Solving Math: $-7 rac{7}{8}+3.7-(-15.9)$

Let's dive into how to evaluate the expression: $-7 \frac{7}{8}+3.7-(-15.9)$. This is a great example of a math problem that combines fractions, decimals, and negative numbers. We'll break it down step-by-step to make sure we understand each part. The key to solving this type of problem is to be meticulous and pay close attention to the details, especially the signs. Let's start by looking at the given expression: $-7 \frac{7}{8}+3.7-(-15.9)$. The first thing we should do is convert the mixed number to an improper fraction. Then, we will handle the subtraction of a negative number, which is the same as adding a positive number. Finally, we'll need to decide whether to convert everything to decimals or to find a common denominator to add/subtract the fraction with decimal numbers. In this case, we'll convert all numbers to decimals. This approach simplifies the calculations and reduces the chances of errors. Following these steps will help you master this type of problem, and you will become more confident in dealing with a range of mathematical expressions.

Step-by-Step Solution

First, we'll convert the mixed number $-7 \frac{7}{8}$ into an improper fraction. To do this, multiply the whole number (-7) by the denominator (8) and then add the numerator (7). That gives us $-7 * 8 = -56$, and $-56 + 7 = -49$. So, the improper fraction is $-\frac{56+7}{8} = -\frac{63}{8}$. Next, it's easier to convert the fraction to a decimal. Dividing 63 by 8 gives us 7.875. So, $-7 \frac{7}{8} = -7.875$. Our expression now looks like this: $-7.875 + 3.7 - (-15.9)$.

Now, let's handle the subtraction of a negative number. Subtracting a negative number is the same as adding the positive version of that number. So, $-(-15.9)$ becomes $+15.9$. Our expression is now: $-7.875 + 3.7 + 15.9$. We can start solving this problem by adding the positive numbers. Let's add $3.7$ and $15.9$ first. $3.7 + 15.9 = 19.6$. So, the expression becomes $-7.875 + 19.6$. Finally, we add $-7.875$ to $19.6$. To do this, we subtract the smaller absolute value from the larger one: $19.6 - 7.875$. When you subtract $7.875$ from $19.6$, you get $11.725$. Therefore, the answer to the expression $-7 \frac{7}{8} + 3.7 - (-15.9)$ is $11.725$. This methodical breakdown ensures that we account for every operation and sign, leading to an accurate result. This kind of detailed, step-by-step approach not only solves the problem but also enhances understanding and reduces the likelihood of making common errors. By practicing this method, you'll become more skilled at handling complex arithmetic expressions with confidence.

Convert Mixed Number to Decimal

Converting the mixed number $-7 \frac{7}{8}$ to a decimal is essential for simplifying the calculation. The mixed number is a combination of a whole number and a fraction. To convert it, first, we convert the fractional part, $\frac{7}{8}$, into a decimal. We can do this by dividing the numerator (7) by the denominator (8). When you perform the division $7 \div 8$, you get $0.875$. Thus, the fraction $\frac{7}{8}$ equals $0.875$. The mixed number is originally negative, so we must also preserve the sign. Thus, $-7 \frac{7}{8}$ becomes $-7.875$. This conversion makes it easier to work with the other decimal numbers in the expression and ensures consistency in our calculations. Understanding how to seamlessly convert mixed numbers to decimals is a foundational skill in arithmetic.

Handle Subtraction of a Negative Number

In the expression, we encounter the subtraction of a negative number, which is denoted as $-(-15.9)$. A fundamental rule in mathematics is that subtracting a negative number is the same as adding its positive counterpart. Therefore, $-(-15.9)$ simplifies to $+15.9$. This rule comes from the concept that a negative sign in front of a number essentially reverses its direction on the number line. When you subtract a negative, you're reversing this reversal, effectively moving in the positive direction. This conversion is crucial for simplifying the expression and ensuring the correct order of operations. Without this understanding, you could easily miscalculate the final result. Always remember that subtracting a negative is equivalent to addition, which is key for solving this kind of problem.

Add and Subtract Decimals

With all the numbers in decimal form, we can perform the addition and subtraction operations. Now our expression is $-7.875 + 3.7 + 15.9$. It is essential to perform the addition and subtraction in the correct order to arrive at the correct answer. We can proceed by adding $3.7$ and $15.9$ together first, which yields $19.6$. Next, subtract $7.875$ from $19.6$. The result of this subtraction gives us $11.725$. These steps highlight the importance of careful alignment of the decimal points when adding or subtracting to get accurate results. Being meticulous with these operations is critical to accurately evaluate the original expression. Correctly calculating the sum and differences of the values in the expression is what finally leads us to the answer.

Conclusion

In conclusion, by converting the mixed number to a decimal, simplifying the subtraction of a negative number, and carefully adding and subtracting the decimal values, we accurately evaluated the expression $-7 \frac{7}{8} + 3.7 - (-15.9)$ to be $11.725$. This problem highlights the importance of understanding and applying the rules of arithmetic correctly. It also emphasizes how crucial it is to pay attention to details, such as signs, order of operations, and decimal place values, to avoid making common errors. By practicing these steps, you will become more proficient in solving a wide array of mathematical problems.

This methodical approach underscores the value of structured problem-solving in mathematics, where each step builds upon the previous one to reach a definitive and correct answer. Mastering these fundamentals is a key component to understanding more complex mathematical concepts.

For more in-depth explanations and examples on similar mathematical problems, you can visit the Khan Academy website. This platform offers extensive resources, including video tutorials and practice exercises, that can further enhance your understanding and skills.