Revision of Second Term’s Topic Basic 6 Mathematics Lesson Note

i. Ratio and Proportion

Quick Review

• Ratio: Compares quantities (3:4)
• Direct Proportion: As one increases, the other increases
• Inverse Proportion: As one increases, the other decreases

Exercise A – Ratio and Proportion

Solve these problems:

1. Simplify the ratio 24:36 Answer: ______
2. Share ₦1,800 between Kemi and Tolu in ratio 2:3 Kemi: ₦______ Tolu: ₦______
3. If 6 books cost ₦1,200, how much do 10 books cost? Answer: ₦______
4. 12 workers complete a task in 8 days. How long will 16 workers take? Answer: ______ days
5. The ratio of boys to girls in a class is 3:2. If there are 15 boys, how many girls are there? Answer: ______ girls

ii. Money (Commercial Arithmetic)

Quick Review

• Profit = S.P – C.P
• Simple Interest = (P × R × T) ÷ 100
• Discount = Marked Price – Selling Price
• Commission = % of Sales

Exercise B – Money Problems

Calculate these business problems:

1. Profit and Loss: C.P = ₦8,000, S.P = ₦9,600 Profit: ₦______ Profit %: ______%
2. Simple Interest: Principal = ₦20,000, Rate = 6%, Time = 3 years Interest: ₦______ Amount: ₦______
3. Discount: Marked Price = ₦5,000, Discount = 15% Discount Amount: ₦______ Selling Price: ₦______
4. Commission: Sales = ₦80,000, Commission Rate = 8% Commission: ₦______
5. Rate Problem: 15 kg of rice costs ₦6,000. What is the cost per kg? Answer: ₦______ per kg

iii. Plane Shapes

Quick Review

• Rectangle: Area = l × w, Perimeter = 2(l + w)
• Circle: Area = πr², Circumference = 2πr
• Triangle: Area = ½ × base × height

Exercise C – Plane Shapes

Find areas and perimeters:

1. Rectangle: length = 15 cm, width = 8 cm Area: ______ cm² Perimeter: ______ cm
2. Circle: radius = 14 cm (use π = 22/7) Area: ______ cm² Circumference: ______ cm
3. Square: side = 12 cm Area: ______ cm² Perimeter: ______ cm
4. Triangle: base = 16 cm, height = 9 cm Area: ______ cm²
5. Trapezium: parallel sides = 10 cm and 6 cm, height = 8 cm Area: ______ cm²

NUMBER BASE

What is Number Base?

Number base tells us how many different digits we use in a counting system.

1. Binary Number (Base 2)

Binary uses only 2 digits: 0 and 1

Place Values in Binary:

… 16  8  4  2  1

… 2⁴ 2³ 2² 2¹ 2⁰

Converting Binary to Denary (Decimal)

Example: 1011₂ to denary

1011₂ = (1×8) + (0×4) + (1×2) + (1×1)

      = 8 + 0 + 2 + 1

      = 11₁₀

Exercise D – Binary to Denary

Convert these binary numbers to denary:

1. 101₂ = ______
2. 1001₂ = ______
3. 1110₂ = ______
4. 10101₂ = ______
5. 11111₂ = ______

1. Denary Number (Base 10)

Denary uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Converting Denary to Binary

Method: Divide by 2 repeatedly, read remainders upward

Example: 13₁₀ to binary

13 ÷ 2 = 6 remainder 1

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Reading upward: 1101₂

Exercise E – Denary to Binary

Convert these denary numbers to binary:

1. 7₁₀ = ______₂
2. 12₁₀ = ______₂
3. 18₁₀ = ______₂
4. 25₁₀ = ______₂
5. 31₁₀ = ______₂

Binary Operations

Addition in Binary

Rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (carry 1)

Example: 101₂ + 110₂

 101

+ 110

—–

 1011

Exercise F – Binary Addition

Add these binary numbers:

1. 10₂ + 11₂ = ______₂
2. 101₂ + 11₂ = ______₂
3. 111₂ + 101₂ = ______₂
4. 1001₂ + 111₂ = ______₂
5. 1010₂ + 101₂ = ______₂

iii. Quantitative Reasoning with Number Base

Exercise G – Pattern Recognition

Complete these patterns:

1. Binary counting: 1₂, 10₂, 11₂, 100₂, ______₂, ______₂
2. Denary to Binary: 1→1₂, 2→10₂, 3→11₂, 4→______₂, 5→______₂
3. Powers of 2: 1, 2, 4, 8, ______, ______

Exercise H – Logic Problems

Use reasoning to solve:

1. What is the largest number you can write with 4 binary digits? Answer: ______₂ = ______₁₀
2. How many different numbers can you represent with 3 binary digits? Answer: ______
3. If 1010₂ = 10₁₀, what is 1011₂ in denary? Answer: ______

Exercise I – Real World Applications

Apply number base knowledge:

1. Computer Memory: A computer uses 8-bit binary. What is the largest denary number it can store? Answer: ______
2. Digital Codes: A security code uses 5 binary digits. How many different codes are possible? Answer: ______
3. Binary Messages: If A=1₂, B=10₂, C=11₂, D=100₂, what does 10₂ 1₂ 11₂ spell? Answer: ______

MIXED REVISION EXERCISES

Exercise J – Combined Problems

Solve these mixed problems:

1. Ratio and Money: Three friends invest in ratio 2:3:5. If the total investment is ₦50,000, how much does each friend invest? Answer: ₦, ₦, ₦______
2. Shapes and Money: A circular garden with radius 7 m needs fencing at ₦150 per meter. What is the total cost? (π = 22/7) Answer: ₦______
3. Binary and Arithmetic: Convert 13₁₀ to binary, then add 101₂ to your answer. 13₁₀ in binary: ______₂ Sum: ______₂

Exercise K – Problem Solving

Think and solve:

1. Business Problem: A trader buys goods for ₦15,000 and wants 25% profit. At what price should he sell? Answer: ₦______
2. Area Problem: A rectangular field (20 m × 15 m) has a circular pond (radius 3 m) inside. What area is available for planting? (π = 3.14) Answer: ______ m²
3. Proportion Problem: If 8 machines produce 240 items in 6 hours, how many items will 12 machines produce in 9 hours? Answer: ______ items

Exercise L – Quick Calculations

Calculate mentally:

1. 2:3 = 10:____
2. 20% of ₦5,000 = ₦____
3. Area of square, side 8 cm = ____ cm²
4. 1100₂ in denary = ____
5. 15₁₀ in binary = ____₂

Answer Key

Exercise A:

1. 2:3, 2. ₦720, ₦1,080, 3. ₦2,000, 4. 6 days, 5. 10 girls

Exercise B:

1. ₦1,600, 20%, 2. ₦3,600, ₦23,600, 3. ₦750, ₦4,250, 4. ₦6,400, 5. ₦400

Exercise C:

1. 120 cm², 46 cm, 2. 616 cm², 88 cm, 3. 144 cm², 48 cm, 4. 72 cm², 5. 64 cm²

Exercise D:

1. 5, 2. 9, 3. 14, 4. 21, 5. 31

Exercise E:

1. 111₂, 2. 1100₂, 3. 10010₂, 4. 11001₂, 5. 11111₂

Exercise F:

1. 101₂, 2. 1000₂, 3. 1100₂, 4. 10000₂, 5. 1111₂

Exercise G:

1. 101₂, 110₂, 2. 100₂, 101₂, 3. 16, 32

Exercise H:

1. 1111₂ = 15₁₀, 2. 8, 3. 11

Exercise I:

1. 255, 2. 32, 3. BAC

Exercise J:

1. ₦10,000, ₦15,000, ₦25,000, 2. ₦6,600, 3. 1101₂, 10010₂

Exercise K:

1. ₦18,750, 2. 271.7 m², 3. 540 items