1. **Problem:** Karanu bought a computer with a marked price of 25000 on hire purchase terms. She paid a deposit of 25% of the cash price and cleared the balance by paying 12 equal monthly instalments of 2000 each. Calculate the rate of compound interest charged per month. 2. **Step 1: Calculate the cash price and deposit** - Deposit = 25% of 25000 = $0.25 \times 25000 = 6250$ - Balance to be paid = $25000 - 6250 = 18750$ 3. **Step 2: Understand the payment terms** - The balance 18750 is paid in 12 monthly instalments of 2000 each. - Total paid in instalments = $12 \times 2000 = 24000$ - The extra amount paid over the balance is interest. 4. **Step 3: Use the compound interest formula for hire purchase** - Let the monthly interest rate be $r$ (in decimal). - The present value (PV) of the 12 instalments is equal to the balance 18750. - The instalments form an annuity: $$PV = R \times \frac{1 - (1+r)^{-n}}{r}$$ where $R=2000$, $n=12$. 5. **Step 4: Set up the equation and solve for $r$** $$18750 = 2000 \times \frac{1 - (1+r)^{-12}}{r}$$ 6. **Step 5: Rearrange** $$\frac{18750}{2000} = \frac{1 - (1+r)^{-12}}{r}$$ $$9.375 = \frac{1 - (1+r)^{-12}}{r}$$ 7. **Step 6: Solve for $r$ numerically** - This is a transcendental equation; solve by trial or using a calculator. - Try $r=0.02$ (2%): Left side $=9.375$, Right side $=\frac{1 - (1.02)^{-12}}{0.02} \approx \frac{1 - 0.7885}{0.02} = \frac{0.2115}{0.02} = 10.575$ (too high) - Try $r=0.025$ (2.5%): Right side $= \frac{1 - (1.025)^{-12}}{0.025} \approx \frac{1 - 0.7401}{0.025} = \frac{0.2599}{0.025} = 10.396$ (still high) - Try $r=0.035$ (3.5%): Right side $= \frac{1 - (1.035)^{-12}}{0.035} \approx \frac{1 - 0.6576}{0.035} = \frac{0.3424}{0.035} = 9.783$ (closer) - Try $r=0.04$ (4%): Right side $= \frac{1 - (1.04)^{-12}}{0.04} \approx \frac{1 - 0.6246}{0.04} = \frac{0.3754}{0.04} = 9.385$ (very close) - Try $r=0.041$ (4.1%): Right side $= \frac{1 - (1.041)^{-12}}{0.041} \approx 9.29$ (slightly less) 8. **Step 7: Conclusion** - The monthly interest rate $r$ is approximately 4%. **Final answer:** The rate of compound interest charged per month is approximately **4%**.