Solving Complex Fraction Division Problems

Let's dive into the world of fractions and tackle some division problems. This article will guide you through solving two complex mathematical expressions involving fractions. We'll break down each step, making it easy to understand even if you find fractions a bit tricky. So, grab your pen and paper, and let's get started!

1) Calculating $\frac{14}{6} \div [\left(-\frac{18}{3}\right) \div \left(-\frac{9}{27}\right)]$

This problem looks intimidating, but don't worry! We'll solve it step-by-step, following the order of operations (PEMDAS/BODMAS). Remember, this means we handle parentheses (or brackets) first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Step 1: Simplify Inside the Brackets

Our main focus is on simplifying the expression inside the brackets: $\left(-\frac{18}{3}\right) \div \left(-\frac{9}{27}\right)$. This involves dividing one fraction by another. To divide fractions, we multiply by the reciprocal of the second fraction. The reciprocal of a fraction is simply flipping the numerator and the denominator.

So, let's rewrite the expression:

$\left(-\frac{18}{3}\right) \div \left(-\frac{9}{27}\right) = \left(-\frac{18}{3}\right) \times \left(-\frac{27}{9}\right)$

Before we multiply, we can simplify the fractions individually. $\frac{18}{3}$ simplifies to 6, and $\frac{27}{9}$ simplifies to 3. So our expression now looks like this:

$(-6) \times (-3)$

A negative number multiplied by a negative number results in a positive number. Therefore,

$(-6) \times (-3) = 18$

Great! We've simplified the expression inside the brackets to 18.

Step 2: Substitute and Solve the Main Division

Now we can substitute the simplified value back into the original expression:

$\frac{14}{6} \div [\left(-\frac{18}{3}\right) \div \left(-\frac{9}{27}\right)] = \frac{14}{6} \div 18$

Again, dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 18 (which can be written as $\frac{18}{1}$) is $\frac{1}{18}$. So, we rewrite the expression:

$\frac{14}{6} \div 18 = \frac{14}{6} \times \frac{1}{18}$

Now we multiply the fractions. To multiply fractions, we multiply the numerators together and the denominators together:

$\frac{14}{6} \times \frac{1}{18} = \frac{14 \times 1}{6 \times 18} = \frac{14}{108}$

Step 3: Simplify the Resulting Fraction

The fraction $\frac{14}{108}$ can be simplified. Both 14 and 108 are divisible by 2. Dividing both the numerator and denominator by 2, we get:

$\frac{14}{108} = \frac{14 \div 2}{108 \div 2} = \frac{7}{54}$

So, the final answer for the first expression is $\frac{7}{54}$.

2) Calculating $\frac{11}{15} \div \left(-\frac{3}{35}\right)$

Now, let's move on to the second problem: $\frac{11}{15} \div \left(-\frac{3}{35}\right)$. This is a more straightforward division of two fractions.

Step 1: Rewrite as Multiplication by the Reciprocal

As before, we divide fractions by multiplying by the reciprocal of the second fraction. The reciprocal of $-\frac{3}{35}$ is $-\frac{35}{3}$. So, we rewrite the expression:

$\frac{11}{15} \div \left(-\frac{3}{35}\right) = \frac{11}{15} \times \left(-\frac{35}{3}\right)$

Step 2: Multiply the Fractions

Now we multiply the numerators and the denominators:

$\frac{11}{15} \times \left(-\frac{35}{3}\right) = \frac{11 \times (-35)}{15 \times 3} = \frac{-385}{45}$

Notice that the result is negative because we are multiplying a positive fraction by a negative fraction.

Step 3: Simplify the Resulting Fraction

The fraction $\frac{-385}{45}$ can be simplified. Both 385 and 45 are divisible by 5. Dividing both the numerator and denominator by 5, we get:

$\frac{-385}{45} = \frac{-385 \div 5}{45 \div 5} = \frac{-77}{9}$

So, the final answer for the second expression is $-\frac{77}{9}$. We can also express this as a mixed number: $-8\frac{5}{9}$.

Key Concepts in Fraction Division

Dividing Fractions: Dividing by a fraction is the same as multiplying by its reciprocal. This is a fundamental rule in fraction arithmetic.

Reciprocal: The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Simply flip the numerator and denominator.

Simplifying Fractions: Always simplify your final answer to its lowest terms. This makes the fraction easier to understand and work with in future calculations.

Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when solving complex expressions involving multiple operations.

Common Mistakes to Avoid

• Forgetting to Use the Reciprocal: A common mistake is to divide fractions directly without multiplying by the reciprocal. Always flip the second fraction before multiplying.
• Incorrect Simplification: Ensure you are dividing both the numerator and denominator by the same number when simplifying fractions.
• Ignoring the Order of Operations: If there are multiple operations, follow the correct order (PEMDAS/BODMAS) to avoid errors.

Practice Makes Perfect

Fraction division might seem daunting at first, but with practice, it becomes much easier. Try solving more problems like these, and soon you'll be a pro at dividing fractions!

Conclusion

We've successfully solved two complex fraction division problems by breaking them down into manageable steps. Remember the key concepts and common mistakes, and keep practicing. You'll be mastering fraction division in no time!

For further learning and practice, consider exploring resources like Khan Academy's Fraction Division Section. They offer excellent explanations and practice exercises to help you build your skills.