Our Quest

To crack this problem, we first need a solid grasp of what compound interest actually is. Unlike simple interest, where interest is calculated only on the principal amount, compound interest is calculated on the principal and the accumulated interest from previous periods. Think of it as interest earning interest – a snowball effect that can significantly boost your returns over time. The formula for compound interest is:

• A = P (1 + R/100)^N

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial sum of money)
• R = the annual interest rate (as a percentage)
• N = the number of years the money is invested or borrowed for

This formula is our key to unlocking the problem. It allows us to calculate the future value of an investment given the principal, interest rate, and time period. But in this case, we need to work backward to find the principal itself. Don't worry, we'll get there!

Deconstructing the Problem: Year by Year

The problem gives us a crucial piece of information: the difference in interest earned between the second year and the first year. To use this, we need to calculate the interest earned in each year separately. This means applying our compound interest knowledge in a step-by-step manner.

Let's denote the principal sum of money as 'P' (this is what we're trying to find). The interest rate is a fixed 8% per annum. Now, let's break down the interest calculation for each year:

• Year 1: The interest earned in the first year is simply the interest on the principal. Using the simple interest concept (which applies for the first year in compound interest), the interest earned is (P * 8 * 1) / 100 = 0.08P. This is straightforward enough, guys.
• Year 2: This is where the magic of compounding comes into play. At the start of the second year, the principal is no longer just 'P'. It's 'P' plus the interest earned in the first year (0.08P). So, the new principal for the second year is P + 0.08P = 1.08P. The interest earned in the second year is then (1.08P * 8 * 1) / 100 = 0.0864P. See how the interest earned is slightly higher than in the first year? That's the power of compounding!

Now we have expressions for the interest earned in both the first and second years, which is a major step forward.

The Crucial Difference: Setting Up the Equation

The problem states that the interest for the second year exceeds the interest for the first year by Rs. 96. This is the key piece of information that allows us to set up an equation and solve for 'P'. We can express this difference mathematically as:

• Interest in Year 2 - Interest in Year 1 = Rs. 96

Substituting the expressions we derived earlier, we get:

• 0. 0864P - 0.08P = 96

This equation is now our battleground. It contains only one unknown variable (P), which means we can use our algebraic skills to isolate it and find its value. This is where the real fun begins!

Solving for the Principal: The Algebraic Finale

Let's simplify the equation: 0.0864P - 0.08P = 96. Combining the 'P' terms, we get 0.0064P = 96. To isolate 'P', we divide both sides of the equation by 0.0064:

• P = 96 / 0.0064

Performing the division, we find that P = 15000. Ta-da! We've found the principal sum of money. The initial amount lent was Rs. 15,000.

Verifying Our Solution: A Sense of Satisfaction

It's always a good practice to verify our solution to make sure it makes sense in the context of the problem. Let's calculate the interest earned in each year using our calculated principal of Rs. 15,000:

• Year 1: Interest = 0.08 * 15000 = Rs. 1200
• Year 2: Principal at the start of Year 2 = 15000 + 1200 = Rs. 16200. Interest = 0.08 * 16200 = Rs. 1296

The difference in interest between Year 2 and Year 1 is 1296 - 1200 = Rs. 96. This matches the information given in the problem, so we can be confident in our solution. We did it, guys!

Conclusion: Compound Interest Demystified

We've successfully navigated the world of compound interest and solved a challenging problem. By breaking down the problem into smaller, manageable steps, understanding the underlying concepts, and applying a little algebra, we were able to find the original sum of money lent. Remember, compound interest is a powerful tool, and understanding how it works can be incredibly beneficial in various financial scenarios. Keep practicing, and you'll become a compound interest master in no time!

Let's tackle a classic compound interest problem! We'll break it down step by step, making it super easy to understand. The problem goes like this: a sum of money is lent out at an 8% per annum compound interest rate. The interest earned in the second year is Rs. 96 more than the interest earned in the first year. Our mission? To find out the initial sum of money that was lent out.

Grasping the Core Concept Compound Interest Explained

Before we jump into calculations, let's quickly recap what compound interest is all about. Compound interest is the interest calculated on the principal amount and also on the accumulated interest of previous periods. It's like interest earning interest! This is different from simple interest, where interest is only calculated on the principal amount. The formula for compound interest, which will be our trusty companion throughout this journey, is:

• A = P (1 + R/100)^N

Where:

• A is the final amount (principal + interest)
• P is the principal amount (the initial sum)
• R is the annual interest rate (as a percentage)
• N is the number of years

But in this scenario, we're not looking for the final amount. We're trying to find the principal (P). So, we'll need to manipulate this formula and use the information given in the problem to our advantage.

Year-by-Year Breakdown Unraveling the Interest Earned

The key to solving this problem lies in understanding how interest accumulates year by year. The problem highlights the difference in interest between the second and first years. So, let's calculate the interest earned in each of those years separately.

Let's assume the principal sum (the amount we're trying to find) is 'P'. The interest rate is 8% per year. Now, let's break it down:

• Interest in Year 1: For the first year, the interest calculation is straightforward. It's simply the interest on the principal amount. So, the interest earned in Year 1 is (P * 8 * 1) / 100 = 0.08P. Easy peasy, right?
• Interest in Year 2: This is where the magic of compounding comes in. At the beginning of the second year, the principal isn't just 'P' anymore. It's 'P' plus the interest earned in Year 1 (0.08P). So, the new principal for Year 2 is P + 0.08P = 1.08P. Now, we calculate the interest on this new principal: (1.08P * 8 * 1) / 100 = 0.0864P. Notice how the interest earned is a bit higher than in Year 1? That's the compounding effect in action!

Now we have expressions for the interest earned in both Year 1 and Year 2, which is a significant step forward in solving our problem. We're getting closer, guys!

Spotting the Crucial Difference Forming the Equation

The problem throws us a lifeline by stating that the interest earned in the second year exceeds the interest earned in the first year by Rs. 96. This difference is our golden ticket to setting up an equation and solving for the principal (P). We can express this mathematically as:

• Interest in Year 2 - Interest in Year 1 = Rs. 96

Substituting the expressions we calculated earlier, we get:

• 0. 0864P - 0.08P = 96

This equation is the heart of our solution. It contains only one unknown (P), which means we can use our algebraic prowess to isolate it and find its value. Let the solving begin!

Solving for the Unknown The Algebraic Showdown

Let's simplify the equation we've formed: 0.0864P - 0.08P = 96. Combining the 'P' terms, we get 0.0064P = 96. To find 'P', we simply divide both sides of the equation by 0.0064:

• P = 96 / 0.0064

Performing the division, we find that P = 15000. Boom! We've cracked it. The initial sum of money lent out was Rs. 15,000.

Double-Checking Our Work A Moment of Validation

It's always wise to double-check our answer to ensure it fits the problem's conditions. Let's calculate the interest earned in each year using our calculated principal of Rs. 15,000:

• Year 1: Interest = 0.08 * 15000 = Rs. 1200
• Year 2: Principal at the start of Year 2 = 15000 + 1200 = Rs. 16200. Interest = 0.08 * 16200 = Rs. 1296

The difference in interest between Year 2 and Year 1 is 1296 - 1200 = Rs. 96. This perfectly matches the information given in the problem! We can confidently say that our solution is correct. High five!

The Takeaway Compound Interest Unveiled

We've successfully unraveled a compound interest problem by breaking it down into manageable steps. We understood the concept of compound interest, calculated interest earned year by year, formed an equation based on the given information, and solved for the principal amount. Remember, compound interest is a powerful financial concept, and mastering it can open doors to various financial applications. Keep practicing, and you'll become a compound interest whiz in no time!

Hey there, math enthusiasts! Let's tackle another intriguing problem involving compound interest. We're presented with a scenario where a sum of money is lent at an 8% per annum compound interest rate. The catch? The interest earned in the second year surpasses the interest earned in the first year by Rs. 96. Our goal? To discover the original sum of money. Let's dive in and solve this puzzle together!

Understanding the Fundamentals What is Compound Interest?

Before we jump into calculations, let's make sure we're all on the same page about compound interest. Compound interest, in simple terms, is interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. This means that interest is earned on interest, making your money grow faster over time. This contrasts with simple interest, where interest is only earned on the original principal amount. The formula for compound interest is:

• A = P (1 + R/100)^N

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• R = the annual interest rate (as a percentage)
• N = the number of years the money is invested or borrowed for

This formula is the foundation of our understanding, but in this problem, we need to work backward. We know the interest difference, and we need to find the principal (P). Let's see how we can do that!

Breaking it Down Year by Year Calculating Interest Incrementally

To solve this problem effectively, we need to analyze the interest earned in each year separately. The problem highlights the difference in interest earned between the second year and the first year. So, let's calculate these values individually.

Let's denote the principal sum of money (the value we're trying to find) as 'P'. The interest rate is a fixed 8% per annum. Now, let's break down the interest calculation for each year:

• Year 1: The interest earned in the first year is simply the interest on the principal. Using the basic interest formula, the interest earned is (P * 8 * 1) / 100 = 0.08P. This is a straightforward calculation, guys.
• Year 2: This is where the power of compounding kicks in. At the beginning of the second year, the principal isn't just 'P' anymore. It's 'P' plus the interest earned in the first year (0.08P). So, the new principal for the second year becomes P + 0.08P = 1.08P. The interest earned in the second year is then (1.08P * 8 * 1) / 100 = 0.0864P. Notice how the interest earned is slightly higher than in the first year? That's the magic of compounding at work!

With these expressions for the interest earned in both the first and second years, we're now in a much stronger position to solve the problem. We're on the right track!

The Key Difference Setting Up the Equation for Success

The problem provides a crucial clue: the interest for the second year exceeds the interest for the first year by Rs. 96. This difference is the key to setting up an equation and solving for our unknown, 'P'. We can express this difference mathematically as:

• Interest in Year 2 - Interest in Year 1 = Rs. 96

Substituting the expressions we derived earlier, we get:

• 0. 0864P - 0.08P = 96

This equation is now our battleground. It contains only one unknown variable (P), which means we can use our algebraic skills to isolate it and find its value. Let the solving begin!

Finding the Principal The Algebraic Solution

Let's simplify the equation: 0.0864P - 0.08P = 96. Combining the 'P' terms, we get 0.0064P = 96. To isolate 'P', we divide both sides of the equation by 0.0064:

• P = 96 / 0.0064

Performing the division, we find that P = 15000. Voila! We've found the principal sum of money. The initial amount lent was Rs. 15,000.

Verification Time Ensuring Our Solution is Correct

It's always a good idea to verify our solution to make sure it aligns with the problem's conditions. Let's calculate the interest earned in each year using our calculated principal of Rs. 15,000:

• Year 1: Interest = 0.08 * 15000 = Rs. 1200
• Year 2: Principal at the start of Year 2 = 15000 + 1200 = Rs. 16200. Interest = 0.08 * 16200 = Rs. 1296

The difference in interest between Year 2 and Year 1 is 1296 - 1200 = Rs. 96. This perfectly matches the information given in the problem, so we can confidently say that our solution is correct. We nailed it, guys!

Conclusion Mastering Compound Interest

We've successfully navigated the world of compound interest and solved a challenging problem. By breaking down the problem into smaller, manageable steps, understanding the underlying concepts, and applying our algebraic skills, we were able to find the original sum of money lent. Remember, compound interest is a powerful financial tool, and understanding how it works can be incredibly beneficial in various scenarios. Keep practicing, and you'll become a compound interest pro in no time!

Hello everyone! Let's tackle a classic compound interest problem that often pops up in math discussions. We'll break it down step-by-step, ensuring a clear understanding of the concepts involved. The problem states: a sum of money is lent at an 8% per annum compound interest rate. If the interest for the second year exceeds that for the first year by Rs. 96, find the original sum of money. Sounds intriguing, right? Let's get started!

Laying the Foundation Understanding Compound Interest

Before we dive into the calculations, it's crucial to have a solid grasp of what compound interest is. Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. In simpler terms, it's interest earning interest. This is different from simple interest, where interest is only calculated on the principal amount. The formula that governs compound interest is:

• A = P (1 + R/100)^N

Where:

• A = the amount of money accumulated after n years, including interest.
• P = the principal amount (the initial amount of money).
• R = the annual interest rate (as a percentage).
• N = the number of years the money is invested or borrowed for.

While this formula is essential, our problem requires us to find the principal (P), given the difference in interest between two years. So, we'll need to adapt our approach and use the information provided strategically. Let's see how!

Analyzing Interest Year by Year Deconstructing the Problem

The problem hinges on the difference in interest earned between the second year and the first year. To utilize this information, we need to calculate the interest earned in each year individually. This involves understanding how the principal changes each year due to compounding.

Let's represent the principal sum of money (the amount we need to find) as 'P'. The interest rate is given as 8% per annum. Now, let's break down the interest calculation for each year:

• Interest for the First Year: In the first year, the interest is calculated directly on the principal. Therefore, the interest earned in the first year is (P * 8 * 1) / 100 = 0.08P. This is a straightforward application of the simple interest concept for the first year.
• Interest for the Second Year: Here's where the compounding effect comes into play. At the start of the second year, the principal is no longer just 'P'. It's 'P' plus the interest earned in the first year (0.08P). So, the new principal for the second year is P + 0.08P = 1.08P. The interest earned in the second year is then (1.08P * 8 * 1) / 100 = 0.0864P. Notice how the interest earned is slightly higher than in the first year? That's the essence of compound interest!

Now, we have expressions for the interest earned in both the first and second years, which is a crucial step towards solving the problem. We're making progress, guys!

The Key Relationship Setting Up the Equation

The problem provides the crucial link we need: the interest for the second year exceeds that for the first year by Rs. 96. This difference allows us to set up an equation and solve for the principal (P). We can express this relationship mathematically as:

• Interest in Year 2 - Interest in Year 1 = Rs. 96

Substituting the expressions we derived earlier, we get:

• 0. 0864P - 0.08P = 96

This equation is the heart of our solution. It contains only one unknown variable (P), which means we can use our algebraic skills to isolate it and find its value. Let's solve it!

Solving for the Principal The Algebraic Maneuvers

Let's simplify the equation: 0.0864P - 0.08P = 96. Combining the 'P' terms, we get 0.0064P = 96. To isolate 'P', we divide both sides of the equation by 0.0064:

• P = 96 / 0.0064

Performing the division, we find that P = 15000. And there we have it! The original sum of money lent was Rs. 15,000.

Verifying Our Solution The Final Check

It's always a good practice to verify our solution to ensure it aligns with the problem's conditions. Let's calculate the interest earned in each year using our calculated principal of Rs. 15,000:

• Year 1: Interest = 0.08 * 15000 = Rs. 1200
• Year 2: Principal at the start of Year 2 = 15000 + 1200 = Rs. 16200. Interest = 0.08 * 16200 = Rs. 1296

The difference in interest between Year 2 and Year 1 is 1296 - 1200 = Rs. 96. This perfectly matches the information given in the problem, confirming that our solution is correct. Success!

Conclusion Mastering Compound Interest Problems

We've successfully solved a compound interest problem by breaking it down into manageable steps. We understood the concept of compound interest, analyzed the interest earned year by year, set up an equation based on the given information, and solved for the principal amount. Remember, compound interest is a fundamental financial concept, and mastering it can be incredibly valuable. Keep practicing, and you'll become a pro at tackling these problems!