1. **Problem statement:** Ismail saves 500 every three months for 4 years in an account with 4% interest compounded semi-annually. After the last deposit, his daughter withdraws half and invests it in West Bank with 6% interest compounded monthly. Two years later, the rate increases to 7% compounded monthly. (a) Find the amount in West Bank 1.5 years after the rate hike. (b) She withdraws 200 monthly starting 19 months after the hike. Find how many times she withdraws. --- 2. **Step 1: Calculate total amount in daughter's account after 4 years** - Deposits: 500 every 3 months for 4 years = $4 \times 4 = 16$ deposits. - Interest rate: 4% compounded semi-annually means 2% every 6 months. - Since deposits are quarterly, convert to semi-annual compounding. Use the future value of an annuity formula with compounding: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where $P=500$, $r=0.02$ (semi-annual rate), $n=8$ (number of semi-annual periods in 4 years). Calculate: $$FV = 500 \times \frac{(1.02)^8 - 1}{0.02} = 500 \times \frac{1.171659 - 1}{0.02} = 500 \times 8.58295 = 4291.48$$ 3. **Step 2: Daughter withdraws half immediately** $$\text{Amount invested in West Bank} = \frac{4291.48}{2} = 2145.74$$ 4. **Step 3: Invested in West Bank at 6% compounded monthly for 2 years** - Monthly rate $r = \frac{6\%}{12} = 0.005$ - Number of months $n = 24$ Calculate amount after 2 years: $$A = 2145.74 \times (1 + 0.005)^{24} = 2145.74 \times 1.12749 = 2418.17$$ 5. **Step 4: Interest rate increases to 7% compounded monthly** - New monthly rate $r = \frac{7\%}{12} = 0.0058333$ - Time after hike = 1.5 years = 18 months Calculate amount 1.5 years after hike: $$A = 2418.17 \times (1 + 0.0058333)^{18} = 2418.17 \times 1.11357 = 2692.88$$ **Answer (a):** The amount in West Bank 1.5 years after the hike is approximately **2692.88**. 6. **Step 5: Withdrawals of 200 every month starting 19 months after hike** - Initial amount at hike + 18 months = 2692.88 - Withdrawals start 1 month later (month 19 after hike) 7. **Step 6: Calculate number of withdrawals until amount depletes** - Monthly interest rate $r=0.0058333$ - Withdrawal amount $W=200$ - Use formula for number of withdrawals $n$: $$n = \frac{\ln\left(\frac{W}{W - A \times r}\right)}{\ln(1 + r)}$$ Calculate: $$n = \frac{\ln\left(\frac{200}{200 - 2692.88 \times 0.0058333}\right)}{\ln(1.0058333)} = \frac{\ln\left(\frac{200}{200 - 15.71}\right)}{0.005816} = \frac{\ln(1.086)}{0.005816} = \frac{0.0825}{0.005816} = 14.18$$ Number of full withdrawals = 14 times. **Answer (b):** She withdrew 200 a total of **14 times** from West Bank. --- Final answers: (a) 2692.88 (b) 14