This lesson is a crucial review and application of all the fraction skills you've learned. Mastery here is key, as fractions appear in many other topics. The focus is on applying the order of operations (BODMAS) to solve complex, multi-step problems.

1. Core Concepts (Short Notes)

• Proper Fraction: Numerator is smaller than the denominator (e.g., 2/5). Represents a value less than 1.

• Improper Fraction: Numerator is larger than or equal to the denominator (e.g., 7/3). Represents a value of 1 or more.

• Mixed Number: A whole number and a proper fraction combined (e.g., 2 ¹/₃).

• Equivalent Fractions: Fractions that represent the same value (e.g., 1/2 = 2/4 = 5/10). Created by multiplying or dividing the numerator and denominator by the same number.

• Reciprocal: A fraction "flipped upside down". The reciprocal of 2/3 is 3/2. This is essential for division.

• BODMAS: The strict order for performing calculations:

  1. Brackets

  2. Of (which means multiplication)

  3. Division

  4. Multiplication

  5. Addition

  6. Subtraction (Division & Multiplication have equal priority, solve from left to right. Same for Addition & Subtraction)

2. Key Methods to Master

• Addition & Subtraction:

  1. Convert all mixed numbers to improper fractions.

  2. Find the Lowest Common Multiple (LCM) of the denominators.

  3. Create equivalent fractions for each term using the LCM as the new denominator.

  4. Add or subtract the numerators only. Keep the denominator the same.

  5. Simplify and convert back to a mixed number if necessary.

• Multiplication (including "Of"):

  1. Convert all mixed numbers to improper fractions.

  2. Multiply all numerators together to get the new numerator.

  3. Multiply all denominators together to get the new denominator.

  4. Simplify the final fraction.

• Division:

  1. Convert all mixed numbers to improper fractions.

  2. Keep, Change, Flip: Keep the first fraction, change the ÷ sign to a × sign, and flip the second fraction (use its reciprocal).

  3. Follow the rules for multiplication.

3. Tips & Tricks for Exams

• First Step ALWAYS: Mixed to Improper: Before you do any calculation, convert every mixed number into an improper fraction. (Whole Number × Denominator) + Numerator. E.g., 2 ⁴/₅ = (2×5+4)/5 = 14/5. This prevents a lot of errors.

• Simplify Before You Multiply: In multiplication problems, look for any numerator and any denominator that share a common factor and cancel them out before multiplying. This keeps the numbers smaller and easier to work with.

• "Of the Remainder" is a Trap: This is a very common exam question style.

  • Example: "A man spent 1/4 of his salary and then 1/3 of the remainder."

  • Step 1: Find the remainder. If he spent 1/4, the remainder is 1 - 1/4 = 3/4.

  • Step 2: Calculate the second amount based on the remainder. 1/3 of 3/4 is 1/3 × 3/4 = 1/4.

  • Don't just calculate 1/3 of the original amount!

• Identify the "Whole": In word problems, be crystal clear about what "1" (the whole unit) represents. Is it the total distance of a journey? The total amount of money? The total area of a field? This helps you set up the problem correctly.

4. Important Points to Remember

• BODMAS is Not a Suggestion, It's a Rule: Always follow the order of operations strictly. A common mistake is doing addition before division if it appears first in the line.

• Common Denominators are ONLY for Adding and Subtracting. You do not need them for multiplication or division.

• Show Your Steps: Clearly show how you found the LCM, the equivalent fractions you created, and any cancellations you made. This is essential for earning partial marks.

• Final Answer Format: If the question used mixed numbers, it's good practice to give your final answer as a simplified mixed number.