1. **State the problem:** We need to find how much more money is needed for Phase 3 to match the total cost of Phases 1 and 2 combined.
2. **Convert Phase 1 cost from base 4 to decimal:**
Phase 1 cost is $(2203)_4$.
Calculate decimal value:
$$2 \times 4^3 + 2 \times 4^2 + 0 \times 4^1 + 3 \times 4^0 = 2 \times 64 + 2 \times 16 + 0 + 3 = 128 + 32 + 3 = 163$$
3. **Convert Phase 2 cost from base 8 to decimal:**
Phase 2 cost is $(124.2)_8$.
Convert integer part $(124)_8$:
$$1 \times 8^2 + 2 \times 8^1 + 4 \times 8^0 = 64 + 16 + 4 = 84$$
Convert fractional part $(0.2)_8$:
$$2 \times 8^{-1} = 2 \times \frac{1}{8} = 0.25$$
Total Phase 2 cost in decimal:
$$84 + 0.25 = 84.25$$
4. **Calculate total cost of Phases 1 and 2:**
$$163 + 84.25 = 247.25$$
5. **Convert Phase 3 days from base 5 to decimal:**
Phase 3 days is $(110)_5$.
Calculate decimal value:
$$1 \times 5^2 + 1 \times 5^1 + 0 \times 5^0 = 25 + 5 + 0 = 30$$ days
6. **Convert Phase 3 daily cost from base 6 to decimal:**
Phase 3 daily cost is $(5.43)_6$.
Convert integer part $(5)_6$:
$$5 \times 6^0 = 5$$
Convert fractional part $(0.43)_6$:
$$4 \times 6^{-1} + 3 \times 6^{-2} = 4 \times \frac{1}{6} + 3 \times \frac{1}{36} = \frac{4}{6} + \frac{3}{36} = 0.6667 + 0.0833 = 0.75$$
Total daily cost in decimal:
$$5 + 0.75 = 5.75$$
7. **Calculate total Phase 3 cost so far:**
$$30 \text{ days} \times 5.75 = 172.5$$
8. **Calculate how much more money is needed for Phase 3 to match Phases 1 and 2 combined:**
$$247.25 - 172.5 = 74.75$$
**Final answer:** Phase 3 needs an additional $74.75$ dollars to match the total cost of Phases 1 and 2 combined.
Phase Costs
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