Mastering the Calculation: A Comprehensive Guide to Solving 724 x 1.075
The seemingly simple calculation of 724 x 1.075 holds significant practical relevance across various fields. From calculating sales tax and interest to understanding percentage increases in various contexts (e.g., population growth, investment returns), multiplying by a decimal exceeding one is a fundamental skill. This article will demystify this specific calculation, addressing common challenges and providing multiple approaches to arrive at the accurate solution. We'll delve into the logic behind the process, offering step-by-step instructions and practical examples to ensure a thorough understanding.
1. Understanding the Problem: 724 x 1.075
The calculation 724 x 1.075 represents a percentage increase. The number 1.075 can be broken down as 1 + 0.075, where '1' represents the original value (100%), and '0.075' represents a 7.5% increase. Therefore, the problem asks us to find the value that is 7.5% greater than 724. Understanding this context helps in verifying the reasonableness of our answer later.
2. Method 1: Direct Multiplication (Long Multiplication)
This is the most straightforward method, relying on the standard long multiplication technique taught in elementary school. While potentially time-consuming for larger numbers, it provides a solid foundation for understanding the process.
Step 1: Set up the problem. Write the numbers vertically, aligning the units digits:
```
724
x 1.075
-------
```
Step 2: Multiply by each digit of the multiplier. Start with the rightmost digit of the multiplier (5) and multiply it by each digit of the multiplicand (724). Then repeat for the next digit (7), then 0, and finally 1. Remember to add zeros as placeholders for each subsequent multiplication:
```
724
x 1.075
-------
3620 (724 x 5)
50680 (724 x 7 x 10)
0000 (724 x 0 x 100)
724000 (724 x 1 x 1000)
-------
```
Step 3: Add the partial products. Sum the results of each multiplication step:
```
3620
50680
0000
724000
-------
778300
```
Step 4: Adjust for the decimal point. Since the multiplier (1.075) has three decimal places, we move the decimal point three places to the left in the final result:
```
778.300
```
Therefore, 724 x 1.075 = 778.3
3. Method 2: Breaking Down the Multiplication
This method simplifies the calculation by breaking down the multiplier (1.075) into smaller, manageable parts.
Step 1: Calculate 7.5% of 724. This is equivalent to multiplying 724 by 0.075:
724 x 0.075 = 54.3
Step 2: Add the 7.5% increase to the original value. This gives us the final answer:
724 + 54.3 = 778.3
4. Method 3: Using a Calculator
The most efficient and practical method for larger or more complex calculations is using a calculator. Simply input "724 x 1.075" and press the equals button. This will instantly provide the correct answer, 778.3. However, understanding the underlying principles remains crucial for problem-solving and error checking.
5. Verifying the Answer
Given that we are calculating a 7.5% increase, our answer (778.3) should be reasonably larger than the original value (724). The difference (778.3 - 724 = 54.3) represents the 7.5% increase, which we calculated separately in Method 2. This verification confirms the accuracy of our result.
Summary
Calculating 724 x 1.075 involves finding the value that is 7.5% greater than 724. We explored three methods: direct multiplication, breaking down the calculation into smaller parts, and using a calculator. Each method provides a pathway to arrive at the correct answer, 778.3. While a calculator offers speed and efficiency, understanding the underlying mathematical principles through direct multiplication or the breakdown method provides a deeper grasp of percentage calculations.
Frequently Asked Questions (FAQs)
1. What if the multiplier was less than 1 (e.g., 724 x 0.9)? This represents a percentage decrease. The same methods can be applied, resulting in a final value less than the original.
2. How would I calculate a different percentage increase, say 12%? You would multiply 724 by 1.12 (1 + 0.12).
3. Is there a way to estimate the answer before calculating? Yes, rounding 724 to 700 and 1.075 to 1.1 provides a rough estimate of 770, giving a reasonable ball-park figure.
4. What are the practical applications of this type of calculation beyond sales tax and interest? This type of calculation is useful for calculating population growth, inflation adjustments, compound interest, and many other scenarios involving percentage changes.
5. What if I make a mistake in the calculation? Double-check your work using a different method (e.g., if you used long multiplication, verify with a calculator) or use estimation to identify potential errors. Understanding the underlying principles will significantly help in identifying and correcting any mistakes.
Note: Conversion is based on the latest values and formulas.
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