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Dividing fractions and mixed numbers by whole numbers is a key skill in secondary mathematics, and it’s one that often appears in GCSE exams. The problem “Show that 2 \frac{2}{3} \div 6 = \frac{4}{9}” is a classic example that tests your understanding of mixed numbers, division, and fraction manipulation. In this blog post, we’ll discuss the logical steps needed to solve this type of problem, highlight common mistakes, and share strategies for success.
Understanding the Problem
At first glance, dividing a mixed number by a whole number may seem tricky, but it’s a process that can be broken down into clear, manageable steps. The question asks you to show that dividing 2 \frac{2}{3} by 6 gives the fraction \frac{4}{9}. This means you need to demonstrate, with clear working, how the calculation leads to the given answer.
Step 1: Converting Mixed Numbers to Improper Fractions
The first step is to convert the mixed number 2 \frac{2}{3} into an improper fraction. Mixed numbers are numbers that have a whole part and a fractional part, and they are easier to work with in calculations when written as improper fractions. This conversion is essential, as it allows you to use the standard rules for dividing fractions.
Step 2: Understanding Division of Fractions
When dividing by a whole number, it’s important to remember that dividing by a number is the same as multiplying by its reciprocal. For example, dividing by 6 is the same as multiplying by \frac{1}{6}. This is a crucial concept, as it transforms the division problem into a multiplication problem, which is often easier to handle.
Step 3: Multiplying Fractions
Once you have rewritten the division as multiplication by the reciprocal, you multiply the numerators together and the denominators together. This is a straightforward process, but it’s important to ensure that all numbers are in the correct form before multiplying.
Step 4: Simplifying the Fraction
After multiplying, you may end up with a fraction that can be simplified. Always check if the numerator and denominator have any common factors that can be cancelled. Simplifying your answer is important, as exam questions often require answers in their simplest form.
Step 5: Comparing with the Given Answer
The question asks you to “show that” the result is \frac{4}{9}. This means your final answer should match the given fraction exactly. If it doesn’t, go back and check your working for errors in conversion, multiplication, or simplification.
Common Mistakes to Avoid
- Not converting the mixed number: Trying to divide a mixed number directly by a whole number without converting it to an improper fraction can lead to confusion and incorrect answers.
- Dividing numerators and denominators separately: Some students mistakenly divide the numerator by the whole number and leave the denominator unchanged, which is incorrect.
- Forgetting to multiply by the reciprocal: Division by a whole number must be rewritten as multiplication by its reciprocal.
- Not simplifying the final fraction: Always check if your answer can be reduced to its simplest form.
- Calculator misuse: While calculators can help, entering mixed numbers or fractions incorrectly can lead to mistakes. Understanding the process is key.
Why This Skill Matters
Dividing fractions and mixed numbers is a skill that appears in many areas of mathematics, from basic arithmetic to algebra and beyond. It’s also useful in everyday life, such as when sharing quantities or working with measurements. If you find this topic challenging, working with a GCSE Maths Tutor for Higher can help you build confidence and accuracy.
The Final Answer
By carefully converting the mixed number to an improper fraction, rewriting the division as multiplication by the reciprocal, and simplifying your result, you can show that
2 \frac{2}{3} \div 6 = \frac{4}{9}
This process not only helps you solve this specific problem but also builds a strong foundation for tackling similar questions in your exams and beyond.
