Let 3x = 5y = 4z = k. Then x = k/3, y = k/5, z = k/4. The ratio is 1/3 : 1/5 : 1/4. Multiply by the LCM of 3, 5, 4 (which is 60) to get 20 : 12 : 15.

Let x be added. (6 + x) / (15 + x) = (20 + x) / (43 + x). Cross-multiplying: (6 + x)(43 + x) = (15 + x)(20 + x) => 258 + 49x + x^2 = 300 + 35x + x^2. Subtract x^2 from both sides: 49x - 35x = 300 - 258 => 14x = 42 => x = 3.

Since a/b = c/d = e/f = 1/2, we have a = b/2, c = d/2, and e = f/2. Substituting these into the numerator: 3(b/2) + 5(d/2) + 7(f/2) = 0.5 * (3b + 5d + 7f). The ratio of numerator to denominator is 0.5, which is 1 : 2.

Let length be 5x and breadth be 4x. Difference = 5x - 4x = x. Given x = 20 m. Length = 100 m, breadth = 80 m. Perimeter = 2 * (L + B) = 2 * (100 + 80) = 2 * 180 = 360 m.

Let the fraction be x. x : (1/27) = (3/11) : (5/9). So, x / (1/27) = (3/11) / (5/9). 27x = (3/11) * (9/5) = 27/55. Dividing both sides by 27 gives x = 1/55.

Let incomes be 3x and 2x, and expenditures be 5y and 3y. 3x - 5y = 1000 and 2x - 3y = 1000. Equating them: 3x - 5y = 2x - 3y => x = 2y. Substitute x in the first equation: 3(2y) - 5y = 1000 => y = 1000. So x = 2000. A's income = 3x = Rs. 6000.

Let numbers be 5x and 9x. (5x + 9) / (9x + 9) = 16 / 27. Cross-multiplying: 27(5x + 9) = 16(9x + 9) => 135x + 243 = 144x + 144 => 9x = 99 => x = 11. The second number is 9x = 9 * 11 = 99.

Let the shares be 5x, 2x, 4x, and 3x. C's share - D's share = 4x - 3x = x. We are given x = 1000. B's share is 2x = 2 * 1000 = Rs. 2000.

10% of x = 20% of y translates to 0.1x = 0.2y. Multiply both sides by 10 to get x = 2y. Therefore, the ratio x / y = 2 / 1, which is 2 : 1.

Let the numbers be 3x and 8x. Their difference is 8x - 3x = 5x. Given 5x = 115, solving gives x = 23. The larger number is 8x = 8 * 23 = 184.