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question:A farm has 100 chickens. 20 percent are Black Copper Marans, and 80 percent of the Black Copper Marans are hens. How many BCM hens are there?
answer:20 percent of 100 chickens is 20/100 * 100 = 20 Black Copper Marans. 80 percent of the Black Copper Marans are hens, so there are 80/100 * 20 = 16 Black Copper Maran hens. 16 The answer is: 16
question:Let O be the origin. A variable plane has a distance of 1 from the origin, and intersects the x-axis, y-axis, and z-axis at A, B, and C, respectively, all distinct from O. Let (p,q,r) be the centroid of triangle ABC. Find [frac{1}{p^2} + frac{1}{q^2} + frac{1}{r^2}.]
answer:ane be ax + by + cz = 1. Then the coordinates of A, B, and C are left( frac{1}{a}, 0, 0 right), left( 0, frac{1}{b}, 0 right), and left( 0, 0, frac{1}{c} right), respectively. The centroid of triangle ABC is then [left( frac{1/a}{3}, frac{1/b}{3}, frac{1/c}{3} right) = left( frac{1}{3a}, frac{1}{3b}, frac{1}{3c} right).]Thus, frac{1}{p^2} = left( frac{3a}{1} right)^2 = 9a^2, frac{1}{q^2} = left( frac{3b}{1} right)^2 = 9b^2, and frac{1}{r^2} = left( frac{3c}{1} right)^2 = 9c^2, so [frac{1}{p^2} + frac{1}{q^2} + frac{1}{r^2} = 9a^2 + 9b^2 + 9c^2 = 9(a^2 + b^2 + c^2).]By the distance formula, [a^2 + b^2 + c^2 = frac{1}{a^2} + frac{1}{b^2} + frac{1}{c^2} = frac{1}{p^2} + frac{1}{q^2} + frac{1}{r^2} = boxed{9}.]The answer is: 9
question:Luca went to a sandwich shop for lunch. The sandwich he bought was normally 8, but he had a coupon for a quarter of the price off. He then upgraded it with sliced avocado for an extra dollar. After adding a drink and a 3 salad, his total lunch bill was x. Luca paid 2 dollars for his drink. What is the value of unknown variable x? What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents Luca's total lunch bill. Let's break down the information given: Normal price of the sandwich: 8 Discounted price of the sandwich: 8 - (1/4 * 8) Price of adding sliced avocado: 1 Price of the drink: 2 Price of the salad: 3 Total lunch bill: x We can set up the equation as follows: Discounted price of the sandwich + Price of adding sliced avocado + Price of the drink + Price of the salad = Total lunch bill 8 - (1/4 * 8) + 1 + 2 + 3 = x Let's simplify and solve for x: 8 - (1/4 * 8) + 1 + 2 + 3 = x 8 - 2 + 1 + 2 + 3 = x 12 = x The value of x is 12. 12 The answer is: 12
question:Melody needs to read 20 pages for her English class, 16 pages for her Science class, x pages for her Civics, and 12 pages for Chinese class. She decided to read one-fourth of the number of pages for each class tomorrow. How many pages will she read tomorrow? If we know the answer to the above question is 14, what is the value of unknown variable x?
answer:Melody needs to read 20 pages for her English class, 16 pages for her Science class, x pages for her Civics class, and 12 pages for her Chinese class. She decided to read one-fourth of the number of pages for each class tomorrow. For her English class, she will read (1/4)*20 = 5 pages. For her Science class, she will read (1/4)*16 = 4 pages. For her Civics class, she will read (1/4)*x = x/4 pages. For her Chinese class, she will read (1/4)*12 = 3 pages. The total number of pages she will read tomorrow is the sum of the number of pages for each class: 5 + 4 + x/4 + 3. We are given that the total number of pages she will read tomorrow is 14, so we can write: 5 + 4 + x/4 + 3 = 14. Simplifying the left side, we get: 12 + x/4 = 14. Subtracting 12 from both sides, we get: x/4 = 2. Multiplying both sides by 4, we get: x = 8. The value of x is 8. 8 The answer is: 8