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question:Mary and James each sit in a row of 7 chairs. They choose their seats at random. What is the probability that they don't sit next to each other?

answer:There are a total of binom{7}{2}=21 ways for Mary and James to choose their seats. If they sit next to each other, there are 6 possible pairs of adjacent seats they could choose. So, the probability that they don't sit next to each other is 1-frac{6}{21}=boxed{frac{5}{7}}. The answer is: frac{5}{7}

question:Derrick measures the length of his yard. The length of Alex's yard is half the size of Derrick's and the length of Brianne's yard is 6 times the size of Alex's. If Brianne's yard is 30 yards long, how long is Derrick's yard, in yards?

answer:If Brianne's yard is 6 times the size of Alex's and Alex's yard is half the size of Derrick's, then Brianne's yard is 6 * (1/2) = 3 times the size of Derrick's yard. If Brianne's yard is 30 yards long, then Derrick's yard is 30 / 3 = 10 yards long. 10 The answer is: 10

question:Given a rectangular prism with length l, width w, and height h, the length of its diagonal can be found using the equation sqrt{l^2 + w^2 + h^2}. If the length (diagonal) is 13 and the length and height are 3 and 12 respectively, what is the width?

answer:We are given that the diagonal of the rectangular prism is 13, the length is 3, and the height is 12. Using the equation for the diagonal, we have sqrt{l^2 + w^2 + h^2} = 13. Substituting the given values, we have sqrt{3^2 + w^2 + 12^2} = 13. Simplifying, we have sqrt{9 + w^2 + 144} = 13. Combining like terms, we have sqrt{w^2 + 153} = 13. Squaring both sides, we have w^2 + 153 = 169. Subtracting 153 from both sides, we have w^2 = 16. Taking the square root of both sides, we have w = pm 4. Since the width cannot be negative in this context, the width of the rectangular prism is boxed{4}.The answer is: 4

question:Angle PQR is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square? [asy] draw((0,0)--(12,0)--(0,5)--cycle); dot((0,0)); dot((12,0)); dot((0,5)); draw((0,0)--(0,5)--(-5,5)--(-5,0)--cycle); draw((0,0)--(0,-12)--(12,-12)--(12,0)); draw((0,5)--(5,17)--(17,12)--(12,0)--cycle); label("P",(0,5),NW); label("Q",(0,0),SE); label("R",(12,0),E); [/asy]

answer:Let s_1, s_2, and s_3 represent the side lengths of the three squares, respectively. Using the Pythagorean Theorem, we have s_1^2 + s_2^2 = s_3^2. We are given that s_1^2 + s_2^2 + s_3^2 = 338. Combining these two equations, we have 2s_3^2 = 338, so s_3^2 = 169. Therefore, s_3 = 13, and the area of the largest square is s_3^2 = boxed{169} square centimeters. The answer is: 169