Evaluate K 2000(1-10%)^3 A Step-by-Step Solution

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Understanding the Problem

The question asks us to evaluate the expression K 2000(1-10%)^3. This expression involves a principal amount (K 2000), a percentage decrease (10%), and an exponent (3). This kind of expression is commonly encountered in financial calculations, particularly when dealing with depreciation or compound interest. In this case, since we have a subtraction within the parentheses, it is likely that we are dealing with a scenario where the initial amount is decreasing over time. Before we jump into the step-by-step solution, let's break down the components of the expression and discuss the underlying mathematical principles.

First, we need to convert the percentage into a decimal. Percentages are simply fractions out of 100, so 10% is equivalent to 10/100, which simplifies to 0.1. This conversion is crucial because we cannot directly perform arithmetic operations with percentages; we must first express them as decimals or fractions. The expression within the parentheses, (1 - 10%), represents the remaining proportion after a 10% decrease. In decimal form, this becomes (1 - 0.1), which equals 0.9. This value, 0.9, is the factor by which the initial amount is multiplied for each period. The exponent of 3 indicates that this reduction is applied three times, which could represent three years, three quarters, or any other three periods depending on the context of the problem. The base amount, K 2000, is the starting value. This could be an initial investment, the original price of an asset, or any other quantity that is subject to decrease. By multiplying this initial amount by (0.9)^3, we are calculating the value after three periods of a 10% decrease. This kind of calculation is fundamental in various financial applications, such as calculating the depreciated value of an asset over time or determining the remaining balance of a loan after several payments.

Now, let's think about the order of operations. In mathematics, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). According to PEMDAS, we must first perform the operation inside the parentheses, then handle the exponent, and finally, perform the multiplication. This ensures that we arrive at the correct answer. If we were to perform the operations in a different order, we would get an incorrect result. For example, if we multiplied 2000 by 1 before subtracting 10%, we would be violating the order of operations and misrepresenting the problem. The exponent operation is also critical. Raising 0.9 to the power of 3 means multiplying 0.9 by itself three times: 0.9 * 0.9 * 0.9. This gives us the cumulative effect of the 10% decrease over the three periods. If we were to simply multiply 0.9 by 3, we would be calculating a 30% decrease, which is different from a 10% decrease applied three times consecutively. This concept is crucial in understanding compound interest and depreciation. By correctly applying the order of operations and understanding the underlying mathematical principles, we can confidently evaluate the expression and arrive at the accurate final value. So, let’s proceed with the step-by-step solution to make sure we get the correct answer.

Step-by-Step Solution

  1. Convert the percentage to a decimal: 10% = 10/100 = 0.1
  2. Evaluate the expression inside the parentheses: (1 - 10%) = (1 - 0.1) = 0.9
  3. Calculate the exponent: (0.9)^3 = 0.9 * 0.9 * 0.9 = 0.729
  4. Multiply by the initial amount: K 2000 * 0.729 = K 1458

Therefore, K 2000(1-10%)^3 = K 1458.

Detailed Breakdown of Each Step

Let's delve deeper into each step of the solution to ensure a complete understanding. In the first step, we converted the percentage to a decimal. This is a fundamental step in almost any mathematical problem involving percentages. The concept behind this conversion is that a percentage represents a proportion out of 100. So, when we say 10%, we mean 10 out of 100, which can be written as the fraction 10/100. To convert this fraction to a decimal, we simply divide the numerator (10) by the denominator (100), which gives us 0.1. This decimal representation is essential for performing arithmetic operations. If we try to perform calculations directly with percentages, we will likely end up with an incorrect answer. For example, if we were to subtract 10 directly from 1, we would get -9, which is meaningless in the context of the problem. Converting to a decimal allows us to accurately represent the proportional decrease.

The second step involved evaluating the expression inside the parentheses: (1 - 0.1) = 0.9. This step represents the proportion of the original amount that remains after the 10% decrease. Think of it this way: if something decreases by 10%, it means that 90% of it remains. The value 0.9 is the decimal equivalent of 90%, so it represents the remaining proportion. This concept is crucial in understanding depreciation and compound interest. In depreciation, the value of an asset decreases over time, and this decrease is often expressed as a percentage. In compound interest, the interest earned is added to the principal, and subsequent interest is calculated on the new principal. In both cases, understanding how to calculate the remaining proportion or the growth factor is essential. The expression (1 - 0.1) gives us this factor directly.

The third step was to calculate the exponent: (0.9)^3 = 0.729. This step is where the concept of compounding comes into play. The exponent of 3 indicates that the 10% decrease is applied three times. This could represent three years of depreciation, three quarters of a loan payment, or any other three periods. When we raise 0.9 to the power of 3, we are essentially multiplying 0.9 by itself three times: 0.9 * 0.9 * 0.9. The result, 0.729, represents the cumulative effect of the 10% decrease over the three periods. It's important to understand that this is not the same as a 30% decrease. A 30% decrease would be represented by multiplying the initial amount by (1 - 0.3) = 0.7. However, a 10% decrease applied three times results in a smaller final value because the decrease is applied to a smaller base each time. This is the essence of compound depreciation or compound interest.

Finally, in the fourth step, we multiplied the initial amount by the result of the exponentiation: K 2000 * 0.729 = K 1458. This final multiplication gives us the value of the amount after the three periods of decrease. In the context of depreciation, this would be the depreciated value of the asset. In the context of a loan, this might represent the remaining balance after three payments. The result, K 1458, is the final answer to the problem. It's crucial to include the unit (K in this case) in the final answer to provide context and ensure that the answer is meaningful. Without the unit, the number 1458 could represent anything, but with the unit K, we know that it represents a monetary value.

Alternative Approaches and Considerations

While the step-by-step solution provided is the most straightforward way to evaluate the expression, there are alternative approaches and considerations that can enhance our understanding of the problem. One approach involves using the formula for compound depreciation, which is a specific case of the compound interest formula. The general formula for compound interest is A = P(1 + r)^n, where A is the final amount, P is the principal amount, r is the interest rate, and n is the number of periods. In the case of depreciation, the rate is negative, so the formula becomes A = P(1 - r)^n. This formula is essentially what we applied in the step-by-step solution, but recognizing this formula can help in solving similar problems more efficiently. Applying the formula directly, we have A = K 2000(1 - 0.1)^3 = K 2000(0.9)^3 = K 1458, which confirms our previous result.

Another consideration is the context of the problem. While the problem is presented as a mathematical expression, it is likely derived from a real-world scenario, such as financial calculations or asset valuation. Understanding the context can provide valuable insights and help in interpreting the result. For example, if the problem represents the depreciation of a piece of equipment over three years, the result, K 1458, would be the book value of the equipment after three years. This information could be used for financial reporting, tax purposes, or investment decisions. Similarly, if the problem represents the decrease in the value of an investment portfolio, the result would indicate the current value of the portfolio after three periods of decline.

Furthermore, it's important to consider the limitations of the model. The formula we used assumes that the decrease is constant over each period. In real-world scenarios, this may not always be the case. For example, the depreciation rate of an asset might change over time due to factors such as wear and tear, technological obsolescence, or market conditions. Similarly, the interest rate on a loan might fluctuate, affecting the remaining balance. In such cases, a more sophisticated model might be required to accurately represent the situation. This could involve using different depreciation methods, such as the double-declining balance method, or incorporating variable interest rates into the calculation.

In addition to these considerations, it's also beneficial to check the reasonableness of the answer. A quick way to do this is to estimate the result mentally. We know that a 10% decrease from K 2000 is K 200, so after one period, the value would be approximately K 1800. After two periods, another 10% decrease would be approximately K 180, so the value would be around K 1620. After three periods, another 10% decrease would be about K 162, bringing the value to around K 1458, which is consistent with our calculated result. This simple estimation technique can help catch any significant errors in the calculation and ensure that the final answer is plausible.

Conclusion

In conclusion, we have successfully evaluated the expression K 2000(1-10%)^3 and found the result to be K 1458. We achieved this by following the order of operations, converting the percentage to a decimal, and carefully performing each step. We also discussed the underlying mathematical principles and the context in which this type of expression is commonly used. By understanding the step-by-step solution, alternative approaches, and the limitations of the model, we can confidently solve similar problems and apply this knowledge to real-world scenarios. The ability to evaluate such expressions is fundamental in various fields, including finance, accounting, and economics. The key takeaways are the importance of the order of operations, the conversion of percentages to decimals, and the concept of compounding. By mastering these concepts, we can effectively analyze and interpret financial data, make informed decisions, and solve complex problems involving percentages and exponents.