Hello, Future Mathematician! Welcome to Number Sense Central!

This chapter is the absolute foundation of mathematics. Fractions, decimals, and percentages (FDP) are just different ways of representing parts of a whole—like three different languages spoken by the same number. Mastering these conversions and calculations will not only boost your exam scores but also help you manage money, understand statistics, and navigate the real world!

Don't worry if fractions felt tricky before. We will break down every concept step-by-step. You've got this!

1. Understanding the Three Forms (C1.4 / E1.4)

Fractions: Parts of a Whole

A fraction represents a division or a portion of a whole quantity.

  • Numerator: The top number (how many parts you have).
  • Denominator: The bottom number (how many parts make up the whole).

We need to know three key types of fractions:

1. Proper Fractions: The numerator is smaller than the denominator. (e.g., \(\frac{3}{4}\)). The value is less than 1.

2. Improper Fractions: The numerator is equal to or larger than the denominator. (e.g., \(\frac{7}{4}\)). The value is 1 or greater.

3. Mixed Numbers: A whole number combined with a proper fraction. (e.g., \(1 \frac{3}{4}\)).

Key Task: Converting between Improper Fractions and Mixed Numbers

Improper to Mixed: Divide the numerator by the denominator.

Example: Convert \(\frac{11}{3}\).

  • 11 divided by 3 is 3, with a remainder of 2.
  • The whole number is 3. The remainder (2) becomes the new numerator.
  • Result: \(3 \frac{2}{3}\).

Mixed to Improper: Multiply the whole number by the denominator, then add the numerator.

Example: Convert \(2 \frac{1}{5}\).

  • \((2 \times 5) + 1 = 11\).
  • Result: \(\frac{11}{5}\).

Decimals: The Power of Ten

Decimals are fractions written using a base-ten system. The position of each digit tells you its value (tenths, hundredths, thousandths, etc.).

Example: 0.25 is read as "twenty-five hundredths," which is the fraction \(\frac{25}{100}\).

Percentages: Out of 100

A percentage (symbol \%) means 'out of 100'. It's simply a fraction where the denominator is fixed at 100.

Did you know? The word "percent" comes from the Latin phrase per centum, meaning "by the hundred."

Key Takeaway 1: Equivalence

All three forms—Fraction, Decimal, Percentage—represent the same value. You must be fluent in converting between them.

2. Recognizing Equivalence and Conversions (C1.4 / E1.4)

You must be able to convert a quantity from one form to another easily.

A. Fraction to Decimal (F → D)

Rule: Divide the numerator by the denominator.

$$\frac{3}{8} = 3 \div 8 = 0.375$$

Note: Some fractions, like \(\frac{1}{3}\), result in recurring decimals (0.333...).

B. Decimal to Percentage (D → P)

Rule: Multiply the decimal by 100.

Trick: Move the decimal point two places to the right and add the % symbol.

$$0.42 \times 100 = 42\%$$

$$1.5 \times 100 = 150\%$$

C. Percentage to Fraction (P → F)

Rule: Write the percentage over 100 and simplify fully.

$$75\% = \frac{75}{100}$$

Simplify (divide numerator and denominator by 25):

$$\frac{75 \div 25}{100 \div 25} = \frac{3}{4}$$

D. Percentage to Decimal (P → D)

Rule: Divide the percentage by 100.

Trick: Move the decimal point two places to the left and remove the % symbol.

$$6.5\% = 6.5 \div 100 = 0.065$$

Quick Review Box: The Conversion Triangle

F → D: Divide
D → P: \(\times 100\)
P → F: Put over 100, Simplify

3. The Four Operations with FDP (C1.6 / E1.6)

Remember, always follow the order of operations (BODMAS/BIDMAS) for complex calculations.

A. Working with Fractions

1. Addition and Subtraction

To add or subtract fractions, the denominators must be the same (a Common Denominator).

Step 1: Find the Lowest Common Multiple (LCM) of the denominators.

Step 2: Convert both fractions to equivalent fractions using the LCM as the new denominator.

Step 3: Add or subtract the numerators, keeping the denominator the same.

Example: Work out \(\frac{1}{4} + \frac{2}{3}\).

  • LCM of 4 and 3 is 12.
  • Conversion: \(\frac{1}{4} = \frac{3}{12}\) and \(\frac{2}{3} = \frac{8}{12}\).
  • Calculation: \(\frac{3}{12} + \frac{8}{12} = \frac{11}{12}\).
2. Multiplication

Rule: Multiply the numerators together and multiply the denominators together.

Step 1: Multiply the tops (numerators).

Step 2: Multiply the bottoms (denominators).

Step 3: Simplify the resulting fraction.

Example: Work out \(\frac{2}{5} \times \frac{3}{4}\).

$$\frac{2 \times 3}{5 \times 4} = \frac{6}{20}$$

Simplify: \(\frac{6}{20} = \frac{3}{10}\).

3. Division

Rule: Keep the first fraction, change the operation to multiplication, and flip the second fraction (Reciprocal).

Memory Aid: K.C.F. (Keep, Change, Flip)

Example: Work out \(\frac{1}{2} \div \frac{3}{5}\).

  • Keep \(\frac{1}{2}\).
  • Change \(\div\) to \(\times\).
  • Flip \(\frac{3}{5}\) to \(\frac{5}{3}\).
  • Calculation: \(\frac{1}{2} \times \frac{5}{3} = \frac{5}{6}\).

B. Working with Decimals

When dealing with decimals in calculations (especially in non-calculator papers), alignment is key.

  • Addition/Subtraction: Always line up the decimal points. Adding zeros at the end (place holders) can help avoid mistakes.
  • Multiplication: Ignore the decimals, multiply the numbers, then count the total number of decimal places in the original numbers and put the decimal back in the answer.
  • Division: If dividing by a decimal, shift the decimal point in the divisor and the dividend until the divisor is a whole number.

Common Mistake to Avoid: When converting a fraction to a decimal, ensure you correctly handle rounding. If a question requires 3 significant figures (3 s.f.), do not round until the *final* answer.

Key Takeaway 2: Fraction Operations

Addition/Subtraction requires common denominators. Multiplication is straightforward. Division uses the KCF rule.

4. Core Percentage Calculations (C1.12 / E1.12)

Percentage problems usually fall into two main categories:

A. Calculating a Given Percentage of a Quantity

This is often referred to as finding the 'part' when you know the 'whole' and the 'percent'.

Example: Find 30% of $600.

Method 1: Using Decimals (Recommended for Calculator use)

Convert the percentage to a decimal and multiply.

$$30\% = 0.30$$

$$0.30 \times 600 = 180$$

Result: $180

Method 2: Using Fractions/Mental Math (Good for Non-Calculator)

Use simple base percentages (10%, 1%, 50%, etc.).

  • \(10\% \text{ of } 600 = 60\).
  • \(30\% = 3 \times 10\%\).
  • \(3 \times 60 = 180\).

B. Expressing One Quantity as a Percentage of Another

This is asking: "What percentage is A of B?"

Rule: Write it as a fraction, then multiply by 100.

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$$

Example: A student scores 45 marks out of a possible 60. What is their percentage score?

$$\frac{45}{60} \times 100\%$$

Simplify the fraction first: \(\frac{45}{60} = \frac{3}{4}\).

$$\frac{3}{4} \times 100\% = 75\%$$

C. Calculating Percentage Increase or Decrease (C1.12.3)

A percentage change tells you how much a value has gone up or down relative to its original value.

Method 1: Two-Step Calculation (Longer but safer)

1. Calculate the actual increase/decrease amount.

2. Add or subtract this amount from the original value.

Method 2: The Multiplier Method (Faster and essential)

A multiplier is a single decimal number you multiply by the original amount to get the final amount.

  • Increase: Start with 100%. Add the percentage. Convert to decimal. (e.g., 20% increase: \(100\% + 20\% = 120\% = 1.2\))
  • Decrease (Discount): Start with 100%. Subtract the percentage. Convert to decimal. (e.g., 15% decrease: \(100\% - 15\% = 85\% = 0.85\))

Example: Increase $450 by 12%.

  • New percentage: \(100\% + 12\% = 112\%\).
  • Multiplier: 1.12.
  • Final calculation: \(450 \times 1.12 = 504\).
  • Result: $504
Key Takeaway 3: Multipliers

Always use the multiplier method for percentage increase/decrease, especially when dealing with money or repeated changes. It is much faster and less prone to rounding errors.

5. Financial Mathematics: Simple and Compound Interest (C1.12.4)

The syllabus requires you to calculate with simple and compound interest. Note that formulas are not given, so you must understand the concepts and processes.

A. Simple Interest (SI)

Simple interest is calculated only on the original amount of money invested or borrowed (the Principal, P). The interest earned each year is the same.

  • P = Principal amount (the starting money)
  • R = Rate (usually the interest percentage per year, as a decimal)
  • T = Time (in years)

The formula usually used is: Total Interest \(= P \times R \times T\). (You calculate the interest, then add it to the principal at the end).

Example: Calculate the simple interest on $5000 at 4% per year for 3 years.

  • Rate R = 0.04.
  • Interest per year: \(5000 \times 0.04 = 200\).
  • Total interest: \(200 \times 3 = 600\).

B. Compound Interest (CI)

Compound interest is interest calculated on the initial principal and also on all the accumulated interest from previous periods.

Analogy: Simple interest is like earning money only on your initial seed money. Compound interest is like earning money on your initial seed *and* on the crops (interest) you grew last year.

We use the multiplier method here, repeated for the number of years.

Compounding Multiplier: \((1 + \frac{Interest Rate}{100})\)

Total Amount \(A\) = Principal \(P\) \(\times\) (Multiplier)Time \(n\)

$$A = P (1 + R)^n$$

Example: Calculate the final amount if $5000 is invested at 4% compound interest for 3 years.

  • Multiplier R = 1.04.
  • Final Amount: \(5000 \times (1.04)^3\).
  • \(5000 \times 1.124864 = 5624.32\).
  • Result: $5624.32 (Total interest earned is \(5624.32 - 5000 = 624.32\)).

6. Extended Content: Reverse Percentages (E1.12.5)

This is a crucial skill only required for Extended students, and it often confuses people! Reverse percentages are used when you are given the final amount after a percentage change and need to find the original starting amount.

The Golden Rule of Reverse Percentages:

$$\text{Original Value} = \frac{\text{Final Value}}{\text{Multiplier}}$$

Step 1: Determine what percentage the Final Value represents (the $100\% + X\%$).

Step 2: Find the decimal multiplier.

Step 3: Divide the Final Value by the Multiplier.

Example 1 (Increase): A shop sells a jacket for $180 after applying a 20% profit margin. What was the original cost price?

  • The final price ($180) represents \(100\% + 20\% = 120\%\).
  • Multiplier: 1.20.
  • Original Cost: \(\frac{180}{1.20} = 150\).
  • Result: The original cost was $150.

Example 2 (Decrease): After a 15% discount, a washing machine costs $765. What was the original price?

  • The final price ($765) represents \(100\% - 15\% = 85\%\).
  • Multiplier: 0.85.
  • Original Cost: \(\frac{765}{0.85} = 900\).
  • Result: The original price was $900.

⚠ Common Mistake in Reverse Percentages ⚠

Do NOT try to find the percentage of the final price. For example, if $180 is 20% more than the original price, the original price is NOT \(180 - (180 \times 0.20)\). This is incorrect! Always divide by the multiplier!

Final Key Takeaway: FDP Summary

Fractions, Decimals, and Percentages are interchangeable representations of value. Conversions allow us to use the easiest form for a given calculation (e.g., decimals for finding a percentage of an amount, or fractions for operations in non-calculator exams). For all money calculations involving growth or decay (interest, profit/loss, depreciation), the multiplier method is the most powerful tool.