Mixed Number Arithmetic Solutions And Step-by-Step Guide

When adding mixed numbers, it's essential to approach the problem methodically to ensure accuracy. In this section, we will delve into the intricacies of adding $8 \frac{11}{12}$ and $9 \frac{5}{18}$, breaking down each step for clarity. The main key in adding mixed numbers lies in understanding how to handle both the whole number parts and the fractional parts effectively. To begin, we focus on the fractional components, which often require finding a common denominator. This foundational step allows us to combine the fractions seamlessly. Subsequently, we add the whole numbers together. The final step involves simplifying the resulting fraction, if necessary, and combining it with the sum of the whole numbers to obtain our final answer. This meticulous process guarantees that we arrive at the correct solution while reinforcing the fundamental principles of mixed number arithmetic. To solve $8 \frac{11}{12} + 9 \frac{5}{18}$, we first need to find a common denominator for the fractions $\frac{11}{12}$ and $\frac{5}{18}$. The least common multiple (LCM) of 12 and 18 is 36. Now, we convert both fractions to have this common denominator. For $\frac{11}{12}$, we multiply both the numerator and the denominator by 3, yielding $\frac{11 \times 3}{12 \times 3} = \frac{33}{36}$. For $\frac{5}{18}$, we multiply both the numerator and the denominator by 2, giving us $\frac{5 \times 2}{18 \times 2} = \frac{10}{36}$. With the fractions now sharing a common denominator, we can add them: $\frac{33}{36} + \frac{10}{36} = \frac{43}{36}$. This is an improper fraction, which we can convert into a mixed number: $\frac{43}{36} = 1 \frac{7}{36}$. Next, we add the whole numbers: $8 + 9 = 17$. Now, we combine the sum of the whole numbers with the mixed number we obtained from adding the fractions: $17 + 1 \frac{7}{36} = 18 \frac{7}{36}$. Therefore, $8 \frac{11}{12} + 9 \frac{5}{18} = 18 \frac{7}{36}$. This detailed step-by-step approach ensures accuracy and clarity in our solution.

Subtracting mixed numbers introduces another layer of complexity in arithmetic operations. When faced with the subtraction $4 \frac{1}{6} - \frac{8}{9}$, it's imperative to address the challenge methodically. The initial step involves recognizing that we are subtracting a fraction from a mixed number, which necessitates careful handling of the whole number and fractional parts. One of the primary techniques in subtracting mixed numbers is to convert the mixed number into an improper fraction. This transformation allows us to work with a single fractional value, simplifying the subtraction process. Simultaneously, we need to find a common denominator for the fractions involved to ensure accurate subtraction. Once the fractions share a common denominator, we can subtract the numerators. If the resulting fraction is improper, we convert it back into a mixed number. Finally, we simplify the mixed number, if possible, to arrive at our final answer. This comprehensive approach not only leads to the correct solution but also reinforces a deeper understanding of fractional arithmetic. To subtract $4 \frac{1}{6} - \frac{8}{9}$, we first convert the mixed number $4 \frac{1}{6}$ into an improper fraction. To do this, we multiply the whole number (4) by the denominator (6) and add the numerator (1), then place the result over the original denominator: $4 \frac{1}{6} = \frac{(4 \times 6) + 1}{6} = \frac{24 + 1}{6} = \frac{25}{6}$. Now we have the subtraction problem $\frac{25}{6} - \frac{8}{9}$. We need to find a common denominator for the fractions $\frac{25}{6}$ and $\frac{8}{9}$. The least common multiple (LCM) of 6 and 9 is 18. We convert both fractions to have this common denominator. For $\frac{25}{6}$, we multiply both the numerator and the denominator by 3: $\frac{25 \times 3}{6 \times 3} = \frac{75}{18}$. For $\frac{8}{9}$, we multiply both the numerator and the denominator by 2: $\frac{8 \times 2}{9 \times 2} = \frac{16}{18}$. Now we can subtract the fractions: $\frac{75}{18} - \frac{16}{18} = \frac{75 - 16}{18} = \frac{59}{18}$. The resulting fraction $\frac{59}{18}$ is an improper fraction. We convert it to a mixed number by dividing 59 by 18. The quotient is 3, and the remainder is 5. Thus, $\frac{59}{18} = 3 \frac{5}{18}$. Therefore, $4 \frac{1}{6} - \frac{8}{9} = 3 \frac{5}{18}$. This detailed breakdown illustrates the step-by-step process, ensuring clarity and precision in our calculations.

When multiplying mixed numbers, a systematic approach is key to achieving accurate results. The multiplication of $6 \frac{5}{8}$ and $3 \frac{1}{2}$ exemplifies this process. The first critical step is to convert both mixed numbers into improper fractions. This transformation simplifies the multiplication process by allowing us to work with single fractional values rather than mixed numbers. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator, placing the result over the original denominator. Once both mixed numbers are converted into improper fractions, we proceed to multiply the fractions. This involves multiplying the numerators together to obtain the new numerator and multiplying the denominators together to obtain the new denominator. The resulting fraction may be improper, in which case we convert it back into a mixed number. Finally, we simplify the mixed number, if possible, to arrive at our final answer. This step-by-step methodology ensures that we handle mixed number multiplication with confidence and precision. To multiply $6 \frac{5}{8} \cdot 3 \frac{1}{2}$, we first convert both mixed numbers into improper fractions. For $6 \frac{5}{8}$, we multiply the whole number (6) by the denominator (8) and add the numerator (5), then place the result over the original denominator: $6 \frac{5}{8} = \frac{(6 \times 8) + 5}{8} = \frac{48 + 5}{8} = \frac{53}{8}$. For $3 \frac{1}{2}$, we multiply the whole number (3) by the denominator (2) and add the numerator (1), then place the result over the original denominator: $3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}$. Now we multiply the two improper fractions: $\frac{53}{8} \cdot \frac{7}{2}$. We multiply the numerators together: $53 \times 7 = 371$. We multiply the denominators together: $8 \times 2 = 16$. So, the result is $\frac{371}{16}$. The resulting fraction $\frac{371}{16}$ is an improper fraction. We convert it to a mixed number by dividing 371 by 16. The quotient is 23, and the remainder is 3. Thus, $\frac{371}{16} = 23 \frac{3}{16}$. Therefore, $6 \frac{5}{8} \cdot 3 \frac{1}{2} = 23 \frac{3}{16}$. This meticulous approach breaks down the multiplication process into manageable steps, ensuring both accuracy and understanding.

Dividing mixed numbers requires a structured approach to ensure accuracy and clarity. When dividing $5 \frac{2}{3}$ by $2 \frac{5}{6}$, the initial crucial step is to convert both mixed numbers into improper fractions. This conversion simplifies the division process, allowing us to work with fractions directly. To convert a mixed number into an improper fraction, we multiply the whole number by the denominator and add the numerator, placing the result over the original denominator. Once both mixed numbers are in improper fraction form, we can proceed with the division. Dividing fractions involves multiplying by the reciprocal of the divisor. In other words, we flip the second fraction (the divisor) and then multiply. After performing the multiplication, we may end up with an improper fraction, which we then convert back into a mixed number. Finally, we simplify the mixed number, if possible, to arrive at the final answer. This methodical process ensures that mixed number division is handled precisely and effectively. To divide $5 \frac{2}{3} \div 2 \frac{5}{6}$, we first convert both mixed numbers into improper fractions. For $5 \frac{2}{3}$, we multiply the whole number (5) by the denominator (3) and add the numerator (2), then place the result over the original denominator: $5 \frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}$. For $2 \frac{5}{6}$, we multiply the whole number (2) by the denominator (6) and add the numerator (5), then place the result over the original denominator: $2 \frac{5}{6} = \frac{(2 \times 6) + 5}{6} = \frac{12 + 5}{6} = \frac{17}{6}$. Now we have the division problem $\frac{17}{3} \div \frac{17}{6}$. To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of $\frac{17}{6}$ is $\frac{6}{17}$. So, we have: $\frac{17}{3} \times \frac{6}{17}$. We multiply the numerators together: $17 \times 6 = 102$. We multiply the denominators together: $3 \times 17 = 51$. Thus, the result is $\frac{102}{51}$. Now we simplify the fraction. We notice that both 102 and 51 are divisible by 51: $\frac{102}{51} = \frac{102 \div 51}{51 \div 51} = \frac{2}{1} = 2$. Therefore, $5 \frac{2}{3} \div 2 \frac{5}{6} = 2$. This step-by-step explanation clarifies the process of dividing mixed numbers, ensuring a thorough understanding of each operation involved.

In summary, mastering arithmetic operations with mixed numbers involves a clear, methodical approach. Whether adding, subtracting, multiplying, or dividing, converting mixed numbers to improper fractions is a fundamental step. Finding common denominators, simplifying fractions, and converting improper fractions back to mixed numbers are crucial skills. By following these steps carefully, one can confidently tackle mixed number arithmetic problems and achieve accurate results.