Step 1: Calculate initial value per part \(x\): \(1(x) + 0.5(2x) + 0.25(4x) = 400 \implies 3x = 400 \implies x = \frac{400}{3}\).
Step 2: New ratio is 4 : 2 : 1.
Step 3: New value = \(1(4x) + 0.5(2x) + 0.25(1x) = 5.25x\).
Step 4: Substitute \(x\): \(5.25 \times \frac{400}{3} = 1.75 \times 400 = 700\).

Step 1: Let Incomes be \(7x, 9x, 12x\) and Spending be \(8y, 9y, 15y\).
Step 2: A's saving = \(7x - 8y = \frac{1}{4}(7x) \implies 8y = \frac{21}{4}x \implies \frac{y}{x} = \frac{21}{32}\).
Step 3: Calculate individual savings: Savings = Income - Spending.
Step 4: A: \(7x - 8(\frac{21}{32}x) = \frac{56}{32}x\). B: \(9x - 9(\frac{21}{32}x) = \frac{99}{32}x\). C: \(12x - 15(\frac{21}{32}x) = \frac{69}{32}x\).
Step 5: Ratio = 56 : 99 : 69.

Step 1: \(A = \frac{1}{2}(B+C)\). Since \(A+B+C=7000\), then \(B+C = 7000-A\).
Step 2: \(A = \frac{1}{2}(7000-A) \implies 2A = 7000-A \implies 3A = 7000 \implies A = \frac{7000}{3}\).
Step 3: \(B = \frac{1}{4}(A+C) \implies B = \frac{1}{4}(7000-B) \implies 5B = 7000 \implies B = 1400\).
Step 4: \(C = 7000 - (\frac{7000}{3} + 1400) = 7000 - 3733.33 = 3266.67\).

Step 1: Find the number of other books: \(\frac{2}{7} \times 1512 = 432\).
Step 2: Let \(x\) be new story books. New ratio: \(\frac{1512 + x}{432} = \frac{15}{4}\).
Step 3: Cross-multiply: \(4(1512 + x) = 15 \times 432\).
Step 4: \(6048 + 4x = 6480 \implies 4x = 432 \implies x = 108\).

Step 1: Let \(a+b=6x, b+c=7x, c+a=8x\).
Step 2: Summing them: \(2(a+b+c) = 21x \implies 2(14) = 21x \implies 28 = 21x \implies x = \frac{4}{3}\).
Step 3: Find \(a+b\): \(6 \times \frac{4}{3} = 8\).
Step 4: Solve for \(c\): \(c = (a+b+c) - (a+b) = 14 - 8 = 6\).

Step 1: Let the number of notes be \(4x, 2x, 5x\).
Step 2: Set up the value equation: \(1(4x) + 5(2x) + 10(5x) = 1280\).
Step 3: Simplify: \(4x + 10x + 50x = 1280 \implies 64x = 1280\).
Step 4: Solve for \(x\): \(x = 20\).
Step 5: Number of $5 notes = \(2x = 2 \times 20 = 40\).

Step 1: Let alcohol be \(4x\) and water be \(3x\).
Step 2: Add 5L water: \(\frac{4x}{3x + 5} = \frac{4}{5}\).
Step 3: Cross-multiply: \(20x = 12x + 20\).
Step 4: Solve for \(x\): \(8x = 20 \implies x = 2.5\).
Step 5: Alcohol = \(4x = 4 \times 2.5 = 10 liters\).

Step 1: According to the properties of equal ratios, if \(\frac{x}{y} = \frac{z}{w} = k\), then any linear combination \(\frac{ax + cz}{ay + cw}\) will also equal \(k\).
Step 2: Here, \(k = \frac{2.5}{1.5}\).
Step 3: Simplify \(k\): \(\frac{25}{15} = \frac{5}{3}\).
Step 4: The ratio is 5 : 3.

Step 1: Let the number be \(10x + y\). Sum of digits = \(x + y\).
Step 2: Given \(\frac{10x + y}{x + y} = 4 \implies 10x + y = 4x + 4y \implies 6x = 3y \implies y = 2x\).
Step 3: Given \(y - x = 3\). Substitute \(y = 2x\): \(2x - x = 3 \implies x = 3\).
Step 4: Find \(y\): \(y = 2(3) = 6\).
Step 5: The number is 36.

Step 1: Use the rule of alligation. Difference between Gold and Alloy = \(19 - 15 = 4\).
Step 2: Difference between Alloy and Copper = \(15 - 9 = 6\).
Step 3: The ratio of Gold to Copper is the inverse of these differences: \(6 : 4\).
Step 4: Simplify: 3 : 2.