Solving Math Problems: Fractions And Decimals Explained

Hey guys! Let's dive into solving some math problems involving fractions and decimals. We'll break down each problem step-by-step, so it’s super easy to follow along. We'll tackle addition, subtraction, and division, making sure you understand the concepts behind each operation. Get ready to sharpen your math skills! This article aims to provide a comprehensive guide to solving these types of mathematical expressions. By understanding the methodologies and principles involved, you'll be well-equipped to tackle similar problems with confidence. Let's get started and make math a little less daunting and a lot more fun!

a. $18 \frac{9}{10}-\left(\frac{-3}{4}\right)$

Let’s start with our first problem, which involves subtracting a negative fraction from a mixed number. This might seem tricky at first, but don’t worry, we'll break it down. First, we need to convert the mixed number $18 \frac{9}{10}$ into an improper fraction. This means we'll multiply the whole number (18) by the denominator (10) and then add the numerator (9). This gives us $(18 * 10) + 9 = 189$. So, our mixed number becomes the improper fraction $\frac{189}{10}$.

Now, we have $\frac{189}{10} - \left(\frac{-3}{4}\right)$. Remember that subtracting a negative number is the same as adding its positive counterpart. So, the problem transforms into $\frac{189}{10} + \frac{3}{4}$. To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 4 is 20. We'll convert both fractions to have this denominator. To convert $\frac{189}{10}$, we multiply both the numerator and the denominator by 2, which gives us $\frac{189 * 2}{10 * 2} = \frac{378}{20}$. For $\frac{3}{4}$, we multiply both the numerator and the denominator by 5, giving us $\frac{3 * 5}{4 * 5} = \frac{15}{20}$.

Now our problem looks like this: $\frac{378}{20} + \frac{15}{20}$. We simply add the numerators and keep the denominator the same: $\frac{378 + 15}{20} = \frac{393}{20}$. Finally, let’s convert this improper fraction back into a mixed number. We divide 393 by 20, which gives us 19 with a remainder of 13. So, the mixed number is $19 \frac{13}{20}$. Thus, $18 \frac{9}{10}-\left(\frac{-3}{4}\right) = 19 \frac{13}{20}$.

Remember, guys, the key here is to convert, find the common denominator, and then add or subtract. Practice makes perfect, so keep at it! This foundational understanding of fraction manipulation is vital for tackling more complex mathematical problems in the future. Understanding these steps thoroughly will help you confidently solve similar problems and build a strong base in arithmetic.

b. $-75.5-|-75.5|$

Next up, we have a problem involving decimals and absolute values. Absolute value, remember, is the distance of a number from zero, so it’s always non-negative. Let's break down the problem $-75.5-|-75.5|$. First, we need to evaluate the absolute value of -75.5, which is written as |-75.5|. The absolute value of -75.5 is simply 75.5 because we're only concerned with the magnitude, not the sign. Now our problem becomes $-75.5 - 75.5$.

This is a straightforward subtraction of two numbers. Since both numbers are negative, we're essentially adding their magnitudes and keeping the negative sign. Think of it like moving 75.5 units to the left on the number line and then moving another 75.5 units to the left. So, we add 75.5 and 75.5, which equals 151. Since both numbers were negative, our final answer is -151. Therefore, $-75.5 - |-75.5| = -151$.

Remember, the absolute value part is crucial here. Always evaluate the absolute value first before performing any other operations. This is a common mistake, so make sure to keep it in mind. This concept of absolute value is crucial not just in basic arithmetic but also in more advanced mathematical fields such as calculus and linear algebra. Mastering these basics now will significantly aid your future studies in mathematics.

c. $\frac{3}{7} \div \frac{5}{8}$

Now let's tackle division with fractions. The problem is $\frac{3}{7} \div \frac{5}{8}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$. Therefore, our division problem becomes a multiplication problem: $\frac{3}{7} * \frac{8}{5}$.

To multiply fractions, we simply multiply the numerators together and the denominators together. So, we have $\frac{3 * 8}{7 * 5} = \frac{24}{35}$. This fraction is already in its simplest form because 24 and 35 have no common factors other than 1. So, our final answer is $\frac{24}{35}$. Therefore, $\frac{3}{7} \div \frac{5}{8} = \frac{24}{35}$.

Division with fractions might seem daunting at first, but the key is to remember to multiply by the reciprocal. This simple trick makes the problem much easier to handle. Understanding reciprocals is important in many areas of mathematics, including algebra and trigonometry. Practicing these types of problems helps solidify your understanding and improve your speed and accuracy. So, keep practicing and you'll be a pro in no time!

d. $-8 \div\left(\frac{-16}{21}\right)$

Our next problem involves dividing a whole number by a negative fraction. We have $-8 \div\left(\frac{-16}{21}\right)$. Just like in the previous problem, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{-16}{21}$ is $\frac{-21}{16}$. So, our problem becomes $-8 * \frac{-21}{16}$.

To make things easier, we can write -8 as a fraction: $\frac{-8}{1}$. Now we have $\frac{-8}{1} * \frac{-21}{16}$. When multiplying fractions, we multiply the numerators together and the denominators together. So, we get $\frac{-8 * -21}{1 * 16} = \frac{168}{16}$. Since both -8 and -21 are negative, their product is positive. Now we need to simplify the fraction $\frac{168}{16}$. We can start by dividing both the numerator and the denominator by their greatest common divisor, which is 8. $\frac{168 \div 8}{16 \div 8} = \frac{21}{2}$.

Finally, let’s convert this improper fraction to a mixed number. We divide 21 by 2, which gives us 10 with a remainder of 1. So, the mixed number is $10 \frac{1}{2}$. Therefore, $-8 \div\left(\frac{-16}{21}\right) = 10 \frac{1}{2}$. This type of problem is an excellent way to practice working with both negative numbers and fractions. Remember, dividing by a negative number changes the sign, so it's crucial to keep track of those signs.

e. $\frac{-4}{7} \div \frac{3}{4}$

Last but not least, we have another division problem involving fractions: $\frac{-4}{7} \div \frac{3}{4}$. Just like before, we'll multiply by the reciprocal. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. So, our problem transforms into $\frac{-4}{7} * \frac{4}{3}$.

Multiplying the numerators and denominators, we get $\frac{-4 * 4}{7 * 3} = \frac{-16}{21}$. This fraction is already in its simplest form because 16 and 21 have no common factors other than 1. So, our final answer is $\frac{-16}{21}$. Therefore, $\frac{-4}{7} \div \frac{3}{4} = \frac{-16}{21}$.

When working with fractions, the process is always the same: multiply by the reciprocal for division and simplify your answer if possible. These basic operations form the foundation for more complex algebraic manipulations. Keep practicing, and you'll become more comfortable with these operations. Remember, a strong understanding of these fundamental concepts is essential for success in higher-level mathematics.

Conclusion

So there you have it, guys! We've walked through solving a variety of math problems involving fractions and decimals. Remember, the key is to break down each problem into smaller, manageable steps. Whether it’s converting mixed numbers to improper fractions, finding common denominators, or multiplying by reciprocals, each step is crucial for getting the right answer. Keep practicing these skills, and you'll become a math whiz in no time! Math is like a muscle – the more you use it, the stronger it gets. By understanding the underlying principles and practicing regularly, you can overcome any mathematical challenge. Keep up the great work, and don't be afraid to ask for help when you need it. Happy solving!