In an examination, the average marks obtained by John in English, Mathematics, Hindi and Drawing together is 50. His average marks in Mathematics, Science, Social Studies and Craft together is 70. If his average marks across all seven subjects is 58, what were his marks in Mathematics?

Introduction / Context:
This problem involves overlapping averages across different sets of subjects. John has one average across four subjects, another average across a different set of four subjects, and a final average across all seven subjects. You are asked to determine his mark in Mathematics, which is common to both four subject groups. This is a standard linear equations and averages problem.

Given Data / Assumptions:
- Subjects: English (E), Mathematics (M), Hindi (H), Drawing (D), Science (Sc), Social Studies (SS), Craft (Cr).
- Average in English, Mathematics, Hindi, Drawing = 50.
- Average in Mathematics, Science, Social Studies, Craft = 70.
- Average across all seven subjects = 58.
- We must find the mark in Mathematics M.

Concept / Approach:
We convert each average into a sum. From the first average we get a sum for E + M + H + D. From the second average we get a sum for M + Sc + SS + Cr. From the overall average we get a sum for all seven subjects together. Adding the first two sums counts Mathematics twice and all other subjects once. By comparing with the total sum, we can isolate and solve for M.

Step-by-Step Solution:
Step 1: From the first average, (E + M + H + D) / 4 = 50, so E + M + H + D = 4 * 50 = 200.Step 2: From the second average, (M + Sc + SS + Cr) / 4 = 70, so M + Sc + SS + Cr = 4 * 70 = 280.Step 3: From the overall average, (E + M + H + D + Sc + SS + Cr) / 7 = 58, so total sum T = 7 * 58 = 406.Step 4: Add the first two sums: (E + M + H + D) + (M + Sc + SS + Cr) = 200 + 280 = 480.Step 5: This combined expression equals (E + H + D + Sc + SS + Cr) + 2M.Step 6: But total sum T is E + H + D + Sc + SS + Cr + M = 406.Step 7: Therefore, 480 = T + M = 406 + M.Step 8: So M = 480 - 406 = 74.Step 9: John scored 74 marks in Mathematics.

Verification / Alternative check:
Now verify quickly. If M = 74, then E + H + D = 200 - 74 = 126 and Sc + SS + Cr = 280 - 74 = 206. Total sum T = 126 + 74 + 206 = 406. Average across seven subjects is 406 / 7 = 58, which matches the given overall average. This confirms that the solution is consistent.

Why Other Options Are Wrong:
Values like 50, 52, 60 or 68 do not satisfy the relation 480 = 406 + M. For example, if M were 60, the sum from the first four subjects would force E + H + D = 140, and the second group sum would force Sc + SS + Cr = 220, giving a total of 420, not 406. Hence these alternative values do not produce the correct overall average.

Common Pitfalls:
Some students try to average the two four subject averages directly or forget that Mathematics is counted twice when adding the two partial sums. Others miscalculate the total for seven subjects from the overall average. Carefully writing each sum and noting that M appears in both expressions avoids these errors.

Final Answer:
John scored 74 marks in Mathematics.