If 50% of the difference between two numbers equals 30% of their sum, then…

Concept

A statement of the form "p% of A equals q% of B" is just a linear equation in disguise: write it as (p/100)·A = (q/100)·B and clear the fractions.

When A and B are the difference and the sum of two unknowns, every term carries exactly one power of an unknown, so the equation is homogeneous. Homogeneous equations fix only the ratio of the two numbers, never their individual values — which is precisely why such a question can ask for a ratio and still have a unique answer.

Applying it here

1. Let the two numbers be x and y with x ≥ y, so their difference is x − y and their sum is x + y.

2. Translate the statement into an equation: 50% of (x − y) = 30% of (x + y), that is 0.5(x − y) = 0.3(x + y).

3. Multiply both sides by 10 to clear the decimals: 5(x − y) = 3(x + y).

4. Expand both sides: 5x − 5y = 3x + 3y.

5. Collect the x terms on one side and the y terms on the other: 5x − 3x = 3y + 5y, so 2x = 8y.

6. Divide both sides by 2y to isolate the ratio: x/y = 4, hence x : y = 4 : 1.

Cross-check

Substitute back with x = 4 and y = 1: the difference is 3 and 50% of it is 1.5, while the sum is 5 and 30% of it is 1.5. The two sides agree, so the ratio 4 : 1 is consistent with the given condition.

A faster route: the condition rearranges to (x − y)/(x + y) = 3/5, and componendo–dividendo then gives x : y = (5 + 3) : (5 − 3) = 8 : 2 = 4 : 1, the same result.

The ratio between the two numbers is 4 : 1.