B) Work Out $4 \frac{1}{2} \div \frac{3}{5}$.Give Your Answer As A Mixed Number In Its Simplest Form.

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**Work out $4 \frac{1}{2} \div \frac{3}{5}$: A Step-by-Step Guide** ===========================================================

What is the Problem?

The problem requires us to divide a mixed number, $4 \frac{1}{2}$, by a fraction, $\frac{3}{5}$. We need to find the quotient in its simplest form.

Why is it Important?

Understanding how to divide mixed numbers by fractions is a crucial skill in mathematics, particularly in algebra and geometry. It helps us to solve real-world problems that involve proportions, ratios, and rates.

Step 1: Convert the Mixed Number to an Improper Fraction

To divide a mixed number by a fraction, we need to convert the mixed number to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

412=(4×2)+12=924 \frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{9}{2}

Step 2: Invert the Fraction and Multiply

To divide a fraction by another fraction, we need to invert the second fraction (i.e., flip the numerator and denominator) and then multiply.

92÷35=92×53\frac{9}{2} \div \frac{3}{5} = \frac{9}{2} \times \frac{5}{3}

Step 3: Multiply the Numerators and Denominators

Now, we multiply the numerators and denominators separately.

9×52×3=456\frac{9 \times 5}{2 \times 3} = \frac{45}{6}

Step 4: Simplify the Fraction

We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

456=45÷36÷3=152\frac{45}{6} = \frac{45 \div 3}{6 \div 3} = \frac{15}{2}

Step 5: Convert the Improper Fraction to a Mixed Number

Finally, we convert the improper fraction to a mixed number.

152=712\frac{15}{2} = 7 \frac{1}{2}

Answer

The final answer is $7 \frac{1}{2}$.

Frequently Asked Questions

Q: What is the difference between dividing a mixed number by a fraction and dividing a fraction by a fraction?

A: When dividing a mixed number by a fraction, we need to convert the mixed number to an improper fraction first. When dividing a fraction by a fraction, we can simply invert the second fraction and multiply.

Q: How do I know if a fraction is in its simplest form?

A: A fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

Q: Can I use a calculator to solve this problem?

A: Yes, you can use a calculator to solve this problem. However, it's always a good idea to understand the underlying math concepts and be able to solve problems manually.

Q: What if the problem involves decimals or percentages?

A: If the problem involves decimals or percentages, you can convert them to fractions and then follow the same steps as before.

Q: Can I use this method to solve problems involving negative numbers?

A: Yes, you can use this method to solve problems involving negative numbers. However, you need to be careful when multiplying and dividing negative numbers.

Conclusion

Dividing a mixed number by a fraction requires us to convert the mixed number to an improper fraction, invert the second fraction, and then multiply. By following these steps, we can find the quotient in its simplest form. Remember to always simplify the fraction and convert it to a mixed number if necessary. With practice and patience, you'll become proficient in solving these types of problems.