1. Problem 6: A relief organization donated 240 kg of maize and 150 kg of beans to needy families. Each family received exactly the same quantity by mass of either maize or beans, and no family received both. Determine the least possible number of needy families. Step 1: We want to divide 240 kg of maize into equal portions and 150 kg of beans into equal portions, with each portion having the same mass for all families. Step 2: The least possible number of families corresponds to the greatest common divisor (GCD) of 240 and 150, because the mass per family must be a divisor of both quantities. Step 3: Find $\gcd(240,150)$. - Prime factorization of 240: $2^4 \times 3 \times 5$ - Prime factorization of 150: $2 \times 3 \times 5^2$ Step 4: The GCD is the product of the lowest powers of common primes: $$\gcd = 2^1 \times 3^1 \times 5^1 = 30$$ Step 5: Therefore, the least possible number of families is: $$\frac{240}{30} + \frac{150}{30} = 8 + 5 = 13$$ 2. Problem 7: Simplify $\frac{x^2 - 4y^2}{x^2 + 4xy + 4y^2}$. Step 1: Recognize the numerator as a difference of squares: $$x^2 - 4y^2 = (x - 2y)(x + 2y)$$ Step 2: Recognize the denominator as a perfect square trinomial: $$x^2 + 4xy + 4y^2 = (x + 2y)^2$$ Step 3: Substitute back: $$\frac{(x - 2y)(x + 2y)}{(x + 2y)^2}$$ Step 4: Cancel one $(x + 2y)$ term: $$\frac{x - 2y}{x + 2y}$$ 3. Problem 8: The area of a sector of a circle is 550 cm$^2$. The sector is curved to form an open cone of radius 7 cm. Calculate the height of the cone. Step 1: Let the radius of the original circle be $R$ and the sector angle be $\theta$ (in radians). Step 2: The area of the sector is: $$\frac{1}{2} R^2 \theta = 550$$ Step 3: When the sector is curved to form a cone, the arc length of the sector becomes the circumference of the cone's base: $$R \theta = 2 \pi r$$ Given $r = 7$ cm, so: $$R \theta = 14 \pi$$ Step 4: From Step 2, express $\theta$: $$\theta = \frac{1100}{R^2}$$ Step 5: Substitute $\theta$ into Step 3: $$R \times \frac{1100}{R^2} = 14 \pi \Rightarrow \frac{1100}{R} = 14 \pi \Rightarrow R = \frac{1100}{14 \pi}$$ Step 6: Calculate $R$: $$R \approx \frac{1100}{43.9823} \approx 25.01 \text{ cm}$$ Step 7: The slant height $l$ of the cone equals the radius of the original circle: $$l = R \approx 25.01 \text{ cm}$$ Step 8: Use Pythagoras theorem to find the height $h$ of the cone: $$h = \sqrt{l^2 - r^2} = \sqrt{25.01^2 - 7^2} = \sqrt{625.5 - 49} = \sqrt{576.5} \approx 24.01 \text{ cm}$$ Final answers: - Problem 6: Least possible number of families is 13. - Problem 7: Simplified expression is $\frac{x - 2y}{x + 2y}$. - Problem 8: Height of the cone is approximately 24.01 cm.