answer:Since triangle GHI is a right triangle with sides 6, 8, and 10, side 10 is the hypotenuse. Thus, the legs are 6 and 8, and the area is: [ text{Area of } triangle GHI = dfrac{1}{2} times 6 times 8 = 24 text{ units}^2 ] For triangle JKL, with sides 9, 40, and 41, it's deduced that 41 is the hypotenuse since (9^2 + 40^2 = 81 + 1600 = 1681 = 41^2). The other two sides are 9 and 40, forming the right angle. Thus, the area is: [ text{Area of } triangle JKL = dfrac{1}{2} times 9 times 40 = 180 text{ units}^2 ] We can now find the ratio of the areas of triangle GHI and triangle JKL: [ text{Ratio of areas} = dfrac{24}{180} = dfrac{4}{30} = dfrac{2}{15} ] Conclusion with boxed answer: [ boxed{dfrac{2}{15}} ]

answer:We are given the equation involving three integer variables ( x, y, z ): [ frac{1}{x} + frac{1}{y} + frac{1}{z} = frac{1}{x+y+z} ] 1. First, we assume that the denominators are non-zero (i.e., ( x, y, z, x+y+z neq 0 )). 2. Multiply by the common denominator ( x y z (x + y + z) ) to clear the fractions. This yields: [ (x + y + z)(xy + yz + zx) = xyz ] 3. Expand the left-hand side and simplify: [ x^2 y + x^2 z + y^2 x + y^2 z + z^2 x + z^2 y + xy^2 + xz^2 + yz^2 = xyz ] 4. Rearrange and combine like terms: [ x^2 (y + z) + y^2 (x + z) + z^2 (x + y) + xy^2 + xz^2 + yz^2 - xyz = 0 ] 5. Notice that the terms can be grouped and factored. Let's isolate the common factor ((y+z)), ((x+z)), and ((x+y)): [ (y + z)(x^2 + x(y + z) + yz) = 0 ] 6. Further factor the expression inside the parentheses: [ (y+z)(x^2 + xy + xz + yz) = (y+z)(x^2 + x(y+z) + yz) ] Given that ( x, y, z) are non-zero, the term ((y+z)(x+z)(x+y)) implies each individual factor should be either zero or a paired cancellation. Evaluating the potential solutions: 7. The non-trivial solutions arise when one of the fractions pairs up, providing a reciprocal equality: [ (y+z) = 0 quad text{or} quad (x+y) = 0 quad text{or} quad (x+z) = 0 ] Thus, ( y = -z ), ( x = -z ), or ( x = -y ). Conclusion: The solutions are exactly those triples ((x, y, z)) where two variables are the negation of each other, such as ((x, y, -x-y)). boxed{(x, y, -x-y)}

answer:Let's denote the time it takes for the first tap to fill the cistern as ( T ) hours. The rate at which the first tap fills the cistern is ( frac{1}{T} ) of the cistern per hour. The rate at which the second tap empties the cistern is ( frac{1}{10} ) of the cistern per hour. When both taps are opened simultaneously, the net rate at which the cistern gets filled is the rate of the first tap minus the rate of the second tap, which is ( frac{1}{T} - frac{1}{10} ). We are given that when both taps are opened simultaneously, the cistern gets filled in ( frac{20}{3} ) hours (which is the decimal 6.666666666666667 converted to a fraction). Therefore, the net rate at which the cistern gets filled when both taps are open is ( frac{1}{frac{20}{3}} = frac{3}{20} ) of the cistern per hour. Now we can set up the equation: [ frac{1}{T} - frac{1}{10} = frac{3}{20} ] To find ( T ), we solve for ( frac{1}{T} ): [ frac{1}{T} = frac{3}{20} + frac{1}{10} ] To add the fractions, we need a common denominator, which is 20: [ frac{1}{T} = frac{3}{20} + frac{2}{20} ] [ frac{1}{T} = frac{5}{20} ] [ frac{1}{T} = frac{1}{4} ] Now, we take the reciprocal of both sides to solve for ( T ): [ T = 4 ] So, the first tap takes boxed{4} hours to fill the cistern.

answer:1. **Calculate the area of the entire hexagon:** The area ( A ) of a regular hexagon with side length ( s ) is given by the formula: [ A = frac{3s^2 sqrt{3}}{2} ] Given ( s = 1 ), the area of the hexagon is: [ A = frac{3(1^2) sqrt{3}}{2} = frac{3sqrt{3}}{2} ] 2. **Determine the area of one of the shaded trapezoids:** The hexagon is divided into four regions of equal width. Each region is a trapezoid. The midpoints ( R, S, T, ) and ( U ) divide the hexagon into these trapezoids. The height of each trapezoid is ( frac{sqrt{3}}{2} ) because the height of the entire hexagon is ( sqrt{3} ) and it is divided into two equal parts. The bases of the trapezoid are the sides of the hexagon and the segments connecting the midpoints. The length of the longer base is ( 1 ) (side of the hexagon), and the length of the shorter base is ( frac{1}{2} ) (half the side of the hexagon). The area ( A_{text{trapezoid}} ) of a trapezoid is given by: [ A_{text{trapezoid}} = frac{1}{2} times (text{Base}_1 + text{Base}_2) times text{Height} ] Substituting the values: [ A_{text{trapezoid}} = frac{1}{2} times left(1 + frac{1}{2}right) times frac{sqrt{3}}{2} = frac{1}{2} times frac{3}{2} times frac{sqrt{3}}{2} = frac{3sqrt{3}}{8} ] 3. **Calculate the total area of the two shaded trapezoids:** Since there are two shaded trapezoids, the total shaded area is: [ A_{text{shaded}} = 2 times frac{3sqrt{3}}{8} = frac{6sqrt{3}}{8} = frac{3sqrt{3}}{4} ] 4. **Determine the fraction of the hexagon that is shaded:** The fraction of the hexagon that is shaded is the ratio of the shaded area to the total area of the hexagon: [ text{Fraction shaded} = frac{A_{text{shaded}}}{A_{text{hexagon}}} = frac{frac{3sqrt{3}}{4}}{frac{3sqrt{3}}{2}} = frac{3sqrt{3}}{4} times frac{2}{3sqrt{3}} = frac{1}{2} ] 5. **Express the fraction in the form ( frac{m}{n} ) and compute ( m + n ):** The fraction ( frac{1}{2} ) is already in its simplest form, where ( m = 1 ) and ( n = 2 ). Therefore: [ m + n = 1 + 2 = 3 ] The final answer is ( boxed{3} )