If 30% of (B - A) = 18% of (A + B), then the ratio A : B will be

Concept: An equation of the form “x% of P = y% of Q” translates directly into the linear equation (x/100)P = (y/100)Q. Once cleared of decimals, expanding and collecting like variable terms on each side reduces it to a simple ratio between the two quantities.

Application:

1. Write the given condition as an equation: 30% of (B − A) = 18% of (A + B), i.e. 0.30(B − A) = 0.18(A + B).

2. Multiply both sides by 100 to clear decimals: 30(B − A) = 18(A + B).

3. Divide both sides by 6 (the HCF of 30 and 18): 5(B − A) = 3(A + B).

4. Expand both brackets: 5B − 5A = 3A + 3B.

5. Collect all A-terms on one side and all B-terms on the other: 5B − 3B = 3A + 5A, which gives 2B = 8A.

6. Divide both sides by 2: B = 4A, so the ratio A : B = 1 : 4.

Cross-check: Take A = 1, B = 4. Then 30% of (B − A) = 30% of 3 = 0.9, and 18% of (A + B) = 18% of 5 = 0.9 — both sides match, confirming A : B = 1 : 4.