📐 CBC Mathematics Handout

📈 Question 32: Gradient of a Line

Points: A(0,2), B(2,4), C(4,6), D(6,8)

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Choose A(0,2) and D(6,8):
$$m = \frac{8 - 2}{6 - 0} = \frac{6}{6} = 1$$

Gradient = 1

Gradient = 1 means the line rises 1 unit for every 1 unit to the right.

      Y
      ↑
      8|                             ● D(6,8)
      7|
      6|                          ● C(4,6)
      5|
      4|                       ● B(2,4)
      3|
      2|                    ● A(0,2)
      1|
      0└────────────────────────────────→ X
       0 1 2 3 4 5 6 7 8 9

📏 Question 33: Rectangle Perimeter

Perimeter = 36 cm, width = 8 cm. Find length.

$$P = 2(L + W)$$

$$36 = 2(L + 8)$$
$$L + 8 = 18$$
$$L = 10\ \text{cm}$$

Length = 10 cm

Perimeter is the total distance around the rectangle.

📊 Question 34: Mean of Numbers

Numbers: 10, 15, 20, 25, 30

$$\text{Mean} = \frac{\text{Sum}}{\text{Count}}$$

Sum = 10 + 15 + 20 + 25 + 30 = 100
Count = 5
Mean = 100 ÷ 5 = 20

Mean = 20

The mean is the average, or “fair share”.

🔢 Question 35: Rounding to Nearest Hundred Thousand

Round 568 349 to the nearest hundred thousand.

The hundred‑thousands digit is 5 (since 568 349 = 5 68 349).
Look at the ten‑thousands digit: 6 (≥5), so round up.
5 becomes 6, all following digits become 0.

600 000

Rounding to nearest 100 000 means the last five digits become zero.

💧 Question 36: Painted Trough Area

⚠️ Diagram missing from the exam paper. The trough is typically a trapezoidal prism or rectangular open tank.
Without the dimensions, the exact area cannot be calculated. However, the method is to add the areas of all outer faces.
If you provide the diagram or measurements, I can compute the exact painted area.

       /‾‾‾‾‾‾‾‾‾‾\      (Example trapezoidal trough)
      /           \
     /             \
    | ‾‾‾‾‾‾‾‾‾‾‾‾‾‾ |
    |________________|

💻 Question 37: Hire Purchase

Cash price = sh. 40 000. Hire purchase price = 20% more. Deposit = sh. 20 000. Balance in 14 equal monthly instalments.

$$\text{HP} = \text{Cash} + 20\% \times \text{Cash}$$

HP = 40 000 × 1.20 = 48 000 shillings.
Balance after deposit = 48 000 – 20 000 = 28 000 shillings.
Monthly instalment = 28 000 ÷ 14 = 2 000 shillings per month.
Extra paid on HP = 48 000 – 40 000 = 8 000 shillings.

(a) 2 000 shillings per month   (b) 8 000 shillings extra

Hire purchase usually adds a percentage or flat fee; here it is 20% of cash price.

🟠 Question 38: Shaded Area (Ring)

⚠️ Diagram missing. A common shaded area problem involves two concentric circles.
For example, if the larger circle radius = 15 cm and smaller radius = 10 cm, the shaded ring area = π(R² – r²).
With those numbers: π(225 – 100) = 125π ≈ 392.7 cm².

        ╭─────────────────────╮
       /  ┌─────────────┐    /    Outer circle radius R
      │  │    ●        │    │    Inner circle radius r
      │  │   Shaded    │    │    Shaded = R²π – r²π
       \  └─────────────┘    /
        ╰─────────────────────╯

🍌 Question 39: Banana Algebra

Kamene has x bananas. Morris has 2 more than Kamene. Mary has 5 less than the total of Kamene and Morris.

Kamene: \(x\)
Morris: \(x + 2\)
Kamene + Morris = \(x + (x+2) = 2x + 2\)
Mary = \((2x + 2) - 5 = 2x - 3\)
Total = Kamene + Morris + Mary = \(x + (x+2) + (2x - 3) = 4x - 1\)

Total bananas = \(4x - 1\)

Always define variables clearly before combining.

🧮 Question 40: Fraction Division

Work out: \(\displaystyle \frac{4}{5} \div 10\frac{1}{5}\)

Convert mixed number to improper fraction: \(10\frac{1}{5} = \frac{51}{5}\)
Division: \(\frac{4}{5} \div \frac{51}{5} = \frac{4}{5} \times \frac{5}{51} = \frac{4}{51}\)

\(\displaystyle \frac{4}{51}\)

Dividing by a fraction is the same as multiplying by its reciprocal.

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📌 All missing diagrams are due to the original exam paper not providing them. With the correct diagrams or measurements, the area problems can be solved precisely.