Calculating Products Of Fractions And Mixed Numbers A Comprehensive Guide

This article delves into the detailed solutions for calculating the products of fractions and mixed numbers. We will explore the step-by-step processes involved in solving these mathematical expressions, ensuring a clear understanding for anyone looking to enhance their arithmetic skills. This comprehensive guide covers the multiplication of fractions, mixed numbers, and decimals, providing a solid foundation for tackling more complex mathematical problems. Mastering these calculations is crucial for various fields, including engineering, finance, and everyday problem-solving.

a) Multiplying Fractions: ${\left(-\frac{5}{6}\right)\left(\frac{9}{25}\right)\left(-\frac{55}{12}\right)}$

To begin, let's dissect the first expression: ${\left(-\frac{5}{6}\right)\left(\frac{9}{25}\right)\left(-\frac{55}{12}\right)}$. When multiplying fractions, the primary step involves multiplying the numerators (the top numbers) and the denominators (the bottom numbers) separately. Before we dive into the multiplication, it's often beneficial to simplify the fractions to make the calculation more manageable. Simplifying involves finding common factors between the numerators and denominators and canceling them out. This process reduces the size of the numbers we are working with, which minimizes the chances of errors and simplifies the final steps.

In this specific case, we have three fractions being multiplied together. Let’s identify if there are any immediate simplifications we can make. Notice that the first fraction, ${-\frac{5}{6}}$, and the second fraction, ${\frac{9}{25}}$, have common factors. The number 5 in the numerator of the first fraction and the number 25 in the denominator of the second fraction share a common factor of 5. Similarly, the number 6 in the denominator of the first fraction and the number 9 in the numerator of the second fraction share a common factor of 3. By canceling these common factors, we can simplify the expression before performing the multiplication.

Detailed Simplification:

1. Rewrite the expression: ${-\frac{5}{6} \times \frac{9}{25} \times -\frac{55}{12}}$
2. Simplify ${-\frac{5}{6}}$ and ${\frac{9}{25}}$: Divide 5 in the first numerator and 25 in the second denominator by 5, resulting in 1 and 5 respectively. Divide 6 in the first denominator and 9 in the second numerator by 3, resulting in 2 and 3 respectively. The expression now looks like: ${-\frac{1}{2} \times \frac{3}{5} \times -\frac{55}{12}}$

Now that we've simplified the initial fractions, let's move on to the next step of further simplifying the expression with the third fraction, ${-\frac{55}{12}}$. Observe that the numerator 3 in the second fraction and the denominator 12 in the third fraction have a common factor of 3. Similarly, the denominator 5 in the second fraction and the numerator 55 in the third fraction share a common factor of 5. Reducing these common factors will further ease the multiplication process.

Further Simplification:

1. Simplify ${\frac{3}{5}}$ and ${-\frac{55}{12}}$: Divide 3 in the second numerator and 12 in the third denominator by 3, resulting in 1 and 4 respectively. Divide 5 in the second denominator and 55 in the third numerator by 5, resulting in 1 and 11 respectively. The expression now becomes: ${-\frac{1}{2} \times \frac{1}{1} \times -\frac{11}{4}}$

Now that all possible simplifications have been made, we are ready to multiply the numerators together and the denominators together. Keep in mind that multiplying two negative numbers results in a positive number. This is a fundamental rule of arithmetic and must be applied correctly to ensure the correct sign in our final answer.

Final Multiplication:

1. Multiply the numerators: ${-1 \times 1 \times -11 = 11}$
2. Multiply the denominators: ${2 \times 1 \times 4 = 8}$
3. Combine the results: ${\frac{11}{8}}$

Thus, the final result of the expression ${\left(-\frac{5}{6}\right)\left(\frac{9}{25}\right)\left(-\frac{55}{12}\right)}$ is ${\frac{11}{8}}$. This fraction can also be expressed as a mixed number. To convert an improper fraction (where the numerator is greater than the denominator) to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, the remainder becomes the new numerator, and the denominator remains the same.

Converting to Mixed Number:

1. Divide 11 by 8: The quotient is 1, and the remainder is 3.
2. Express as a mixed number: ${1\frac{3}{8}}$

Therefore, the simplified answer can also be written as ${1\frac{3}{8}}$. This comprehensive approach ensures not only the correct answer but also a clear understanding of the underlying principles of fraction multiplication and simplification.

b) Multiplying Mixed Numbers, Decimals, and Fractions: ${\left(-3\frac{6}{7}\right)\left(\frac{1}{81}\right)(2.3)(-0.5) - \frac{5}{21}}$

In this section, we will tackle the expression ${\left(-3\frac{6}{7}\right)\left(\frac{1}{81}\right)(2.3)(-0.5) - \frac{5}{21}}$, which involves multiplying a mixed number, a fraction, and decimals, followed by a subtraction. The first step in simplifying this expression is to convert the mixed number into an improper fraction and the decimals into fractions. This conversion makes it easier to perform the multiplication and ensures we are working with consistent forms of numbers.

Converting Mixed Number and Decimals to Fractions:

1. Convert ${-3\frac{6}{7}}$ to an improper fraction: Multiply the whole number (3) by the denominator (7), which gives 21. Add the numerator (6) to this result, giving 27. Place this over the original denominator (7). The improper fraction is ${-\frac{27}{7}}$.
2. Convert 2.3 to a fraction: 2.3 can be written as ${\frac{23}{10}}$ since there is one decimal place.
3. Convert -0.5 to a fraction: -0.5 can be written as ${-\frac{1}{2}}$ as it is a common decimal-fraction equivalent.

Now that we have converted the mixed number and decimals into fractions, the expression looks like this: ${\left(-\frac{27}{7}\right)\left(\frac{1}{81}\right)\left(\frac{23}{10}\right)\left(-\frac{1}{2}\right) - \frac{5}{21}}$. The next step is to multiply the fractions together. Before doing so, it is advantageous to simplify the fractions by canceling out common factors between the numerators and denominators. This process will reduce the complexity of the multiplication and make the calculations more manageable.

Simplifying and Multiplying Fractions:

1. Identify and cancel common factors: Notice that 27 and 81 have a common factor of 27. Dividing both by 27 simplifies them to 1 and 3, respectively. The expression now becomes: ${\left(-\frac{1}{7}\right)\left(\frac{1}{3}\right)\left(\frac{23}{10}\right)\left(-\frac{1}{2}\right) - \frac{5}{21}}$.
2. Multiply the fractions: Multiply the numerators together: ${(-1) \times 1 \times 23 \times (-1) = 23}$. Multiply the denominators together: ${7 \times 3 \times 10 \times 2 = 420}$. The product of the fractions is ${\frac{23}{420}}$.

So far, we have simplified the multiplication part of the expression. The expression now reads ${\frac{23}{420} - \frac{5}{21}}$. The next step is to subtract the fraction ${\frac{5}{21}}$ from ${\frac{23}{420}}$. To do this, we need to find a common denominator. The least common multiple (LCM) of 420 and 21 will be the easiest common denominator to work with. Since 420 is divisible by 21 (420 = 21 * 20), 420 is the least common multiple.

Finding a Common Denominator and Subtracting:

1. Find the common denominator: The LCM of 420 and 21 is 420.
2. Convert ${\frac{5}{21}}$ to an equivalent fraction with a denominator of 420: Multiply both the numerator and the denominator of ${\frac{5}{21}}$ by 20 (since 420 / 21 = 20). This gives us ${\frac{5 \times 20}{21 \times 20} = \frac{100}{420}}$.
3. Subtract the fractions: Now we subtract ${\frac{100}{420}}$ from ${\frac{23}{420}}$: ${\frac{23}{420} - \frac{100}{420} = \frac{23 - 100}{420} = \frac{-77}{420}}$.

We have now arrived at the fraction ${-\frac{77}{420}}$. As a final step, we should simplify this fraction by finding the greatest common divisor (GCD) of 77 and 420 and dividing both the numerator and denominator by it. The GCD of 77 and 420 is 7. Dividing both the numerator and the denominator by 7 gives us the simplified fraction.

Simplifying the Final Fraction:

1. Find the GCD of 77 and 420: The GCD is 7.
2. Divide both the numerator and the denominator by 7: ${-\frac{77 ÷ 7}{420 ÷ 7} = -\frac{11}{60}}$.

Thus, the final answer to the expression ${\left(-3\frac{6}{7}\right)\left(\frac{1}{81}\right)(2.3)(-0.5) - \frac{5}{21}}$ is ${-\frac{11}{60}}$. This detailed breakdown illustrates the importance of converting mixed numbers and decimals to fractions, simplifying before multiplying, finding common denominators for subtraction, and reducing the final fraction to its simplest form.

c) Multiplying Decimals and Fractions: ${(3.5)\frac{10}{14}(1.1)\frac{9}{18}(1.23)}$

In this section, we will address the expression ${(3.5)\frac{10}{14}(1.1)\frac{9}{18}(1.23)}$, which involves the multiplication of decimals and fractions. To effectively solve this, we will first convert all decimals to fractions and then simplify before performing the multiplication. Converting decimals to fractions allows us to work with a consistent format, making the multiplication process more straightforward.

Converting Decimals to Fractions:

1. Convert 3.5 to a fraction: 3.5 can be written as ${\frac{35}{10}}$, which simplifies to ${\frac{7}{2}}$.
2. Convert 1.1 to a fraction: 1.1 can be written as ${\frac{11}{10}}$.
3. Convert 1.23 to a fraction: 1.23 can be written as ${\frac{123}{100}}$.

Now that we have converted the decimals to fractions, the expression looks like this: ${\left(\frac{7}{2}\right)\left(\frac{10}{14}\right)\left(\frac{11}{10}\right)\left(\frac{9}{18}\right)\left(\frac{123}{100}\right)}$. Before we multiply these fractions, it's beneficial to simplify them to reduce the complexity of the calculation. Simplifying involves identifying common factors between the numerators and denominators and canceling them out.

Simplifying Fractions Before Multiplication:

1. Simplify ${\frac{10}{14}}$: Both 10 and 14 have a common factor of 2. Dividing both by 2 gives us ${\frac{5}{7}}$.
2. Simplify ${\frac{9}{18}}$: Both 9 and 18 have a common factor of 9. Dividing both by 9 gives us ${\frac{1}{2}}$.

Now our expression is: ${\left(\frac{7}{2}\right)\left(\frac{5}{7}\right)\left(\frac{11}{10}\right)\left(\frac{1}{2}\right)\left(\frac{123}{100}\right)}$. We can further simplify by canceling common factors between different fractions.

Further Simplification:

1. Cancel the 7 in ${\frac{7}{2}}$ and ${\frac{5}{7}}$: The 7 in the numerator of the first fraction and the 7 in the denominator of the second fraction cancel each other out.
2. Simplify ${\frac{5}{10}}$: The 5 in the numerator of the second fraction and the 10 in the denominator of the third fraction have a common factor of 5. Dividing both by 5 gives us ${\frac{1}{2}}$.

After these simplifications, the expression becomes: ${\left(\frac{1}{2}\right)\left(\frac{1}{1}\right)\left(\frac{11}{2}\right)\left(\frac{1}{2}\right)\left(\frac{123}{100}\right)}$. Now we are ready to multiply the fractions together. This involves multiplying all the numerators together and all the denominators together.

Multiplying the Fractions:

1. Multiply the numerators: ${1 \times 1 \times 11 \times 1 \times 123 = 1353}$.
2. Multiply the denominators: ${2 \times 1 \times 2 \times 2 \times 100 = 800}$.
3. Combine the results: The product is ${\frac{1353}{800}}$.

The final step is to simplify the resulting fraction if possible. In this case, 1353 and 800 do not have any common factors other than 1, so the fraction ${\frac{1353}{800}}$ is already in its simplest form. However, we can convert this improper fraction to a mixed number to better understand its value.

Converting to a Mixed Number:

1. Divide 1353 by 800: The quotient is 1, and the remainder is 553.
2. Express as a mixed number: ${1\frac{553}{800}}$.

Therefore, the final answer to the expression ${(3.5)\frac{10}{14}(1.1)\frac{9}{18}(1.23)}$ is ${\frac{1353}{800}}$ or, as a mixed number, ${1\frac{553}{800}}$. This comprehensive solution demonstrates the steps involved in converting decimals to fractions, simplifying fractions before multiplication, multiplying the simplified fractions, and converting an improper fraction to a mixed number.

d) Discussion Category: Mathematics

This discussion falls squarely into the category of mathematics, specifically within the sub-disciplines of arithmetic and pre-algebra. The problems presented involve fundamental arithmetic operations such as multiplication, division, addition, and subtraction, applied to fractions, decimals, and mixed numbers. These concepts are foundational to more advanced mathematical topics and are typically introduced in elementary and middle school mathematics curricula. Understanding how to manipulate and calculate with fractions, decimals, and mixed numbers is crucial for success in algebra, geometry, calculus, and various other quantitative fields.

Furthermore, the exercises emphasize the importance of simplifying expressions, which is a core skill in algebra. Simplifying fractions and expressions not only makes calculations easier but also helps in recognizing patterns and relationships within mathematical problems. The ability to convert between different forms of numbers (e.g., decimals to fractions, mixed numbers to improper fractions) is another critical skill highlighted in these examples. This flexibility in representation is essential for solving a wide range of mathematical problems efficiently.

The problems also touch on the concept of the order of operations, although implicitly. In expression b), the multiplication operations must be performed before the subtraction. This adherence to the order of operations (often remembered by the acronym PEMDAS/BODMAS) is vital for obtaining correct results in any mathematical calculation involving multiple operations.

In a broader context, these types of calculations are not only important for academic mathematics but also for real-world applications. For instance, understanding fractions and decimals is necessary for tasks such as measuring ingredients in cooking, calculating proportions in construction, managing finances, and understanding statistical data. Therefore, mastering these fundamental mathematical skills has practical significance beyond the classroom.

In summary, the discussion category for the given mathematical problems is mathematics, encompassing areas such as arithmetic, pre-algebra, and fundamental algebraic skills. The problems serve as a valuable exercise in mastering basic operations with fractions, decimals, and mixed numbers, which are essential for both academic and practical applications.