Simplify the mixed-number expression 5(2/7) − 3(1/14) − 2(1/14) − 1(1/7) and express the result in the form ? + 3(1/17); what is the value of the unknown mixed number represented by ?

Introduction / Context:
This question tests the ability to work with mixed numbers, convert them to improper fractions, perform subtractions, and then re express a result in terms of another mixed number. Such problems appear in aptitude exams to check fraction manipulation skills and careful handling of signs. The structure ? + 3(1/17) encourages candidates to think algebraically about the final result.

Given Data / Assumptions:
- Expression: 5(2/7) − 3(1/14) − 2(1/14) − 1(1/7).
- This entire value equals ? + 3(1/17).
- All mixed numbers are real positive or negative numbers written as whole part plus fractional part.
- We must find the mixed number represented by ?.

Concept / Approach:
First, convert each mixed number to an improper fraction. Then perform the subtractions step by step, making sure to use a common denominator. After finding the final result as a single fraction, convert the right hand side into an expression involving ? and 3(1/17). Because 3(1/17) is known, we subtract it from the total value to isolate ?. Finally, we convert that fraction back to a mixed number and match it with the given options.

Step-by-Step Solution:
Convert 5(2/7) to an improper fraction: (5*7 + 2)/7 = 37/7. Convert 3(1/14) to an improper fraction: (3*14 + 1)/14 = 43/14. Convert 2(1/14) to an improper fraction: (2*14 + 1)/14 = 29/14. Convert 1(1/7) to an improper fraction: (1*7 + 1)/7 = 8/7. Rewrite 37/7 and 8/7 with denominator 14: 37/7 = 74/14, 8/7 = 16/14. Now total expression = 74/14 − 43/14 − 29/14 − 16/14. Combine numerators: 74 − 43 − 29 − 16 = 74 − 88 = −14. So the value of the expression is −14/14 = −1. We have −1 = ? + 3(1/17). Convert 3(1/17) to improper fraction: (3*17 + 1)/17 = 52/17. Write −1 as −17/17. Then ? = −17/17 − 52/17 = −69/17. Convert −69/17 to a mixed number: 69 ÷ 17 = 4 remainder 1, so ? = −4(1/17).

Verification / Alternative check:
We can check by adding ? and 3(1/17). ? = −69/17 and 3(1/17) = 52/17, so their sum is (−69 + 52)/17 = −17/17 = −1. This matches the value of the original expression, confirming that ? is indeed −4(1/17).

Why Other Options Are Wrong:
Option 5(2/17) is positive and would make the sum with 3(1/17) positive, not −1.
Option −2(1/17) corresponds to −35/17, which gives −35/17 + 52/17 = 17/17 = 1, not −1.
Option 3(3/17) corresponds to a positive number, again incompatible with the negative total of the original expression.

Common Pitfalls:
Typical errors include converting mixed numbers to improper fractions incorrectly, forgetting to use a common denominator, or mishandling the negative sign when combining fractions. Another common mistake is to misinterpret the equation structure and not realise that the entire left hand side equals ? plus 3(1/17). Working slowly through each conversion step helps avoid these issues.

Final Answer:
The value of the unknown mixed number ? is -4 (1/17).