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Difficulty: Very hardOverlapping Sets and Venn Diagrams

At a software firm, an audit of 200 developers evaluated proficiency in three programming languages: Python, Java, and C++.

- 120 developers are proficient in Python.
- 105 developers are proficient in Java.
- 95 developers are proficient in C++.
- 15 developers are proficient in all three languages.
- 10 developers are proficient in none of the three languages.
- The ratio of the number of developers proficient in BOTH Python and Java ONLY to the number of developers proficient in BOTH Python and C++ ONLY is 3:23 : 2.
- The number of developers proficient in ONLY Java is equal to the number of developers proficient in ONLY C++.

How many developers are proficient in ONLY Python?

Answer: 55 developers

Answer

55 developers are proficient in ONLY Python.
By decomposing the 3-set Venn diagram into 7 mutually exclusive regions, the given ratio d:e=3:2d : e = 3 : 2 gives d=3kd = 3k and e=2ke = 2k. Comparing the totals for Java (105105) and C++ (9595) under the condition that Java-only equals C++-only shows that k=10k = 10. Thus, 3030 developers are in Python and Java only, 2020 are in Python and C++ only, and 1515 are in all three. Subtracting these three regions from the total 120120 Python developers yields 120−30−20−15=55120 - 30 - 20 - 15 = 55 proficient in Python only.

Step-by-Step Solution

1
Assign variables to the Venn diagram regions and incorporate given ratios.
Represent d=∣P∩J only∣=3kd = |P \cap J \text{ only}| = 3k and e=∣P∩C only∣=2ke = |P \cap C \text{ only}| = 2k, with g=15g = 15 and None=10\text{None} = 10.
Establishing explicit variables for non-overlapping regions simplifies system solving.
2
Set up set total equations for ∣J∣|J| and ∣C∣|C| to solve for kk.
∣J∣  ⟹  b+3k+f=90|J| \implies b + 3k + f = 90 and ∣C∣  ⟹  b+2k+f=80|C| \implies b + 2k + f = 80. Subtracting yields k=10k = 10.
Since b=cb = c (Java only = C++ only), subtracting the two set equations eliminates bb and ff, directly giving kk.
3
Calculate the region values dd and ee, then solve for a=∣P only∣a = |P \text{ only}|.
d=30d = 30, e=20e = 20. Then a=120−(30+20+15)=55a = 120 - (30 + 20 + 15) = 55.
Subtracting all other regions of set PP from the total proficient in Python isolates those proficient ONLY in Python.

Key Concept

Three-Set Overlapping Sets and Region Decomposition
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Topics

Overlapping Sets and Venn Diagrams

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