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

Introduction

In this article, we will simplify the given expression, which involves fractions, mixed numbers, and division. We will break down the expression into smaller parts, simplify each part, and then combine them to get the final result.

Understanding the Expression

The given expression is:

$\left(2 \frac{1}{4} - 1 \frac{3}{8}\right) \div \frac{3}{10} + \frac{1}{3}$

This expression involves several operations:

1. Subtraction of two mixed numbers
2. Division of a fraction by another fraction
3. Addition of a fraction

Step 1: Convert Mixed Numbers to Improper Fractions

To simplify the expression, we need to convert the mixed numbers to improper fractions. A mixed number is a combination of a whole number and a fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

Let's convert the mixed numbers in the expression:

• $2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$
• $1 \frac{3}{8} = \frac{(1 \times 8) + 3}{8} = \frac{8 + 3}{8} = \frac{11}{8}$

Step 2: Subtract the Two Improper Fractions

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

$\frac{9}{4} - \frac{11}{8}$

To subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 4 and 8 is 8. So, we can rewrite the fractions with a denominator of 8:

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

Now we can subtract the fractions:

$\frac{18}{8} - \frac{11}{8} = \frac{18 - 11}{8} = \frac{7}{8}$

**Step 3: Divide the Result by $\frac{3}{10}$

Now we need to divide the result by $\frac{3}{10}$. To divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction:

$\frac{7}{8} \div \frac{3}{10} = \frac{7}{8} \times \frac{10}{3}$

To multiply fractions, we multiply the numerators and the denominators:

$\frac{7 \times 10}{8 \times 3} = \frac{70}{24}$

Step 4: Simplify the Result

The result is $\frac{70}{24}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 70 and 24 is 2. So, we can simplify the fraction:

$\frac{70}{24} = \frac{70 \div 2}{24 \div 2} = \frac{35}{12}$

**Step 5: Add $\frac{1}{3}$

Finally, we need to add $\frac{1}{3}$ to the result:

$\frac{35}{12} + \frac{1}{3}$

To add fractions, we need to have the same denominator. The least common multiple (LCM) of 12 and 3 is 12. So, we can rewrite the fractions with a denominator of 12:

$\frac{35}{12} = \frac{35}{12}$ (no change) $\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$

Now we can add the fractions:

$\frac{35}{12} + \frac{4}{12} = \frac{35 + 4}{12} = \frac{39}{12}$

Step 6: Simplify the Final Result

The final result is $\frac{39}{12}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 39 and 12 is 3. So, we can simplify the fraction:

$\frac{39}{12} = \frac{39 \div 3}{12 \div 3} = \frac{13}{4}$

Conclusion

In this article, we simplified the given expression by breaking it down into smaller parts, converting mixed numbers to improper fractions, subtracting fractions, dividing fractions, and adding fractions. The final result is $\frac{13}{4}$.

Final Answer

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Introduction

In our previous article, we simplified the given expression, which involved fractions, mixed numbers, and division. We broke down the expression into smaller parts, simplified each part, and then combined them to get the final result. In this article, we will provide a Q&A section to help you understand the process better.

Q&A

Q: What is the first step in simplifying the expression?

A: The first step is to convert the mixed numbers to improper fractions. A mixed number is a combination of a whole number and a fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

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

A: To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. For example, $2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$.

Q: What is the next step after converting the mixed numbers to improper fractions?

A: The next step is to subtract the two improper fractions. To subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 4 and 8 is 8. So, we can rewrite the fractions with a denominator of 8.

Q: How do we subtract fractions with different denominators?

A: To subtract fractions with different denominators, we need to have the same denominator. We can do this by finding the least common multiple (LCM) of the denominators and rewriting the fractions with the LCM as the denominator.

Q: What is the result of subtracting the two improper fractions?

A: The result of subtracting the two improper fractions is $\frac{7}{8}$.

Q: What is the next step after subtracting the two improper fractions?

A: The next step is to divide the result by $\frac{3}{10}$. To divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction.

Q: How do we divide a fraction by another fraction?

A: To divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction. For example, $\frac{7}{8} \div \frac{3}{10} = \frac{7}{8} \times \frac{10}{3}$.

Q: What is the result of dividing the result by $\frac{3}{10}$?

A: The result of dividing the result by $\frac{3}{10}$ is $\frac{70}{24}$.

Q: How do we simplify the result?

A: We can simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 70 and 24 is 2. So, we can simplify the fraction: $\frac{70}{24} = \frac{70 \div 2}{24 \div 2} = \frac{35}{12}$.

Q: What is the final step in simplifying the expression?

A: The final step is to add $\frac{1}{3}$ to the result. To add fractions, we need to have the same denominator. The least common multiple (LCM) of 12 and 3 is 12. So, we can rewrite the fractions with a denominator of 12.

Q: How do we add fractions with different denominators?

A: To add fractions with different denominators, we need to have the same denominator. We can do this by finding the least common multiple (LCM) of the denominators and rewriting the fractions with the LCM as the denominator.

Q: What is the final result of adding $\frac{1}{3}$ to the result?

A: The final result of adding $\frac{1}{3}$ to the result is $\frac{39}{12}$.

Q: How do we simplify the final result?

A: We can simplify the final result by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 39 and 12 is 3. So, we can simplify the fraction: $\frac{39}{12} = \frac{39 \div 3}{12 \div 3} = \frac{13}{4}$.

Conclusion

In this article, we provided a Q&A section to help you understand the process of simplifying the given expression. We covered topics such as converting mixed numbers to improper fractions, subtracting fractions, dividing fractions, and adding fractions. We hope this Q&A section has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is $\boxed{\frac{13}{4}}$.