Find the ratio of two numbers whose terms differ by 40 and whose value is equal to 2/7.

Introduction / Context:
This aptitude question tests the concept of ratios, specifically how to construct a ratio when its numerical value is given and the difference between its terms is known. Such questions are very common in competitive exams because they combine basic algebra with the idea of comparing two related quantities in a simple ratio form.

Given Data / Assumptions:
• The value (measure) of the ratio is 2/7.
• Let the two terms of the ratio be the first number and the second number.
• The difference between these two terms is 40.
• We assume both numbers are positive and the larger term is associated with the larger part of the ratio.

Concept / Approach:
If two numbers are in the ratio 2:7, then they can be written as 2k and 7k for some positive constant k. The difference between the numbers is then 7k - 2k = 5k. By equating this algebraic difference to the given actual difference, we can solve for k and then calculate the actual numbers. Finally, we can express the required ratio using those actual numbers, which will simplify back to the same 2:7 form but with concrete terms that differ by 40.

Step-by-Step Solution:
Step 1: Let the two numbers be 2k and 7k because the ratio is 2:7.Step 2: Their difference is 7k - 2k = 5k.Step 3: We are told that the difference between the two numbers is 40, so set 5k = 40.Step 4: Solve for k: k = 40 / 5 = 8.Step 5: Substitute back to get the actual numbers: first number = 2k = 2 * 8 = 16; second number = 7k = 7 * 8 = 56.Step 6: Hence the required ratio whose terms differ by 40 and whose value is 2/7 is 16:56.

Verification / Alternative check:
Check the ratio value: 16/56 simplifies by dividing numerator and denominator by 8. We get 2/7, which matches the required measure. Check the difference: 56 - 16 = 40, which matches the given difference. Both conditions are satisfied, so 16:56 is consistent and correct. No other simple multiple of 2:7 gives a difference of exactly 40, which confirms uniqueness in this context.

Why Other Options Are Wrong:
Option 14:56 gives a ratio value of 14/56 = 1/4, not 2/7, and the difference is 42, not 40, so this does not satisfy either condition.
Option 15:56 gives a ratio 15/56, which does not reduce to 2/7, and the difference is 41, so it fails both the ratio value and difference conditions.
Option 16:72 has difference 56, not 40, and 16/72 reduces to 2/9, not 2/7, so it is also incorrect.

Common Pitfalls:
A common mistake is to subtract the parts of the ratio directly (7 - 2 = 5) and think that 5 is the difference of the actual numbers instead of 5k. Another error is to forget to multiply both terms of the ratio by the same constant and thus not link the algebraic difference 5k to the real difference 40. Some students also pick 2:7 as the final answer without checking the actual difference in terms, which would be incorrect because 2 and 7 differ only by 5, not 40.

Final Answer:
The required ratio with value 2/7 whose terms differ by 40 is 16:56.