1. **State the problem:** We are given the difference between Simple Interest (SI) and Compound Interest (CI) for 3 years as 453, with a principal of 2500 and an unknown rate $x$% per annum. We need to find $x$. 2. **Recall formulas:** - Simple Interest for 3 years: $$SI = \frac{P \times R \times T}{100} = \frac{2500 \times x \times 3}{100} = 75x$$ - Compound Interest for 3 years (compounded annually): $$CI = P \left(1 + \frac{R}{100}\right)^3 - P = 2500 \left(1 + \frac{x}{100}\right)^3 - 2500$$ 3. **Difference between CI and SI:** Given difference = $$CI - SI = 453$$ 4. **Set up the equation:** $$2500 \left(1 + \frac{x}{100}\right)^3 - 2500 - 75x = 453$$ 5. **Simplify:** $$2500 \left(1 + \frac{x}{100}\right)^3 - 2500 = 453 + 75x$$ Divide both sides by 2500: $$\left(1 + \frac{x}{100}\right)^3 - 1 = \frac{453 + 75x}{2500}$$ 6. **Let $y = \frac{x}{100}$, then:** $$ (1 + y)^3 - 1 = \frac{453 + 75x}{2500}$$ 7. **Expand $(1 + y)^3$:** $$1 + 3y + 3y^2 + y^3 - 1 = 3y + 3y^2 + y^3 = \frac{453 + 75x}{2500}$$ 8. **Substitute back $y = \frac{x}{100}$:** $$3 \times \frac{x}{100} + 3 \times \left(\frac{x}{100}\right)^2 + \left(\frac{x}{100}\right)^3 = \frac{453 + 75x}{2500}$$ 9. **Multiply both sides by 2500 to clear denominators:** $$2500 \left(3 \frac{x}{100} + 3 \frac{x^2}{10000} + \frac{x^3}{1000000}\right) = 453 + 75x$$ 10. **Calculate each term:** $$2500 \times 3 \frac{x}{100} = 75x$$ $$2500 \times 3 \frac{x^2}{10000} = 0.75 x^2$$ $$2500 \times \frac{x^3}{1000000} = 0.0025 x^3$$ So, $$75x + 0.75 x^2 + 0.0025 x^3 = 453 + 75x$$ 11. **Subtract $75x$ from both sides:** $$0.75 x^2 + 0.0025 x^3 = 453$$ 12. **Rewrite:** $$0.0025 x^3 + 0.75 x^2 - 453 = 0$$ 13. **Multiply entire equation by 400 to clear decimals:** $$x^3 + 300 x^2 - 181200 = 0$$ 14. **Try possible integer roots from options:** - For $x=15$: $$15^3 + 300 \times 15^2 - 181200 = 3375 + 300 \times 225 - 181200 = 3375 + 67500 - 181200 = -110325 \neq 0$$ - For $x=17$: $$17^3 + 300 \times 17^2 - 181200 = 4913 + 300 \times 289 - 181200 = 4913 + 86700 - 181200 = -89487 \neq 0$$ - For $x=23$: $$23^3 + 300 \times 23^2 - 181200 = 12167 + 300 \times 529 - 181200 = 12167 + 158700 - 181200 = -4333 \neq 0$$ 15. **Since none of the options satisfy the equation exactly, check which is closest:** $x=23$ gives the smallest absolute value (-4333), so $x=23$ is the best approximate solution. **Final answer:** $x = 23$ (Option C)