6) Simplify:$\[ \frac{1}{3} \times \frac{1}{5} \text{ Of }\left(\frac{1}{2}+\frac{1}{4}\right) \div \frac{4}{5} \\]1) Arrange \[$\frac{7}{12}, \frac{5}{6}, \frac{3}{4}\$\] In Ascending Order.8) Convert \[$4 \frac{3}{4}\$\]

Introduction

In mathematics, simplifying expressions and converting fractions are essential skills that help us solve problems efficiently. These skills are crucial in various mathematical operations, including multiplication, division, addition, and subtraction. In this article, we will explore how to simplify expressions and convert fractions, focusing on the given problems.

Problem 1: Simplifying an Expression

Simplify the Expression

${ \frac{1}{3} \times \frac{1}{5} \text{ of }\left(\frac{1}{2}+\frac{1}{4}\right) \div \frac{4}{5} }$

To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $\frac{1}{2}+\frac{1}{4}$
2. Multiply $\frac{1}{3}$ and $\frac{1}{5}$
3. Divide the result by $\frac{4}{5}$

Step 1: Evaluate the Expression Inside the Parentheses

$\frac{1}{2}+\frac{1}{4} = \frac{2}{4}+\frac{1}{4} = \frac{3}{4}$

Step 2: Multiply $\frac{1}{3}$ and $\frac{1}{5}$

$\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}$

Step 3: Divide the Result by $\frac{4}{5}$

$\frac{1}{15} \div \frac{4}{5} = \frac{1}{15} \times \frac{5}{4} = \frac{1}{12}$

Therefore, the simplified expression is $\frac{1}{12}$.

Problem 2: Arranging Fractions in Ascending Order

Arrange the Fractions in Ascending Order

$\frac{7}{12}, \frac{5}{6}, \frac{3}{4}$

To arrange these fractions in ascending order, we need to compare their values. We can do this by converting each fraction to a decimal or by finding the least common multiple (LCM) of the denominators.

Method 1: Converting Fractions to Decimals

$\frac{7}{12} = 0.5833$ $\frac{5}{6} = 0.8333$ $\frac{3}{4} = 0.75$

Method 2: Finding the LCM of the Denominators

The LCM of 12, 6, and 4 is 12.

$\frac{7}{12} = \frac{7}{12}$ $\frac{5}{6} = \frac{10}{12}$ $\frac{3}{4} = \frac{9}{12}$

Therefore, the fractions in ascending order are $\frac{7}{12}, \frac{3}{4}, \frac{5}{6}$.

Problem 3: Converting a Mixed Number to an Improper Fraction

Convert the Mixed Number to an Improper Fraction

$4 \frac{3}{4}$

To convert a mixed number to an improper fraction, we need to multiply the whole number by the denominator and add the numerator.

Step 1: Multiply the Whole Number by the Denominator

$4 \times 4 = 16$

Step 2: Add the Numerator

$16 + 3 = 19$

Step 3: Write the Result as an Improper Fraction

$\frac{19}{4}$

Therefore, the mixed number $4 \frac{3}{4}$ is equivalent to the improper fraction $\frac{19}{4}$.

Conclusion

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a fraction?

A: To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. We can then divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, such as $4 \frac{3}{4}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{19}{4}$.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, we need to multiply the whole number by the denominator and add the numerator. We can then write the result as an improper fraction.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the given numbers. We can find the LCM by listing the multiples of each number and finding the smallest number that appears in all the lists.

Q: How do I compare fractions?

A: To compare fractions, we need to find a common denominator. We can then compare the numerators to determine which fraction is larger.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two or more numbers is the largest number that divides each of the given numbers without leaving a remainder. We can find the GCD by listing the factors of each number and finding the largest number that appears in all the lists.

Q: How do I simplify an expression with multiple operations?

A: To simplify an expression with multiple operations, we need to follow the order of operations. We can then simplify the expression by combining like terms and eliminating any unnecessary operations.

Q: What are some common mistakes to avoid when simplifying expressions and converting fractions?

A: Some common mistakes to avoid when simplifying expressions and converting fractions include:

• Not following the order of operations
• Not finding the greatest common divisor (GCD) when simplifying fractions
• Not converting mixed numbers to improper fractions when necessary
• Not finding the least common multiple (LCM) when comparing fractions
• Not combining like terms when simplifying expressions

Q: How can I practice simplifying expressions and converting fractions?

A: You can practice simplifying expressions and converting fractions by working through examples and exercises. You can also use online resources and practice tests to help you prepare for exams and assessments.

Conclusion

In conclusion, simplifying expressions and converting fractions are essential skills in mathematics. By following the order of operations and using various techniques, we can simplify complex expressions and convert mixed numbers to improper fractions. By practicing these skills, we can become proficient in mathematics and solve problems efficiently.