Mastering Mixed Fraction Multiplication Step-by-Step Solutions

When it comes to multiplying fractions, understanding the process thoroughly is essential. This section focuses on the multiplication of a proper fraction by a mixed fraction. Let’s break down the expression ${\frac{2}{3} \times 2\frac{3}{5}}$. The first step in solving this is to convert the mixed fraction into an improper fraction. A mixed fraction consists of a whole number and a proper fraction. To convert it, you multiply the whole number by the denominator of the fraction and then add the numerator. The result becomes the new numerator, and the denominator stays the same. In this case, 2${\frac{3}{5}}$ becomes (2 * 5 + 3) / 5 = 13/5. Now, the expression is ${\frac{2}{3} \times \frac{13}{5}}$. To multiply fractions, you simply multiply the numerators together and the denominators together. So, (2 * 13) / (3 * 5) equals 26/15. The final step is to simplify the fraction if possible. 26/15 is an improper fraction, meaning the numerator is greater than the denominator. We convert it back to a mixed fraction by dividing 26 by 15. The quotient is 1, and the remainder is 11. So, the simplified mixed fraction is 1${\frac{11}{15}}$. Therefore, ${\frac{2}{3} \times 2\frac{3}{5} = 1\frac{11}{15}}$.

Understanding these steps is crucial for mastering fraction multiplication. This process ensures accuracy and builds a solid foundation for more complex mathematical operations involving fractions. The ability to convert between mixed and improper fractions is a key skill, and practicing this will greatly improve your confidence in handling such problems. Remember, the key to success in mathematics is consistent practice and a thorough understanding of the basic concepts.

Moving on to the next example, let's tackle ${\frac{2}{3} \times 1\frac{1}{4}}$. This problem provides another opportunity to practice multiplying a proper fraction by a mixed fraction. As we did before, the initial step is to convert the mixed fraction into an improper fraction. Here, we have 1${\frac{1}{4}}$. To convert this, we multiply the whole number 1 by the denominator 4, which gives us 4, and then add the numerator 1. This results in 5. So, the improper fraction is 5/4. Now, our expression becomes ${\frac{2}{3} \times \frac{5}{4}}$. To multiply these fractions, we multiply the numerators together (2 * 5 = 10) and the denominators together (3 * 4 = 12). This gives us 10/12. The next crucial step is to simplify the fraction. Both 10 and 12 are divisible by 2. Dividing both the numerator and the denominator by 2, we get 5/6. This fraction is in its simplest form because 5 and 6 have no common factors other than 1. Therefore, ${\frac{2}{3} \times 1\frac{1}{4} = \frac{5}{6}}$.

Simplifying fractions is a vital skill in mathematics, ensuring that the answer is expressed in its most concise form. This not only makes the answer easier to understand but also demonstrates a thorough understanding of fractional concepts. Mastering simplification involves recognizing common factors and applying division to both the numerator and denominator until the fraction is irreducible. Regular practice with different examples will help you become proficient in this skill.

Let’s dive into the third problem: ${\frac{1}{2} \times 1\frac{4}{6}}$. Again, we begin by converting the mixed fraction into an improper fraction. We have 1${\frac{4}{6}}$. Multiply the whole number 1 by the denominator 6, which gives us 6, and then add the numerator 4. This results in 10. So, the improper fraction is 10/6. Our expression now looks like ${\frac{1}{2} \times \frac{10}{6}}$. To multiply these fractions, we multiply the numerators (1 * 10 = 10) and the denominators (2 * 6 = 12), resulting in 10/12. Simplifying this fraction is essential. Both 10 and 12 are divisible by 2. Dividing both the numerator and the denominator by 2, we get 5/6. The fraction 5/6 is in its simplest form, as 5 and 6 have no common factors other than 1. Thus, ${\frac{1}{2} \times 1\frac{4}{6} = \frac{5}{6}}$.

Simplifying fractions after multiplication is a critical step in obtaining the correct answer in its simplest form. It demonstrates a deeper understanding of fraction manipulation and ensures accuracy. Practice simplification techniques regularly to enhance your skills and make solving fraction problems more efficient. This step not only provides the correct answer but also showcases mathematical clarity and precision.

Now, let's explore the multiplication of two mixed fractions in the problem: ${1\frac{2}{8} \times 1\frac{2}{5}}$. The first step, as with previous examples, is to convert both mixed fractions into improper fractions. Starting with 1${\frac{2}{8}}$, we multiply the whole number 1 by the denominator 8, which equals 8, and then add the numerator 2, resulting in 10. So, 1${\frac{2}{8}}$ becomes 10/8. Next, we convert 1${\frac{2}{5}}$. We multiply the whole number 1 by the denominator 5, which equals 5, and add the numerator 2, resulting in 7. So, 1${\frac{2}{5}}$ becomes 7/5. Now, our expression is ${\frac{10}{8} \times \frac{7}{5}}$. Multiplying the numerators (10 * 7) gives us 70, and multiplying the denominators (8 * 5) gives us 40. Therefore, we have 70/40. This fraction needs to be simplified. Both 70 and 40 are divisible by 10. Dividing both by 10, we get 7/4. Since 7/4 is an improper fraction, we convert it back into a mixed fraction. Dividing 7 by 4, we get a quotient of 1 and a remainder of 3. Thus, 7/4 is equivalent to 1${\frac{3}{4}}$. Hence, ${1\frac{2}{8} \times 1\frac{2}{5} = 1\frac{3}{4}}$.

Multiplying mixed fractions requires careful conversion to improper fractions and subsequent simplification. Mastering this process ensures accurate solutions and enhances your understanding of fraction manipulation. This skill is crucial for more advanced mathematical concepts and practical applications. Regular practice with different types of fraction problems will solidify your proficiency in this area.

Finally, let's tackle the problem: ${1\frac{1}{4} \times 5\frac{1}{3}}$. This example provides an excellent opportunity to reinforce the steps for multiplying mixed fractions. We begin by converting both mixed fractions into improper fractions. First, we convert 1${\frac{1}{4}}$. Multiply the whole number 1 by the denominator 4, which equals 4, and add the numerator 1, resulting in 5. So, 1${\frac{1}{4}}$ becomes 5/4. Next, we convert 5${\frac{1}{3}}$. Multiply the whole number 5 by the denominator 3, which equals 15, and add the numerator 1, resulting in 16. So, 5${\frac{1}{3}}$ becomes 16/3. Now, our expression is ${\frac{5}{4} \times \frac{16}{3}}$. Multiplying the numerators (5 * 16) gives us 80, and multiplying the denominators (4 * 3) gives us 12. Therefore, we have 80/12. This fraction needs simplification. Both 80 and 12 are divisible by 4. Dividing both by 4, we get 20/3. Since 20/3 is an improper fraction, we convert it back into a mixed fraction. Dividing 20 by 3, we get a quotient of 6 and a remainder of 2. Thus, 20/3 is equivalent to 6${\frac{2}{3}}$. Therefore, ${1\frac{1}{4} \times 5\frac{1}{3} = 6\frac{2}{3}}$.

Consistent practice with converting mixed fractions to improper fractions and simplifying the results is key to mastering fraction multiplication. This process ensures accuracy and builds confidence in handling more complex problems. The ability to efficiently perform these steps is a valuable skill in mathematics and its practical applications.