1. **Problem 42: Calculate the total number of females and males given mature counts and percentages.** Given: - 57% are female, 43% are male. - 70% of females are mature. - 39% of males are mature. - Total mature = 957. 2. **Set variables:** Let total population be $N$. Number of females = $0.57N$. Number of males = $0.43N$. 3. **Calculate mature females and males:** Mature females = $0.70 \times 0.57N = 0.399N$. Mature males = $0.39 \times 0.43N = 0.1677N$. 4. **Sum of mature individuals:** $$0.399N + 0.1677N = 0.5667N = 957$$ 5. **Solve for $N$:** $$N = \frac{957}{0.5667} \approx 1688.5$$ 6. **Calculate number of females and males:** Females = $0.57 \times 1688.5 \approx 962.4$. Males = $0.43 \times 1688.5 \approx 726.1$. --- 7. **Problem 43: Simplify the product** $$\left( \frac{4P^5 x^4}{2M^8} \right) \cdot \left( \frac{5M^4}{7P^6 x^3} \right)$$ 8. **Multiply numerators and denominators:** Numerator: $4P^5 x^4 \times 5M^4 = 20P^5 M^4 x^4$ Denominator: $2M^8 \times 7P^6 x^3 = 14M^8 P^6 x^3$ 9. **Write combined fraction:** $$\frac{20P^5 M^4 x^4}{14M^8 P^6 x^3}$$ 10. **Simplify coefficients:** $$\frac{20}{14} = \frac{10}{7}$$ 11. **Simplify variables using exponent rules:** $$P^{5-6} = P^{-1} = \frac{1}{P}$$ $$M^{4-8} = M^{-4} = \frac{1}{M^4}$$ $$x^{4-3} = x^{1} = x$$ 12. **Final simplified expression:** $$\frac{10}{7} \times \frac{x}{P M^4} = \frac{10x}{7 P M^4}$$