The difference between two numbers is 5 and six times the smaller number is 6 less than four times the greater number; find the values of the two numbers.

Introduction / Context:
This problem is a standard example of forming and solving simultaneous linear equations from a word description. Arithmetic reasoning questions like this test the ability to translate relationships stated in words into algebraic equations and then solve them in a structured way. The information about a difference and a comparison of multiples of the numbers is sufficient to determine both values exactly.

Given Data / Assumptions:
- There are two real numbers, one greater and one smaller.
- The difference between the greater number and the smaller number is 5.
- Six times the smaller number is 6 less than four times the greater number.
- We assume the numbers are positive, consistent with typical aptitude questions, but the algebra works for general real numbers.

Concept / Approach:
We introduce variables to represent the greater and the smaller number, then carefully convert each English statement into an algebraic equation. The phrase difference between two numbers is 5 becomes one equation, and the phrase six times the smaller lacks by 6 from four times the greater becomes the second equation. Solving this pair of simultaneous equations gives unique values for the two numbers. This is a systematic approach widely applicable to many linear word problems.

Step-by-Step Solution:
Step 1: Let the greater number be x and the smaller number be y. Step 2: The difference condition gives x - y = 5. Step 3: The statement six times the smaller lacks by 6 from four times the greater means that four times the greater is six more than six times the smaller. So we write 4 * x = 6 * y + 6. Step 4: Rearrange the second equation as 4x - 6y = 6. Step 5: From x - y = 5, express x in terms of y as x = y + 5. Step 6: Substitute x = y + 5 into 4x - 6y = 6 to get 4(y + 5) - 6y = 6. Step 7: Expand and simplify: 4y + 20 - 6y = 6 gives -2y + 20 = 6, so -2y = -14 and therefore y = 7. Step 8: Substitute y = 7 back into x = y + 5 to obtain x = 12. Step 9: Thus the greater number is 12 and the smaller number is 7.

Verification / Alternative check:
Check the difference: 12 - 7 = 5, which matches the first condition. Check the second condition: six times the smaller is 6 * 7 = 42 and four times the greater is 4 * 12 = 48. The phrase lacks by 6 means 42 is 6 less than 48, which is correct. Therefore both conditions are satisfied exactly by the pair (12, 7), confirming our solution. No other option pair given in the choices simultaneously satisfies both equations.

Why Other Options Are Wrong:
Option 13 and 8: The difference is 5, but 6 * 8 = 48 and 4 * 13 = 52, and 48 is not exactly 6 less than 52.
Option 15 and 10: The difference is 5, but 6 * 10 = 60 and 4 * 15 = 60, so there is no difference of 6 between these multiples.
Option 14 and 9: The difference is again 5, but 6 * 9 = 54 and 4 * 14 = 56, which differ by only 2, not 6.
Option 17 and 12: These differ by 5, but 6 * 12 = 72 and 4 * 17 = 68, making the smaller multiple greater than the larger multiple, contrary to the condition given.

Common Pitfalls:
One common mistake is misreading the phrase lacks by 6 and writing 6y = 4x + 6 instead of 4x = 6y + 6. Another error is reversing the difference equation as y - x = 5, which changes the sign and leads to incorrect values. Some learners also try to guess and check options without forming equations, which can be time consuming and error prone for more complex problems. Writing variables, forming equations carefully, and then solving them systematically is the most reliable method.

Final Answer:
The two numbers are 12 and 7.