Fraction, Decimal, & Percent Calculations

1. The Three Faces of a Number: F, D, and P

Fractions, decimals, and percentages (FDP) are simply different ways to represent the same value. Being able to switch between them effortlessly is key to solving many problems efficiently. Often, the easiest way to compare or calculate with a mix of FDP is to convert them all to a single format.

Let's start by watching a video that walks through the essential conversion methods.

Summary of Key Conversions

Based on the video and standard rules, here is a summary of the conversions you must know.

Converting a Fraction to a Percentage: Method Selection

As you saw in the video, converting a fraction to a percentage has a couple of methods. Choosing the right one can save you time.

• Method 1 (Denominator is a factor of 100): If the denominator is 2, 4, 5, 10, 20, 25, or 50, it's fastest to create an equivalent fraction with a denominator of 100. The numerator is then the percentage.
  • Example: $\frac{13}{20}=\frac{13\times 5}{20\times 5}=\frac{65}{100}=65\%$
• Method 2 (Universal Method): Multiply the fraction by 100. This always works.
  • Example: $\frac{3}{8}\to \frac{3}{8}\times 100=\frac{300}{8}=37.5\%$

2. The Special Case: Converting Recurring Decimals to Fractions

Some fractions, like $\frac{1}{3}$, produce recurring decimals (0.333...). For your IGCSE Extended paper, you need to be able to do the reverse: convert a recurring decimal back into a fraction. This is a classic "show that" question where you must demonstrate the correct algebraic method.

This process involves a clever algebraic trick to eliminate the infinitely repeating part of the decimal.

Let's summarise the method for $0.1\dot{5}$ (which means 0.1555...):

1. Set up the equation: Let $x=\mathrm{0.1555...}$
2. Shift the decimal: We need to create two equations where the part after the decimal point is identical.
   • Multiply by 10: $10x=\mathrm{1.555...}$
   • Multiply by 100: $100x=\mathrm{15.555...}$
3. Subtract: Now, $100x$ and $10x$ both have .555... as their decimal part. Subtracting them will cancel this out. $$\begin{array}{rl}100x& =\mathrm{15.555...}\\ -10x& =\mathrm{1.555...}\\ 90x& =14\end{array}$$
4. Solve for x: $$x=\frac{14}{90}=\frac{7}{45}$$

The key is to multiply by powers of 10 until you find two results that have the exact same recurring decimal part, then subtract them.

3. Fluency with Fraction Calculations

For many IGCSE questions, especially on the non-calculator paper or when method marks are at stake, you must perform fraction arithmetic accurately and show your steps.

Key Rules for Fraction Arithmetic:

• Mixed Numbers: For multiplying and dividing, always convert mixed numbers to improper ("top-heavy") fractions first. For adding and subtracting, this is the safest method to avoid errors.
  • $2\frac{3}{4}=\frac{(2\times 4)+3}{4}=\frac{11}{4}$
• Addition and Subtraction: You must find a common denominator.
  • $\frac{2}{3}+\frac{1}{5}=\frac{10}{15}+\frac{3}{15}=\frac{13}{15}$
• Multiplication: Multiply the numerators and multiply the denominators. Crucial tip: Cancel any common factors between a numerator and a denominator before you multiply. This keeps the numbers small and simplifies the calculation.
  • $\frac{4}{9}\times \frac{3}{8}=\frac{{\cancel{4}}^1}{{\cancel{9}}^3}\times \frac{{\cancel{3}}^1}{{\cancel{8}}^2}=\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$
• Division: Keep the first fraction, Flip the second (find its reciprocal), and Change the operation to multiplication.
  • $\frac{2}{5}÷\frac{3}{4}=\frac{2}{5}\times \frac{4}{3}=\frac{8}{15}$

4. Percentage Calculations in Context

Percentages are used everywhere, from calculating discounts to analysing statistics. Being fast and accurate with them is essential. The most efficient way to handle percentage calculations is by using decimal multipliers.

The Multiplier Method:

This is the fastest and most reliable method for most percentage problems.

• To find a percentage of an amount: Convert the percentage to a decimal and multiply.
  • Find 23% of 150: $0.23\times 150=34.5$
• To increase by a percentage: Add the percentage to 100%, convert to a decimal, and multiply.
  • Increase 80 by 15%: $100\%+15\%=115\%\to 1.15\times 80=92$
• To decrease by a percentage: Subtract the percentage from 100%, convert to a decimal, and multiply.
  • Decrease 120 by 30%: $100\%-30\%=70\%\to 0.70\times 120=84$
• Reverse Percentages (Finding the original): This is a classic exam question. Do not simply reverse the operation (e.g., add back the 20% that was taken off). You must divide by the multiplier.
  • A price is £60 after a 20% discount. Find the original price.
  • A 20% discount means the price is $100\%-20\%=80\%$ of the original.
  • Original Price $\times$ 0.80 = £60
  • Original Price = £60 $÷$ 0.80 = £75

Key Takeaways

• Interchangeability is Key: Fractions, decimals, and percentages are different languages for the same number. To solve problems, convert them to a common format—often decimals or percentages.
• Master the Methods: Memorise the rules for each conversion and calculation. For fraction arithmetic, showing your working is non-negotiable for securing method marks.
• Use Multipliers for Percentages: For increase, decrease, and reverse percentage problems, using decimal multipliers is faster and less error-prone than two-step methods.
• Recurring Decimals Require Algebra: Remember the "let x equal..." method. This is a "show that" skill where the process is more important than the final answer.

In our next lesson, we will continue building your core number skills by tackling Standard Form. This is another way of representing numbers that is essential for handling very large or very small quantities in science-related questions.