A student scores 22 marks more in French than in German.\nHer German marks are 28% of the sum of her French and German marks.\nWhat are her marks in French?

Introduction / Context:
This problem is a classic example of using percentages and linear equations with two related subjects. The marks in French and German are connected by both a difference condition and a percentage condition. Solving such questions enhances algebraic thinking and is common in aptitude and school level exams.

Given Data / Assumptions:

• Let German marks be G and French marks be F.
• The student scores 22 marks more in French than in German, so F = G + 22.
• German marks are 28% of the sum of French and German marks, so G = 28% of (F + G).
• We must find the value of F, the French marks.

Concept / Approach:
We translate both conditions into algebraic equations. One equation comes from the difference F = G + 22. The second equation comes from the percentage statement G = 0.28 * (F + G). We then substitute the expression for F from the first equation into the second equation, solve for G, and finally compute F using F = G + 22. This two equation system is straightforward once written properly.

Step-by-Step Solution:
Step 1: From the difference condition, write F = G + 22. Step 2: From the percentage condition, write G = 0.28 * (F + G). Step 3: Substitute F from Step 1 into Step 2: G = 0.28 * ((G + 22) + G) = 0.28 * (2G + 22). Step 4: Expand the right side: G = 0.28 * 2G + 0.28 * 22 = 0.56G + 6.16. Step 5: Bring terms with G together: G - 0.56G = 6.16, which gives 0.44G = 6.16. Step 6: Solve for G: G = 6.16 / 0.44 = 14. Step 7: Use F = G + 22, so F = 14 + 22 = 36.

Verification / Alternative check:
Check both conditions with F = 36 and G = 14. The difference condition: French minus German = 36 - 14 = 22, which is correct. The percentage condition: F + G = 36 + 14 = 50. German marks as a percentage of total = 14 / 50 * 100 = 28%, which matches the statement. Both checks confirm the solution.

Why Other Options Are Wrong:
14: This is the value of German marks, not French marks.
18: Leads to F + G values and differences that do not satisfy the given equations.
42: F = 42 would not yield German marks that are exactly 28% of the total when combined with the difference condition.
28: This again fails one or both of the given relationships when substituted.

Common Pitfalls:
Learners sometimes reverse the percentage condition, treating French as 28% of the total instead of German. Others may forget to substitute correctly or mismanage the algebra when combining the two equations. Another frequent error is rounding too early; here the arithmetic yields exact integers if the steps are done carefully.

Final Answer:
The student scored 36 marks in French.