Simplifying Complex Fractions And Expressions A Step-by-Step Guide

In the realm of mathematics, the ability to simplify complex fractions and expressions is a fundamental skill. These expressions, often appearing daunting at first glance, can be systematically reduced to their simplest forms using a combination of arithmetic operations and algebraic manipulations. This article serves as a comprehensive guide to mastering the art of simplification, providing step-by-step explanations, illustrative examples, and practical tips to enhance your understanding and proficiency. Mastering the simplification of complex fractions and expressions is crucial for success in various mathematical disciplines, including algebra, calculus, and beyond.

(a) Simplifying a Complex Fraction

Our first task involves simplifying the following expression:

$$\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}\right) \div\left(\frac{2}{5}-\frac{5}{9}+\frac{3}{5}-\frac{7}{18}\right)$$

To tackle this, we'll break it down into manageable steps. Understanding the order of operations (PEMDAS/BODMAS) is paramount here, ensuring we perform calculations in the correct sequence.

Step 1: Simplify the Numerator

The numerator of our complex fraction is a sum and difference of fractions: $\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}$. To combine these fractions, we need to find the least common multiple (LCM) of the denominators (3, 4, 5, and 6). The LCM of these numbers is 60. Now, we rewrite each fraction with a denominator of 60:

• $\frac{1}{3} = \frac{1 imes 20}{3 imes 20} = \frac{20}{60}$
• $\frac{1}{4} = \frac{1 imes 15}{4 imes 15} = \frac{15}{60}$
• $\frac{1}{5} = \frac{1 imes 12}{5 imes 12} = \frac{12}{60}$
• $\frac{1}{6} = \frac{1 imes 10}{6 imes 10} = \frac{10}{60}$

Now we can substitute these equivalent fractions back into the numerator expression:

$\frac{20}{60} - \frac{15}{60} + \frac{12}{60} - \frac{10}{60}$

Combining the numerators, we get:

$\frac{20 - 15 + 12 - 10}{60} = \frac{7}{60}$

Therefore, the simplified numerator is $\frac{7}{60}$. Finding the LCM is a crucial step in adding or subtracting fractions, ensuring we can combine them accurately.

Step 2: Simplify the Denominator

Next, we focus on the denominator: $\frac{2}{5}-\frac{5}{9}+\frac{3}{5}-\frac{7}{18}$. Similar to the numerator, we need to find the LCM of the denominators (5, 9, and 18). The LCM of these numbers is 90. We rewrite each fraction with a denominator of 90:

• $\frac{2}{5} = \frac{2 imes 18}{5 imes 18} = \frac{36}{90}$
• $\frac{5}{9} = \frac{5 imes 10}{9 imes 10} = \frac{50}{90}$
• $\frac{3}{5} = \frac{3 imes 18}{5 imes 18} = \frac{54}{90}$
• $\frac{7}{18} = \frac{7 imes 5}{18 imes 5} = \frac{35}{90}$

Substituting these equivalent fractions into the denominator expression:

$\frac{36}{90} - \frac{50}{90} + \frac{54}{90} - \frac{35}{90}$

Combining the numerators, we get:

$\frac{36 - 50 + 54 - 35}{90} = \frac{5}{90}$

This fraction can be further simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 5:

$\frac{5}{90} = \frac{5 ext{ ÷ } 5}{90 ext{ ÷ } 5} = \frac{1}{18}$

So, the simplified denominator is $\frac{1}{18}$. Simplifying fractions after combining them is essential to obtain the most reduced form.

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now we have the simplified numerator (rac{7}{60}) and the simplified denominator (rac{1}{18}). Dividing fractions is the same as multiplying by the reciprocal of the divisor:

$\frac{7}{60} \div \frac{1}{18} = \frac{7}{60} imes \frac{18}{1}$

Before multiplying, we can simplify by canceling common factors. Both 60 and 18 are divisible by 6:

$\frac{7}{60} imes \frac{18}{1} = \frac{7}{10 imes 6} imes \frac{3 imes 6}{1} = \frac{7}{10} imes \frac{3}{1}$

Now, multiply the numerators and the denominators:

$\frac{7 imes 3}{10 imes 1} = \frac{21}{10}$

Therefore, the simplified form of the original expression is $\frac{21}{10}$. Remember to simplify before multiplying whenever possible to avoid dealing with large numbers.

(b) Simplifying an Expression with Multiplication and Division

Our second task is to simplify the following expression:

$$\left(\frac{13}{21} \div \frac{29}{42}\right) imes \frac{3}{-5}$$

Again, we'll proceed step-by-step, adhering to the order of operations.

Step 1: Perform the Division within the Parentheses

We begin by dividing $\frac{13}{21}$ by $\frac{29}{42}$. Dividing by a fraction is equivalent to multiplying by its reciprocal:

$\frac{13}{21} \div \frac{29}{42} = \frac{13}{21} imes \frac{42}{29}$

Before multiplying, we look for opportunities to simplify. 21 and 42 share a common factor of 21:

$\frac{13}{21} imes \frac{42}{29} = \frac{13}{1} imes \frac{2}{29}$

Now we multiply the numerators and the denominators:

$\frac{13 imes 2}{1 imes 29} = \frac{26}{29}$

So, the simplified result of the division is $\frac{26}{29}$. Recognizing and canceling common factors simplifies calculations significantly.

Step 2: Multiply the Result by the Remaining Fraction

Now we multiply $\frac{26}{29}$ by $\frac{3}{-5}$:

$\frac{26}{29} imes \frac{3}{-5} = \frac{26 imes 3}{29 imes -5}$

There are no common factors to cancel in this case, so we proceed with multiplying:

$\frac{26 imes 3}{29 imes -5} = \frac{78}{-145}$

We can rewrite this fraction with the negative sign in the numerator or in front of the entire fraction:

$\frac{78}{-145} = -\frac{78}{145}$

Therefore, the simplified form of the expression is $-\frac{78}{145}$. Maintaining the correct sign is crucial when dealing with negative numbers.

Key Concepts and Strategies

• Order of Operations (PEMDAS/BODMAS): Always follow the order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to ensure correct simplification.
• Least Common Multiple (LCM): Find the LCM of the denominators when adding or subtracting fractions to create equivalent fractions with a common denominator.
• Greatest Common Divisor (GCD): Simplify fractions by dividing both the numerator and denominator by their GCD.
• Reciprocal for Division: Dividing by a fraction is the same as multiplying by its reciprocal.
• Simplifying Before Multiplying: Look for common factors to cancel before multiplying fractions to simplify calculations.
• Sign Rules: Remember the rules for multiplying and dividing positive and negative numbers.

Common Mistakes to Avoid

• Forgetting the Order of Operations: Always adhere to PEMDAS/BODMAS to avoid incorrect results.
• Incorrectly Finding the LCM: A wrong LCM will lead to incorrect equivalent fractions and an incorrect answer.
• Failing to Simplify Fractions: Always simplify fractions to their lowest terms for the most accurate answer.
• Sign Errors: Pay close attention to signs when multiplying and dividing negative numbers.
• Incorrectly Inverting Fractions for Division: Remember to invert the second fraction when dividing fractions.

Practice Problems

To solidify your understanding, try simplifying the following expressions:

1. $$\left(\frac{2}{3} + \frac{1}{4}\right) \div \left(\frac{5}{6} - \frac{1}{2}\right)$$

2. $$\left(\frac{7}{10} \div \frac{14}{15}\right) imes \frac{2}{3}$$

3. $$\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4} - \frac{1}{6}}$$

Conclusion

Simplifying complex fractions and expressions is a vital skill in mathematics. By understanding the fundamental concepts, employing the correct strategies, and practicing regularly, you can master this skill and confidently tackle even the most challenging problems. Consistent practice and a methodical approach are key to success in simplifying mathematical expressions. Remember to break down complex problems into smaller, manageable steps, and always double-check your work to ensure accuracy. With dedication and effort, you can unlock the power of simplification and excel in your mathematical endeavors. The ability to simplify complex mathematical expressions is not just a skill; it's a gateway to deeper mathematical understanding and problem-solving prowess.