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Learning Resources

Whole Numbers

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 07 Pathway: N/A

First Principles

Objective: Master the correct sequence of operations (BODMAS) to accurately evaluate multi-step mathematical expressions involving whole numbers.

Interactive Operation Inspector

Evaluate the expression step-by-step: (4 + 2) × 3 − 8 ÷ 2

(4 + 2) × 3 − 8 ÷ 2
Click the operation that must be done first according to BODMAS.

Building the Concept from First Principles

(a) Concrete Scenario: Imagine Mama Jane running a vegetable stall in Gikomba market. A customer buys 2 bunches of sukuma wiki at KSh 20 each, and also buys a cabbage priced at KSh 50 with a special discount of KSh 10 off. To find the total bill, we write: \[ \text{Total} = 2 \times 20 + (50 - 10) \] We cannot simply work left-to-right without rules; we must first compute the discounted cabbage price inside the brackets \((50 - 10 = 40)\), then calculate the cost of the sukuma wiki \(2 \times 20 = 40\), and finally add the amounts: \(40 + 40 = 80\) shillings.

(b) Logical Insight: Mathematical operations have different inherent binding strengths. Multiplication represents repeated addition, so it takes precedence over single additions. Brackets act as grouping boundaries that demand immediate full evaluation before their result can interact with outside numbers.

(c) The Hierarchy of Operations:
B — Brackets first.
O — Orders (indices/powers or roots).
D & M — Division and Multiplication (equal priority; evaluate from left to right).
A & S — Addition and Subtraction (equal priority; evaluate from left to right).

Key Formulas

The BODMAS Hierarchy

When an arithmetic expression contains multiple operations, evaluate in the following strict order:

  1. Brackets: Compute expressions inside parentheses \( (\dots) \), brackets \( [\dots] \), or braces \( \{\dots\} \) first.
  2. Orders: Evaluate powers and square roots, e.g., \( 3^2 = 9 \).
  3. Division & Multiplication: Evaluate from left to right as they appear.
  4. Addition & Subtraction: Evaluate from left to right as they appear.

Equal Priority & Left-to-Right Rule

Division does not take priority over multiplication, nor does addition over subtraction. When operations of equal priority appear together, evaluate strictly from left to right:

\[ a \div b \times c = (a \div b) \times c \]\[ a - b + c = (a - b) + c \]

Distributive Property vs. BODMAS Brackets

An expression like \( a(b + c) \) can be evaluated by solving inside the brackets first: \[ a \times (b + c) \] or by expanding using distribution: \[ a \times b + a \times c \]

Worked Examples

Example 1 (Easy): Single Bracket Operation

Evaluate: \( 18 + 4 \times (10 - 6) \)

  1. Step 1 (Brackets): Work out the expression inside the bracket first.
    \[ 10 - 6 = 4 \]
    The expression becomes: \( 18 + 4 \times 4 \)
  2. Step 2 (Multiplication): Multiply before adding.
    \[ 4 \times 4 = 16 \]
    The expression becomes: \( 18 + 16 \)
  3. Step 3 (Addition): Complete the addition.
    \[ 18 + 16 = 34 \]

Final Answer: \( 34 \)

Example 2 (Medium): Mixed Operations Without Brackets

Evaluate: \( 50 - 6 \times 4 + 24 \div 3 \)

  1. Step 1 (Multiplication & Division from left to right):
    First multiplication: \( 6 \times 4 = 24 \)
    Then division: \( 24 \div 3 = 8 \)
    The expression simplifies to: \( 50 - 24 + 8 \)
  2. Step 2 (Addition & Subtraction from left to right):
    Work from left to right. First subtract:
    \[ 50 - 24 = 26 \]
    Then add:
    \[ 26 + 8 = 34 \]

Final Answer: \( 34 \)

Example 3 (Hard): Nested Groups & Multi-Step Operations

Evaluate: \( 100 - 3 \times [2 + (15 - 7)] + 36 \div 4 \)

  1. Step 1 (Innermost Brackets): Evaluate \( (15 - 7) \).
    \[ 15 - 7 = 8 \]
    The expression becomes: \( 100 - 3 \times [2 + 8] + 36 \div 4 \)
  2. Step 2 (Outer Brackets): Evaluate \( [2 + 8] \).
    \[ 2 + 8 = 10 \]
    The expression becomes: \( 100 - 3 \times 10 + 36 \div 4 \)
  3. Step 3 (Multiplication & Division):
    \[ 3 \times 10 = 30 \quad \text{and} \quad 36 \div 4 = 9 \]
    The expression becomes: \( 100 - 30 + 9 \)
  4. Step 4 (Addition & Subtraction left to right):
    \[ 100 - 30 = 70 \]
    \[ 70 + 9 = 79 \]

Final Answer: \( 79 \)

Common Mistakes

Misconception 1: Strict Left-to-Right Reading Order

Mistake: Calculating \( 5 + 3 \times 4 \) as \( (5 + 3) \times 4 = 8 \times 4 = 32 \).

Why it feels right: Natural languages like English or Kiswahili are read left to right, making it intuitive to calculate numbers in the exact sequence they are written.

Correction: Multiplication binds more tightly than addition. Always evaluate multiplication first: \[ 5 + (3 \times 4) = 5 + 12 = 17 \]

Misconception 2: Treating Addition as Higher Priority than Subtraction

Mistake: Evaluating \( 12 - 4 + 3 \) by doing addition first: \( 4 + 3 = 7 \), then \( 12 - 7 = 5 \).

Why it feels right: In the acronym BODMAS, 'A' comes before 'S', leading learners to believe addition must always precede subtraction.

Correction: Addition and subtraction have equal ranking and must be solved from left to right: \[ (12 - 4) + 3 = 8 + 3 = 11 \]

Misconception 3: Dropping Brackets Prematurely

Mistake: Writing \( 20 - (5 + 3) \) as \( 20 - 5 + 3 = 18 \).

Correction: The brackets require the interior sum to be found first: \( 5 + 3 = 8 \), giving \( 20 - 8 = 12 \). Alternatively, distributing the minus sign yields \( 20 - 5 - 3 = 12 \).

Real World

Matatu Fare Collection: A conductor collects KSh 50 from each of the 14 regular passengers and KSh 30 from each of 4 students with discount passes. Before paying the petrol bill of KSh 350, the net revenue is represented as: \[ \text{Cash Remaining} = (14 \times 50 + 4 \times 30) - 350 = (700 + 120) - 350 = 820 - 350 = 470 \text{ KSh} \] Without proper grouping, incorrect billing occurs.
Wholesale Cereal Packaging: An agricultural cooperative in Eldoret packages 5 bags of maize weighing 90 kg each and 3 bags weighing 50 kg each. The total shipment is distributed equally among 6 local retailers: \[ \text{Maize per retailer} = (5 \times 90 + 3 \times 50) \div 6 = (450 + 150) \div 6 = 600 \div 6 = 100 \text{ kg} \]
M-Pesa Transaction Reconciliations: A shopkeeper sends KSh 1,500 each to 3 suppliers and pays a transaction tariff of KSh 25 per transaction. The total money deducted from the account is: \[ 3 \times (1500 + 25) = 3 \times 1525 = 4575 \text{ KSh} \] Notice how multiplying after adding inside the bracket ensures tariffs on all 3 transactions are included!

Practice

Evaluate the arithmetic expression: \[ 14 + 6 \times 5 \] (Type only the number, e.g., 42)
Review the concepts above.
Evaluate the following expression: \[ (28 - 8) \div 4 \] (Type only the number, e.g., 42)
Review the concepts above.
Evaluate the value of: \[ 45 - 3 \times 8 + 12 \div 2 \] (Type only the number, e.g., 42)
Review the concepts above.
A fruit vendor in Machakos buys 4 crates of mangoes with 20 mangoes in each crate. She discovers that 15 mangoes are spoilt and discards them. She then sells all the remaining good mangoes at KSh 10 each. How much money in KSh does she receive in total? (Type only the number, e.g., 42)
Review the concepts above.
Evaluate the multi-step expression: \[ 80 - 2 \times (15 - 3) + 48 \div 6 \] (Type only the number, e.g., 42)
Review the concepts above.
Juma buys 3 bags of fertilizer at KSh 2,500 each and 4 packets of seeds at KSh 450 each. He receives a discount of KSh 800 off the total bill. He then shares the final cost equally with his farming partner. How many shillings does Juma pay? (Type only the number, e.g., 42)
Review the concepts above.