Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third…

Concept: To find a ratio A : B from a linear relation between percentages of A and B, convert every percentage term to a fraction, clear the denominators by multiplying through by a common factor, and collect all A-terms and B-terms on opposite sides. The resulting equation of the form p·A = q·B directly gives A : B = q : p.

Application:

1. Write the given condition as an equation: 5% of A + 4% of B = (2/3)(6% of A + 8% of B).

2. Convert every percentage to a fraction: 5% = 1/20, 4% = 1/25, 6% = 3/50, 8% = 2/25. So (1/20)A + (1/25)B = (2/3)[(3/50)A + (2/25)B].

3. Simplify the right-hand side: (2/3)(3/50) = 1/25 and (2/3)(2/25) = 4/75. So (1/20)A + (1/25)B = (1/25)A + (4/75)B.

4. Collect A-terms on the left and B-terms on the right: (1/20 − 1/25)A = (4/75 − 1/25)B.

5. Simplify each bracket: 1/20 − 1/25 = 1/100, and 4/75 − 1/25 = 1/75. So (1/100)A = (1/75)B.

6. Solve for the ratio: A/B = 100/75 = 4/3, i.e., A : B = 4 : 3.

Cross-check: Verification: with A = 4, B = 3 — LHS = 5% of 4 + 4% of 3 = 0.20 + 0.12 = 0.32; RHS = (2/3)(6% of 4 + 8% of 3) = (2/3)(0.24 + 0.24) = (2/3)(0.48) = 0.32. Both sides match, confirming A : B = 4 : 3.

Result: Hence, the ratio A : B = 4 : 3.