CIE IGCSE Mathematics (0580) The Four Operations Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) The Four Operations Study Notes
LEARNING OBJECTIVE
- The Four Operations
Key Concepts:
- Calculations with Integers, Fractions, and Decimals
The Four Operations
Calculations with Integers, Fractions, and Decimals
The Four Operations
The four basic operations in arithmetic are:
- Addition (+): combining quantities
- Subtraction (−): finding the difference or removing
- Multiplication (×): repeated addition or scaling
- Division (÷): splitting into equal parts
Order of Operations
When an expression includes more than one operation, the order must be followed correctly:
The correct sequence is:
- Brackets
- Orders (powers and roots)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
This is often remembered using the acronym BODMAS.
Negative Numbers
Negative numbers are numbers less than zero. They are used in contexts like temperature, finance, and elevation.
Important rules:
- \( -3 + 5 = 2 \)
- \( -4 – 7 = -11 \)
- \( -3 \times 2 = -6 \)
- \( -12 \div 4 = -3 \)
Calculations with Fractions
Fractions can be:
- Proper: numerator < denominator (e.g. \( \frac{2}{3} \))
- Improper: numerator >= denominator (e.g. \( \frac{7}{4} \))
- Mixed number: combination of whole and fraction (e.g. \( 1\frac{3}{4} \))
Rules:
- To add/subtract: convert to common denominators
- To multiply: multiply numerators and denominators
- To divide: multiply by the reciprocal of the second fraction
Calculations with Decimals
Use place value when adding or subtracting decimals. Line up the decimal points correctly.
For multiplication and division, ignore the decimal, compute, and then place the decimal point according to place value.
Mixed Numbers and Improper Fractions
Mixed numbers are often converted to improper fractions before calculation:
\( 2\frac{1}{3} = \frac{7}{3} \)
Practical Situations (e.g. Temperatures)
In contexts like temperature, negative numbers are used frequently.
- If the temperature changes from \( 5^\circ C \) to \( -3^\circ C \), the change is: \( -3 – 5 = -8^\circ C \)
- Going from \( -6^\circ C \) to \( 4^\circ C \): \( 4 – (-6) = 4 + 6 = 10^\circ C \) increase
Example:
The temperature at 6 AM was \( -5^\circ \text{C} \). By noon, it rose by \( 9^\circ \text{C} \). What is the temperature at noon?
▶️ Answer/Explanation
Step 1: Start with the initial temperature
\( -5^\circ \text{C} \)
Step 2: Add the change
\( -5 + 9 = 4 \)
Final Answer:
The temperature at noon is \( 4^\circ \text{C} \).
Example:
Calculate: \( 3\frac{1}{2} – 1\frac{3}{4} \)
▶️ Answer/Explanation
Step 1: Convert to improper fractions:
\( 3\frac{1}{2} = \frac{7}{2} \), \( 1\frac{3}{4} = \frac{7}{4} \)
Step 2: Find a common denominator (LCM of 2 and 4 is 4):
\( \frac{7}{2} = \frac{14}{4} \), so: \( \frac{14}{4} – \frac{7}{4} = \frac{7}{4} \)
Step 3: Convert to a mixed number (optional):
\( \frac{7}{4} = 1\frac{3}{4} \)
Example:
Evaluate: \( (2.5 + 1.2) \times 3 – 4.2 \div 2 \)
▶️ Answer/Explanation
Step 1: Evaluate inside the brackets:
\( 2.5 + 1.2 = 3.7 \)
Step 2: Multiply:
\( 3.7 \times 3 = 11.1 \)
Step 3: Divide:
\( 4.2 \div 2 = 2.1 \)
Step 4: Subtract:
\( 11.1 – 2.1 = 9 \)
Example:
Priya walks \( \frac{3}{4} \) km in the morning and \( 1.25 \) km in the evening. During the day, the temperature rose from \( -2^\circ \text{C} \) to \( 6^\circ \text{C} \). She also bought 3 pens at ₹12.50 each and a notebook for ₹55.
(a) What is the total distance Priya walked that day?
(b) What was the total rise in temperature?
(c) What was her total spending?
▶️ Answer/Explanation
(a) Total Distance Walked
Convert \( \frac{3}{4} \) km to decimal: \( \frac{3}{4} = 0.75 \) km
Total distance = \( 0.75 + 1.25 = 2 \) km
(b) Temperature Rise
From \( -2^\circ \text{C} \) to \( 6^\circ \text{C} \): Rise = \( 6 – (-2) = 6 + 2 = 8^\circ \text{C} \)
(c) Total Spending
3 pens at ₹12.50 = \( 3 \times 12.50 = 37.50 \)
Total = \( 37.50 + 55 =\) ₹\(92.50 \)