1. Problem 3: Nimesh's TV purchase and installment payments. Nimesh bought a TV worth 180000 rupees, paid half as down payment, and agreed to pay the rest in 10 equal monthly installments with 20% annual interest on the reducing loan balance. We need to check if having 110000 rupees allows him to pay the down payment and two installments. Formula for monthly installment with reducing balance interest: $$\text{Monthly interest rate} = \frac{20\%}{12} = \frac{0.20}{12} = 0.0166667$$ Loan amount after down payment: $$180000 \times \frac{1}{2} = 90000$$ Monthly installment calculation uses amortization formula: $$EMI = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where $P=90000$, $r=0.0166667$, $n=10$. Calculate $(1+r)^n$: $$ (1+0.0166667)^{10} \approx 1.1800$$ Calculate numerator: $$90000 \times 0.0166667 \times 1.1800 = 90000 \times 0.0196667 = 1770$$ Calculate denominator: $$1.1800 - 1 = 0.1800$$ So, $$EMI = \frac{1770}{0.1800} = 9833.33$$ Each installment is approximately 9833.33 rupees. Total needed for down payment and two installments: $$90000 + 2 \times 9833.33 = 90000 + 19666.66 = 109666.66$$ Nimesh has 110000 rupees, which is more than 109666.66. Therefore, the statement is true. 2. Problem 4(a): Ashen's coins problem. Ashen has 100 rupees in 5 rupee and 10 rupee coins. Number of 5 rupee coins is twice the number of 10 rupee coins. Let $x$ = number of 5 rupee coins, $y$ = number of 10 rupee coins. Equations: $$5x + 10y = 100$$ $$x = 2y$$ Substitute $x=2y$ into first equation: $$5(2y) + 10y = 100$$ $$10y + 10y = 100$$ $$20y = 100$$ $$y = 5$$ Then, $$x = 2 \times 5 = 10$$ So, Ashen has 10 five-rupee coins and 5 ten-rupee coins. 3. Problem 4(b): Solve inequality $$45p + 750 \geq 1100$$ Subtract 750 from both sides: $$45p \geq 350$$ Divide both sides by 45: $$p \geq \frac{350}{45} = 7.777...$$ Minimum integral value of $p$ is 8. Final answers: - Nimesh can pay the down payment and two installments with 110000 rupees. - Ashen has 10 five-rupee coins and 5 ten-rupee coins. - Minimum integral $p$ satisfying inequality is 8.