Tax Discount Interest

1. **Problem 19:** Given the tax function $$T(x) = 50000 + 0.30(x - 500000)$$ and total tax paid $$T(x) = 80000$$, find the annual income $$x$$. 2. **Formula:** The tax function is linear, where $$T(x)$$ is tax paid and $$x$$ is income. 3. **Set up equation:** $$80000 = 50000 + 0.30(x - 500000)$$. 4. **Solve for $$x$$:** $$80000 - 50000 = 0.30(x - 500000)$$ $$30000 = 0.30(x - 500000)$$ $$\frac{30000}{0.30} = x - 500000$$ $$100000 = x - 500000$$ $$x = 600000$$ 5. **Answer:** The annual income is **600000**. 1. **Problem 20:** Calculate total cost for 51 units with 15% discount, unit price 200. 2. **Formula:** Total cost = unit price × quantity × (1 - discount rate). 3. **Calculate:** For 51 units, discount = 15% = 0.15 Total cost = 200 × 51 × (1 - 0.15) = 200 × 51 × 0.85 = 8670 4. **Answer:** Cost for 51 units is **8670**. 1. **Problem 21:** Find maturity value of 80000 loan at 6% simple annual interest for 3 months. 2. **Formula:** Simple interest $$I = P \times r \times t$$, maturity value $$A = P + I$$, where $$t$$ in years. 3. **Calculate:** $$t = \frac{3}{12} = 0.25$$ years $$I = 80000 \times 0.06 \times 0.25 = 1200$$ $$A = 80000 + 1200 = 81200$$ 4. **Answer:** Maturity value is **81200**. 1. **Problem 22:** Find future value of 15000 investment at 5% simple interest for 18 months. 2. **Formula:** $$A = P(1 + rt)$$, $$t$$ in years. 3. **Calculate:** $$t = \frac{18}{12} = 1.5$$ years $$A = 15000(1 + 0.05 \times 1.5) = 15000(1 + 0.075) = 15000 \times 1.075 = 16125$$ 4. **Answer:** Future value is **16125**. 1. **Problem 23:** Find years for 50000 to grow to 70000 at 8% simple interest. 2. **Formula:** $$A = P(1 + rt)$$ solve for $$t$$. 3. **Calculate:** $$70000 = 50000(1 + 0.08t)$$ $$\frac{70000}{50000} = 1 + 0.08t$$ $$1.4 = 1 + 0.08t$$ $$0.4 = 0.08t$$ $$t = \frac{0.4}{0.08} = 5$$ years 4. **Answer:** It takes **5 years**. 1. **Problem 24:** Find annual simple interest rate for 1500 interest on 10000 principal after 30 months. 2. **Formula:** $$I = Prt$$ solve for $$r$$. 3. **Calculate:** $$t = \frac{30}{12} = 2.5$$ years $$1500 = 10000 \times r \times 2.5$$ $$r = \frac{1500}{10000 \times 2.5} = 0.06 = 6\%$$ 4. **Answer:** Annual interest rate is **6%**. 1. **Problem 25:** Find simple annual rate to earn 40000 interest from 200000 investment over 4 years. 2. **Formula:** $$I = Prt$$ solve for $$r$$. 3. **Calculate:** $$40000 = 200000 \times r \times 4$$ $$r = \frac{40000}{200000 \times 4} = 0.05 = 5\%$$ 4. **Answer:** Rate is **5%**. 1. **Problem 26:** Find future value of 5000 compounded quarterly at 4% for 1 year. 2. **Formula:** $$A = P(1 + \frac{r}{n})^{nt}$$ where $$n=4$$ quarters/year. 3. **Calculate:** $$A = 5000(1 + \frac{0.04}{4})^{4 \times 1} = 5000(1 + 0.01)^4 = 5000(1.01)^4$$ $$= 5000 \times 1.04060401 = 5203.02$$ 4. **Answer:** Future value is **5203.02**. 1. **Problem 27:** Find maturity value of 10000 bond at 6% compounded monthly for 3 years. 2. **Formula:** $$A = P(1 + \frac{r}{n})^{nt}$$, $$n=12$$. 3. **Calculate:** $$A = 10000(1 + \frac{0.06}{12})^{12 \times 3} = 10000(1 + 0.005)^{36} = 10000(1.005)^{36}$$ $$= 10000 \times 1.196681 = 11966.81$$ 4. **Answer:** Maturity value is **11966.81**. 1. **Problem 28:** Given $$FV = 50000(1.004)^{60}$$ and $$t=5$$ years, find nominal annual rate and compounding frequency. 2. **Formula:** $$FV = P(1 + \frac{r}{n})^{nt}$$, here $$1.004 = 1 + \frac{r}{n}$$, $$nt=60$$. 3. **Calculate:** Since $$nt=60$$ and $$t=5$$, $$n = \frac{60}{5} = 12$$ compounding periods per year (monthly). $$\frac{r}{n} = 0.004 \Rightarrow r = 0.004 \times 12 = 0.048 = 4.8\%$$ 4. **Answer:** Nominal annual rate is **4.8% compounded monthly**.