answer:Since AT/DT=2, we have AT = 2DT, and let DT = x, then AT = 2x. For BT/ET=3, let ET=y, then BT = 3y. From Menelaus' theorem with transversals overline{AD} and overline{BE}, the ratios on line segments satisfy: [ frac{AT}{TD} cdot frac{DB}{BC} cdot frac{CE}{EA} = 1 ] [ frac{BT}{TE} cdot frac{EC}{CA} cdot frac{AD}{DB} = 1 ] Given AT/DT = 2, AT = 2AD, TD = AD. Therefore, frac{AD}{DT} = 2. Placing values from ratios into Menelaus' theorem: [ 2 cdot frac{DB}{BC} cdot frac{CE}{EA} = 1 ] [ 3 cdot frac{EC}{CA} cdot frac{AD}{DB} = 1 ] Rearranging the equations to focus on components we need: [ frac{DB}{BC} = frac{1}{2} left( frac{EA}{CE} right) ] [ frac{EC}{CA} = frac{1}{3} left( frac{DB}{AD} right) ] With AD = 2DB, substitute: [ frac{EC}{CA} = frac{1}{3} left( frac{DB}{2DB} right) = frac{1}{6} ] Set BC = DB + CD. Using the ratios and solving for what we need, utilizing components and their relationships: [ CD = BC - BD = 2BD - BD = BD ] Consequently, ( frac{CD}{BD} = frac{BD}{BD} = 1). Hence, the value is: [boxed{1}]

answer:Let's assume the amount invested by B is x. According to the problem, A invests 3 times as much as B, so A's investment is 3x. It is also given that A invests 2/3 of what C invests. So if C's investment is y, then A's investment (which is 3x) is equal to 2/3 of y. So we have: 3x = (2/3)y From this, we can solve for y in terms of x: y = (3/2) * 3x y = (9/2)x Now, the profit is divided among A, B, and C in the ratio of their investments. So the total investment is: Total Investment = A's investment + B's investment + C's investment Total Investment = 3x + x + (9/2)x Total Investment = (8/2)x + (9/2)x Total Investment = (17/2)x The share of each partner is then: A's share = (3x / Total Investment) * Profit B's share = (x / Total Investment) * Profit C's share = ((9/2)x / Total Investment) * Profit We want to find C's share, so we plug in the values: C's share = ((9/2)x / (17/2)x) * 55000 C's share = (9/17) * 55000 Now we calculate C's share: C's share = 9/17 * 55000 C's share = 9 * 3235.29411764706 (approx) C's share = Rs. 29117.65 (approx) So, C's share of the profit is approximately Rs. boxed{29117.65} .

answer:First, let's find the area of the rectangle. The area of a rectangle is found by multiplying its length by its width: Area of rectangle = length * width Area of rectangle = 32 cm * 10 cm Area of rectangle = 320 cm² The area of the square is five times the area of the rectangle: Area of square = 5 * Area of rectangle Area of square = 5 * 320 cm² Area of square = 1600 cm² Since the area of a square is equal to the side length squared (Area = side²), we can find the side length of the square by taking the square root of the area: side = √Area of square side = √1600 cm² side = 40 cm Now that we know the side length of the square, we can find the perimeter of the square. The perimeter of a square is found by adding up all four sides: Perimeter of square = 4 * side Perimeter of square = 4 * 40 cm Perimeter of square = 160 cm Therefore, the perimeter of the square is boxed{160} cm.

answer:Let's call Julio's current age J. In 14 years, James will be 11 + 14 = 25 years old. According to the problem, in 14 years, Julio's age will be twice James' age. So, in 14 years, Julio will be 2 * 25 = 50 years old. To find out Julio's current age, we subtract 14 years from his age in 14 years: J + 14 = 50 J = 50 - 14 J = 36 So, Julio is currently boxed{36} years old.