Calculate The Following Expression:$\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{8}{10} + \frac{1}{3}$

Introduction

Mathematical expressions can be complex and challenging to solve, especially when they involve fractions, decimals, and mixed numbers. In this article, we will focus on calculating the expression $\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{8}{10} + \frac{1}{3}$, which requires a deep understanding of mathematical operations and techniques.

Understanding the Expression

Before we dive into solving the expression, let's break it down and understand what it means. The expression consists of three main parts:

1. Subtraction: We need to subtract $1 \frac{3}{8}$ from $2 \frac{1}{4}$.
2. Division: We need to divide the result of the subtraction by $\frac{8}{10}$.
3. Addition: We need to add $\frac{1}{3}$ to the result of the division.

Step 1: Convert Mixed Numbers to Improper Fractions

To simplify the expression, we need to convert the mixed numbers to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

• $2 \frac{1}{4}$ can be converted to an improper fraction as follows: $\frac{(2 \times 4) + 1}{4} = \frac{9}{4}$
• $1 \frac{3}{8}$ can be converted to an improper fraction as follows: $\frac{(1 \times 8) + 3}{8} = \frac{11}{8}$

Step 2: Subtract the Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can subtract them.

$\frac{9}{4} - \frac{11}{8} = \frac{9 \times 2}{4 \times 2} - \frac{11}{8} = \frac{18}{8} - \frac{11}{8} = \frac{7}{8}$

Step 3: Convert the Division to a Fraction

To simplify the expression, we need to convert the division to a fraction. We can do this by inverting the divisor and changing the division sign to a multiplication sign.

$\frac{7}{8} \div \frac{8}{10} = \frac{7}{8} \times \frac{10}{8}$

Step 4: Multiply the Fractions

Now that we have converted the division to a fraction, we can multiply the fractions.

$\frac{7}{8} \times \frac{10}{8} = \frac{7 \times 10}{8 \times 8} = \frac{70}{64}$

Step 5: Simplify the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator.

The GCD of 70 and 64 is 2.

$\frac{70}{64} = \frac{70 \div 2}{64 \div 2} = \frac{35}{32}$

Step 6: Add the Fraction

Finally, we need to add $\frac{1}{3}$ to the result of the division.

$\frac{35}{32} + \frac{1}{3} = \frac{35 \times 3}{32 \times 3} + \frac{1 \times 32}{3 \times 32} = \frac{105}{96} + \frac{32}{96}$

Step 7: Add the Fractions

To add the fractions, we need to find a common denominator, which is 96.

$\frac{105}{96} + \frac{32}{96} = \frac{105 + 32}{96} = \frac{137}{96}$

Conclusion

In this article, we have solved the complex mathematical expression $\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{8}{10} + \frac{1}{3}$ using a step-by-step approach. We have converted mixed numbers to improper fractions, subtracted the fractions, converted the division to a fraction, multiplied the fractions, simplified the fraction, and finally added the fractions. The final result is $\frac{137}{96}$.

Final Answer

$$

Q: What is the first step in solving a complex mathematical expression?

A: The first step in solving a complex mathematical expression is to break it down into smaller parts and understand what it means. This involves identifying the different operations involved, such as addition, subtraction, multiplication, and division, and determining the order in which they should be performed.

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

A: To convert a mixed number to an improper fraction, you need to multiply the whole number part by the denominator and add the numerator. For example, to convert $2 \frac{1}{4}$ to an improper fraction, you would multiply 2 by 4 and add 1, resulting in $\frac{9}{4}$.

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

A: A proper fraction is a fraction where the numerator is less than the denominator, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, $\frac{1}{2}$ is a proper fraction, while $\frac{3}{2}$ is an improper fraction.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, you need to find a common denominator and then subtract the numerators. For example, to subtract $\frac{1}{2}$ from $\frac{3}{4}$, you would find a common denominator of 4 and then subtract the numerators, resulting in $\frac{2}{4}$.

Q: What is the rule for dividing fractions?

A: The rule for dividing fractions is to invert the divisor and change the division sign to a multiplication sign. For example, to divide $\frac{1}{2}$ by $\frac{3}{4}$, you would invert the divisor and change the division sign to a multiplication sign, resulting in $\frac{1}{2} \times \frac{4}{3}$.

Q: How do I multiply fractions?

A: To multiply fractions, you simply multiply the numerators and denominators separately. For example, to multiply $\frac{1}{2}$ by $\frac{3}{4}$, you would multiply the numerators and denominators separately, resulting in $\frac{3}{8}$.

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

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD. For example, to simplify $\frac{12}{18}$, you would find the GCD of 12 and 18, which is 6, and then divide both numbers by 6, resulting in $\frac{2}{3}$.

Q: What is the final answer to the expression $\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{8}{10} + \frac{1}{3}$?

A: The final answer to the expression $\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{8}{10} + \frac{1}{3}$ is $\frac{137}{96}$.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving complex mathematical expressions. We have covered topics such as converting mixed numbers to improper fractions, subtracting fractions with different denominators, dividing fractions, multiplying fractions, finding the greatest common divisor, and simplifying fractions. We hope that this article has been helpful in providing a better understanding of these concepts and how to apply them to solve complex mathematical expressions.