If 25% of a first number is subtracted from a second number, the second number becomes four sixths (that is, two thirds) of its original value. What is the ratio of the first number to the second number?

Introduction / Context:This problem involves two unknown numbers related by a percentage subtraction condition. It tests your ability to translate percentage statements into algebraic equations and then derive a ratio of two quantities.

Given Data / Assumptions:

• Let the first number be A.• Let the second number be B.• Twenty five percent of A is subtracted from B.• After subtraction, B becomes four sixths of its original value, that is two thirds of B.• We must find the ratio A : B.

Concept / Approach:Percentages can be turned into fractions. Twenty five percent is 25/100 = 1/4. Four sixths simplifies to 2/3. We set up an equation for the new value of B after subtracting one quarter of A, express it as 2/3 of B and then solve for the relationship between A and B. Finally we write the ratio in simplest form.

Step-by-Step Solution:Step 1: Amount subtracted from B is 25% of A = (1/4) * A.Step 2: New value of B after subtraction = B − (A/4).Step 3: According to the statement, this equals four sixths (two thirds) of B.Step 4: So B − A/4 = (2/3) * B.Step 5: Rearrange: B − (2/3)B = A/4.Step 6: Left side gives (1/3)B = A/4.Step 7: Cross multiply: 4B = 3A.Step 8: So A : B = 4 : 3.

Verification / Alternative check:Pick simple numbers satisfying the ratio 4 : 3. Let A = 4 and B = 3. Then 25% of A = 1. New value of B after subtraction = 3 − 1 = 2. Two thirds of original B is 2. This matches the condition, so the ratio is validated.

Why Other Options Are Wrong:Ratios 2:3, 3:2 and 5:3 do not satisfy the equation B − 25% of A = (2/3)B when you test them with sample integer values, so they do not fit the given condition.

Common Pitfalls:Candidates sometimes treat four sixths as 4/6 without simplifying to 2/3, which does not change the answer but may confuse arithmetic. Another error is subtracting 25% of B instead of 25% of A.

Final Answer:The required ratio of the first number to the second number is 4 : 3.