A car costing shs. 800,000 depreciates by 20% in the first year and 5% in the second year. Thereafter, it depreciates by 10% annually for the next three years. Calculate how much less it will cost at

The car depreciates by 20% in the first year.

$$\text{Value after 1st year} = \text{shs. } 800,000 \times (1 - 0.20) = \text{shs. } 800,000 \times 0.80 = \text{shs. } 640,000$$

The car depreciates by 5% in the second year.

$$\text{Value after 2nd year} = \text{shs. } 640,000 \times (1 - 0.05) = \text{shs. } 640,000 \times 0.95 = \text{shs. } 608,000$$

The car depreciates by 10% for each of these three years.

$$\text{Value after 5th year} = \text{shs. } 608,000 \times (1 - 0.10)^3 = \text{shs. } 608,000 \times (0.90)^3$$ $$\text{Value after 5th year} = \text{shs. } 608,000 \times 0.729 = \text{shs. } 443,232$$

$$\text{Difference} = \text{Initial cost} - \text{Value after 5th year}$$ $$\text{Difference} = \text{shs. } 800,000 - \text{shs. } 443,232 = \text{shs. } 356,768$$

The car will cost shs. 356,768 less at the end of the fifth year.

$$\boxed{\text{shs. } 356,768}$$

Q2. b)

The formula for simple interest is $SI = P \times R \times T$.
Given: Principal ($P$) = shs. 3,000, Rate ($R$) = 15% = 0.15, Time ($T$) = 2 years.

$$SI = \text{shs. } 3,000 \times 0.15 \times 2 = \text{shs. } 900$$

The formula for the amount with compound interest is $A = P(1 + R)^T$.

$$A = \text{shs. } 3,000 (1 + 0.15)^2 = \text{shs. } 3,000 (1.15)^2$$ $$A = \text{shs. } 3,000 \times 1.3225 = \text{shs. } 3,967.50$$

Compound Interest ($CI$) is $A - P$.

$$CI = \text{shs. } 3,967.50 - \text{shs. } 3,000 = \text{shs. } 967.50$$

$$\text{Difference} = CI - SI = \text{shs. } 967.50 - \text{shs. } 900 = \text{shs. } 67.50$$

The compound interest will exceed simple interest by shs. 67.50.

$$\boxed{\text{shs. } 67.50}$$

Q3. a)

Joseph pays 24 equal monthly installments of shs. 1,250.

$$\text{Total installments} = 24 \times \text{shs. } 1,250 = \text{shs. } 30,000$$

The hire purchase price is the sum of the deposit and the total installments.

$$\text{Hire Purchase Price} = \text{Deposit} + \text{Total installments}$$ $$\text{Hire Purchase Price} = \text{shs. } 7,200 + \text{shs. } 30,000 = \text{shs. } 37,200$$

The hire purchase price of the camera is shs. 37,200.

$$\boxed{\text{shs. } 37,200}$$

Q3. b)

Let the cash price be $C$.
The hire purchase price (HPP) is 24% higher than the cash price.

$$HPP = C + 0.24C = 1.24C$$

From part (a), we know the HPP is shs. 37,200.

$$\text{shs. } 37,200 = 1.24C$$ $$C = \frac{\text{shs. } 37,200}{1.24}$$ $$C = \text{shs. } 30,000$$

The cash price of the camera is shs. 30,000.

$$\boxed{\text{shs. } 30,000}$$