A customer buys a television costing Sh. 24,600 on hire purchase. She pays a deposit of Sh. 10,200 and the balance is paid in 18 equal monthly installments. If the interest rate is 2.1% per month, cal

The loan amount is the cash price minus the deposit.

$$\text{Loan Amount (PV)} = \text{Cash Price} - \text{Deposit}$$ $$PV = \text{Sh. } 24,600 - \text{Sh. } 10,200$$ $$PV = \text{Sh. } 14,400$$

We use the present value of an annuity formula to find the monthly installment ($PMT$).
The formula is:

$$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$

Rearranging to solve for $PMT$:

$$PMT = PV \times \frac{r}{1 - (1 + r)^{-n}}$$

Given:
$PV = \text{Sh. } 14,400$
$r = 2.1\% \text{ per month} = 0.021$
$n = 18 \text{ months}$

Substitute the values into the formula:

$$PMT = 14,400 \times \frac{0.021}{1 - (1 + 0.021)^{-18}}$$ $$PMT = 14,400 \times \frac{0.021}{1 - (1.021)^{-18}}$$

Calculate $(1.021)^{-18}$:

$$(1.021)^{-18} \approx 0.690596608$$

Now substitute this back:

$$PMT = 14,400 \times \frac{0.021}{1 - 0.690596608}$$ $$PMT = 14,400 \times \frac{0.021}{0.309403392}$$ $$PMT = 14,400 \times 0.067869600$$ $$PMT \approx \text{Sh. } 977.32$$

a) The amount paid for each installment is:

$$\boxed{\text{Sh. } 977.32}$$

The total amount paid is the sum of the deposit and all installments.

$$\text{Total Amount Paid} = \text{Deposit} + (\text{Number of Installments} \times \text{Amount per Installment})$$ $$\text{Total Amount Paid} = \text{Sh. } 10,200 + (18 \times \text{Sh. } 977.32)$$ $$\text{Total Amount Paid} = \text{Sh. } 10,200 + \text{Sh. } 17,591.76$$ $$\text{Total Amount Paid} = \text{Sh. } 27,791.76$$

b) The total amount paid is:

$$\boxed{\text{Sh. } 27,791.76}$$