What Is -27.375 Divided By 7.5?

When we talk about mathematics, we often encounter problems that require us to perform basic arithmetic operations. One such operation is finding the quotient, which is the result of dividing one number by another. In this case, we need to determine the quotient of -27.375 and 7.5. This might seem like a straightforward division problem, but the presence of a negative number and decimal points can sometimes make people pause. Let's break down how to solve this, ensuring clarity and accuracy. Understanding division with decimals and negative numbers is a fundamental skill in mathematics, applicable in various real-world scenarios, from calculating average costs to understanding financial reports. The beauty of mathematics lies in its systematic approach, where each step builds upon the last, leading to a definitive answer. So, let's embark on this journey of calculation, where we will uncover the precise value when -27.375 is divided by 7.5. We'll explore the process step-by-step, making sure that even those who might find decimals a bit intimidating can follow along and feel confident in their ability to solve similar problems. The goal here isn't just to find the answer but to build a solid understanding of the underlying principles, empowering you to tackle more complex mathematical challenges in the future. Mathematics is a language, and understanding how to manipulate numbers, especially with different signs and decimal places, is like mastering its vocabulary and grammar.

Understanding the Division Process

The problem asks us to find the quotient of -27.375 and 7.5. This means we need to perform the division: -27.375 ÷ 7.5. The first thing to note is that we are dividing a negative number by a positive number. In mathematics, when you divide a negative number by a positive number, the result will always be negative. This is a crucial rule to remember and apply right from the start. So, we know our final answer will have a negative sign. Now, let's focus on the numerical division: 27.375 ÷ 7.5. When dealing with decimal division, a common strategy is to eliminate the decimal in the divisor (the number you are dividing by). To do this, we can multiply both the dividend (the number being divided) and the divisor by a power of 10 that will make the divisor a whole number. In this case, the divisor is 7.5, which has one decimal place. To make it a whole number, we multiply by 10. So, 7.5 * 10 = 75. We must apply the same operation to the dividend, 27.375. Multiplying 27.375 by 10 gives us 273.75. So, our division problem transforms into 273.75 ÷ 75. This step simplifies the division process significantly, allowing us to work with whole numbers and a single decimal place in the dividend. It's like rearranging the problem to make it more manageable without changing its fundamental value. This technique is a cornerstone of decimal arithmetic, making complex calculations approachable and systematic. By converting the problem into an equivalent one with a whole number divisor, we pave the way for a clear and accurate calculation.

Performing the Decimal Division

Now that we have transformed the problem into 273.75 ÷ 75, we can proceed with the division. Think of this as a long division problem. We are dividing 273.75 by 75. First, we look at the whole number part of the dividend, 273. How many times does 75 go into 273? We can estimate: 75 * 1 = 75, 75 * 2 = 150, 75 * 3 = 225, 75 * 4 = 300. So, 75 goes into 273 three times. We write down '3' as the first digit of our quotient. Next, we multiply 3 by 75, which equals 225. Then, we subtract 225 from 273: 273 - 225 = 48. Now, we bring down the next digit from the dividend, which is 7. Before we bring down the 7, we encounter the decimal point in the dividend. In long division, when you bring down a digit that is after the decimal point, you place the decimal point in the quotient directly above it. So, we place the decimal point after the '3' in our quotient. Now we have the number 487 to work with (after bringing down the 7). How many times does 75 go into 487? Let's continue our estimation: 75 * 5 = 375, 75 * 6 = 450, 75 * 7 = 525. So, 75 goes into 487 six times. We write down '6' after the decimal point in our quotient. Multiply 6 by 75, which equals 450. Subtract 450 from 487: 487 - 450 = 37. Finally, we bring down the last digit of the dividend, which is 5. We now have the number 375. How many times does 75 go into 375? We found this earlier: 75 * 5 = 375. So, 75 goes into 375 five times. We write down '5' as the last digit of our quotient. Multiply 5 by 75, which equals 375. Subtract 375 from 375: 375 - 375 = 0. Since we have a remainder of 0, our division is complete. The numerical result of 273.75 ÷ 75 is 3.65. This careful execution of long division, paying close attention to decimal placement and step-by-step subtraction, ensures accuracy.

Finalizing the Answer

We have successfully performed the numerical division and found that 27.375 divided by 7.5 equals 3.65. However, we must remember our initial observation about the signs. We were asked to find the quotient of a negative number (-27.375) and a positive number (7.5). As established earlier, dividing a negative number by a positive number results in a negative quotient. Therefore, we need to apply the negative sign to our calculated numerical result. So, the quotient of -27.375 and 7.5 is -3.65. Let's quickly review the options provided: A. -3.65, B. -0.365, C. 0.365, D. 3.65. Our calculated answer, -3.65, matches option A. This confirms our step-by-step calculation and our understanding of the rules of division with signed numbers. It’s always a good practice to double-check your work, especially when dealing with negative numbers and decimals. One way to quickly check is to estimate. We know -27 divided by 7 is roughly -4, so -3.65 is a very reasonable answer. The process involved understanding the definition of a quotient, handling decimal division by converting it to a whole number divisor, performing accurate long division, and correctly applying the rules for signs in multiplication and division. These are fundamental skills in mathematics that are essential for success in higher-level problem-solving and critical thinking. The ability to systematically break down a problem and apply learned principles is what makes mathematics a powerful tool for understanding the world around us. This exercise reinforces the importance of precision and attention to detail in mathematical computations, ensuring that we arrive at the correct and meaningful conclusion.

Conclusion

In conclusion, the quotient of -27.375 and 7.5 is -3.65. We arrived at this answer by first understanding that a negative divided by a positive yields a negative result. Then, we simplified the division by removing the decimal from the divisor, transforming the problem into 273.75 divided by 75. Through careful long division, we found the numerical value to be 3.65. Finally, by reintroducing the negative sign, we obtained the correct quotient of -3.65. This problem highlights the importance of understanding signed numbers and decimal arithmetic in mathematics. Mastering these concepts is crucial for a solid mathematical foundation and for tackling more complex problems encountered in various academic and real-world situations.

For further exploration into the world of mathematics and number properties, you can visit resources like Khan Academy or Math is Fun.