Welcome to 'The Four Operations'!
Hello IGCSE Maths students! This chapter is all about the four fundamental building blocks of Mathematics: Addition, Subtraction, Multiplication, and Division. Don't underestimate this topic! Mastering these operations, especially when dealing with negative numbers, fractions, and the correct order of steps, is absolutely crucial for success throughout your entire IGCSE course. Think of this as training to be a master builder—you need to know exactly how your basic tools work before you can construct a skyscraper!
Section 1: Operating with Integers (The Whole Numbers)
Integers are all the whole numbers, including positive numbers, negative numbers, and zero (..., -3, -2, -1, 0, 1, 2, 3, ...). Calculating with negative numbers is often where students trip up, so let's make the rules crystal clear.
1.1 Addition and Subtraction of Integers
Imagine a number line, or think of money in a bank account (positive is money you have, negative is debt).
-
Adding a positive number: Move to the right on the number line (or increase your money).
Example: \(5 + 3 = 8\) -
Subtracting a positive number: Move to the left (or decrease your money).
Example: \(5 - 7 = -2\) -
Adding a negative number: This is the same as subtracting the positive version.
Rule: \(a + (-b) = a - b\)
Example: \(10 + (-4) = 10 - 4 = 6\) -
Subtracting a negative number: This is the same as adding the positive version.
Rule: \(a - (-b) = a + b\)
Example: \(-3 - (-5) = -3 + 5 = 2\)
Memory Aid: The Double Sign Rule
When two signs are next to each other (separated only by a bracket, if at all):
- Same signs make a plus (\(++ \to +\) or \(-- \to +\))
- Different signs make a minus (\(+ - \to -\) or \(- + \to -\))
1.2 Multiplication and Division of Integers
These rules rely on the signs of the two numbers being multiplied or divided:
-
Same signs: The answer is always Positive.
Example: \(-4 \times -2 = 8\) or \(10 \div 5 = 2\) -
Different signs: The answer is always Negative.
Example: \(-6 \times 3 = -18\) or \(15 \div (-3) = -5\)
Quick Review: Key Takeaways for Integers
The key is converting subtraction of a negative into addition, and remembering the positive/negative rules for multiplication and division.
Practical Example: If the temperature in London is \(-5^{\circ}C\) and it rises by \(8^{\circ}C\), the new temperature is \(-5 + 8 = 3^{\circ}C\).
Section 2: The Priority Rule (BODMAS/PEMDAS)
When a calculation involves more than one operation (addition, multiplication, etc.), we cannot just work from left to right! You must follow the correct order of operations to get the right answer.
2.1 What is BODMAS? (Or PEMDAS)
This mnemonic (memory aid) tells you the hierarchy of operations.
Brackets (or Parentheses)
Orders (or Exponents/Powers/Roots)
Division and Multiplication (Work from left to right)
Addition and Subtraction (Work from left to right)
Important Note: Division and Multiplication are at the same level of importance. You solve whichever one appears first when reading the sum from left to right. The same applies to Addition and Subtraction.
2.2 Step-by-Step Example
Work out the value of \(10 + 2 \times (12 - 4) \div 8\).
-
Brackets: Start inside the brackets.
\(12 - 4 = 8\)
The expression becomes: \(10 + 2 \times 8 \div 8\) - Orders: None in this example (no powers or roots).
-
Division and Multiplication (from left to right):
First, we see Multiplication: \(2 \times 8 = 16\).
The expression becomes: \(10 + 16 \div 8\)
Next, we see Division: \(16 \div 8 = 2\).
The expression becomes: \(10 + 2\) -
Addition and Subtraction:
\(10 + 2 = 12\)
Final Answer: 12
Common Mistake to Avoid:
NEVER do Addition before Multiplication or Division, unless the addition is inside brackets. If you did \(10+2\) first, you would get \(12 \times 8 \div 8 = 12\), which worked here *by coincidence* because of the division at the end. Try \(10 + 2 \times 3\). Correct answer is \(10 + 6 = 16\). Wrong answer is \((10+2) \times 3 = 36\).
Key Takeaway: BODMAS is your roadmap. Always check which operation has the highest priority before calculating!
Section 3: Operations with Fractions
Working with fractions (including proper fractions, improper fractions, and mixed numbers) requires specific techniques for each operation.
3.1 The Golden Rule for Mixed Numbers
Before you add, subtract, multiply, or divide mixed numbers (like \(1 \frac{1}{2}\)), you must first convert them into improper fractions (where the numerator is larger than the denominator, like \(\frac{3}{2}\)).
Step-by-Step Conversion Example:
To convert \(2 \frac{3}{4}\):
- Multiply the whole number by the denominator: \(2 \times 4 = 8\)
- Add the numerator: \(8 + 3 = 11\)
- The improper fraction is \(\frac{11}{4}\)
3.2 Addition and Subtraction of Fractions
You can only add or subtract fractions if they have the same denominator (a common denominator).
Step-by-Step:
- Find the Lowest Common Multiple (LCM) of the denominators. This is your new common denominator.
- Convert both fractions using the new denominator.
- Add or subtract the numerators, keeping the denominator the same.
- Simplify the final answer (if possible) or convert back to a mixed number.
Example: \(\frac{1}{3} + \frac{1}{6}\)
The LCM of 3 and 6 is 6.
\(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)
\(\frac{2}{6} + \frac{1}{6} = \frac{3}{6}\)
Simplify: \(\frac{3}{6} = \frac{1}{2}\)
3.3 Multiplication of Fractions
This is the easiest operation! No common denominator is needed.
Rule: Multiply the numerators together, and multiply the denominators together.
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Example: \(\frac{2}{5} \times \frac{3}{4}\)
\(2 \times 3 = 6\) (New numerator)
\(5 \times 4 = 20\) (New denominator)
Result: \(\frac{6}{20}\). Simplify to \(\frac{3}{10}\).
3.4 Division of Fractions
Remember the phrase: Keep, Change, Flip (KCF).
Step-by-Step:
- Keep the first fraction as it is.
- Change the division sign to a multiplication sign.
- Flip the second fraction (find its reciprocal).
- Multiply the resulting fractions (as shown in 3.3).
Example: \(\frac{3}{4} \div \frac{9}{10}\)
\(\frac{3}{4} \times \frac{10}{9}\)
Multiply: \(\frac{3 \times 10}{4 \times 9} = \frac{30}{36}\)
Simplify (divide top and bottom by 6): \(\frac{5}{6}\)
Did you know?
The word reciprocal means 1 divided by that number. For a fraction \(\frac{a}{b}\), the reciprocal is simply \(\frac{b}{a}\). If you have a whole number (like 5), its reciprocal is \(\frac{1}{5}\).
Key Takeaway for Fractions: Always convert mixed numbers first, find a common denominator for adding/subtracting, and use KCF for division.
Section 4: Operations with Decimals
Decimals are simply fractions written in a different form (like \(\frac{1}{2} = 0.5\)). Calculations with decimals are straightforward, but you must be careful with alignment and the decimal point placement.
4.1 Addition and Subtraction of Decimals
The key here is aligning the decimal points vertically.
Example: Calculate \(15.3 + 2.78\).
\(
\begin{array}{l}
\quad 15.30 \\
+ \quad 2.78 \\
\hline
\quad 18.08
\end{array}
\)
(Adding a zero to 15.3 helps align the place values properly.)
4.2 Multiplication of Decimals
When multiplying, ignore the decimal points initially, and multiply as if they were whole numbers.
Step-by-Step:
- Count the total number of decimal places (d.p.) in the original numbers.
- Multiply the numbers without the decimal points.
- Insert the decimal point in your answer so it has the same total number of decimal places counted in step 1.
Example: Calculate \(0.4 \times 1.2\).
- 0.4 has 1 d.p. 1.2 has 1 d.p. Total d.p. = 2.
- Multiply \(4 \times 12 = 48\).
- Place the decimal point so there are 2 d.p.: \(0.48\).
4.3 Division of Decimals (Non-Calculator)
If you are dividing by a decimal, always adjust the problem so you are dividing by an integer.
Rule: Multiply both the dividend (the number being divided) and the divisor (the number you are dividing by) by 10, 100, or 1000 until the divisor is a whole number.
Example: Work out \(1.2 \div 0.03\).
We need to multiply 0.03 by 100 to make it the integer 3.
We must do the same to 1.2: \(1.2 \times 100 = 120\).
The calculation becomes: \(120 \div 3 = 40\).
Key Takeaway for Decimals: Align points for adding/subtracting. Count total decimal places for multiplication. Eliminate the decimal in the divisor for division.
Section 5: Using Operations in Practical Situations
In your IGCSE exam, these operations will often appear in word problems, covering concepts like money, length, mass, and crucially, temperature changes.
5.1 Temperature and Negative Numbers
Temperature problems are classic applications of integer operations.
Scenario 1: Finding a change in temperature.
If the temperature starts at \(-2^{\circ}C\) and rises to \(10^{\circ}C\), the change is \(10 - (-2) = 10 + 2 = 12^{\circ}C\).
Scenario 2: Finding a new temperature.
If the temperature is \(4^{\circ}C\) and drops by \(6.5^{\circ}C\), the new temperature is \(4 - 6.5 = -2.5^{\circ}C\).
5.2 Mixed Operations and Problem Solving
Often, a single problem requires multiple steps involving different types of numbers (fractions, decimals) and must adhere to BODMAS.
Example Problem: A baker uses \(1 \frac{1}{2}\) kg of flour per cake. If he has 12 kg of flour, how many cakes can he make?
This is a division problem: Total flour \(\div\) Flour per cake.
- Convert mixed number to improper fraction: \(1 \frac{1}{2} = \frac{3}{2}\)
- Set up the division: \(12 \div \frac{3}{2}\)
- Apply KCF (remember \(12 = \frac{12}{1}\)): \(\frac{12}{1} \times \frac{2}{3}\)
- Multiply: \(\frac{12 \times 2}{1 \times 3} = \frac{24}{3}\)
- Calculate the answer: \(24 \div 3 = 8\).
Encouragement: Don't worry if these calculations look complicated when combined. Break every problem down into small, manageable steps, and always check your signs and your BODMAS rules!
Summary of Operations Checklist
Use this table to quickly review the core rules before tackling any complex problem:
Operation Checklist
- Integers (+/-): Use the number line or double sign rule. \(-(-)\) becomes \(+\).
- Integers (\(\times/\div\)): Same signs \(\to\) Positive. Different signs \(\to\) Negative.
- Order of Operations: Follow BODMAS/PEMDAS strictly. Brackets first!
- Fractions (Mixed Numbers): Convert to improper fractions before calculating.
- Fractions (+/-): Must find a Common Denominator.
- Fractions (\(\div\)): Use Keep, Change, Flip (KCF).
- Decimals (\(\times\)): Count total decimal places in the factors to place the point in the product.