A Scientific Research Study Examined a Large Number of Young Foxes, that is, Foxes Between 1 Year and 2 Years Old. GMAT Problem Solving

Question: A scientific research study examined a large number of young foxes, that is, foxes between 1 year and 2 years old. The study found that 80% of the young foxes caught a rabbit at least once, and 60% caught a songbird at least once If 10% of the young foxes never caught either a rabbit or a songbird, then what percentage of young foxes were successful in catching at least one rabbit and at least one songbird?

1. 40%
2. 50%
3. 60%
4. 80%
5. 90%

“A scientific research study examined a large number of young foxes, that is, foxes between 1 year and 2 years old.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

This sum can be solved by using the Venn diagram. Let us consider two circles overlapping themselves. The left one is rabbits and the right one is songbirds. The intersection part is common for both rabbits and songbirds. Now as the question states,
Let the number of rabbits be 80-(R+S)
And the number of songbirds be 60-(R+S)
And the common part be (R+S)
Hence the equation becomes,
60 + 80 - (Both) = 90
This implies 140 - Both = 90
This implies Both = 50.
Hence the percentage of young foxes who were successful in catching at least one rabbit and at least one songbird is 50.

Correct Answer: B

Approach Solution 2: 

This can be solved using the probability OR rule. Let us consider R to be the event that a young fox catches at least one rabbit. Let us consider S to be the event that a young fox catches at least one songbird.
Then using algebraic probability notation, we know P(R) = 0.8 and P(S) = 0.6.
It is to be noted that P((not R) and (not S)) = 0.1
The complement of [(not R) and (not S)] would be [R or S], so by the complement rule, P(R or S) = 1 – 0.1 = 0.9.
Hence the question inquiring P(R and S). Hence the OR rule implies that
P(R or S) = P(R) + P(S) – P(R and S)
0.9 = 0.6 + 0.8 – P(R and S)
0.9 = 1.4 – P(R and S)
0.9 + P(R and S) = 1.4
P(R and S) = 0.5
Hence the percentage of young foxes who were successful in catching at least one rabbit and at least one songbird is 50.

Correct Answer: B

Approach Solution 3: 

Consider two circles that cross one other. Consider R to be the occurrence of at least one rabbit being caught by a young fox. Consider S to be the occurrence of at least one songbird being caught by a young fox. As mentioned in the question,
The number of songbirds should be 60-(R+S) and the number of rabbits should be 80-(R+S).
The common component is (R+S).
Consequently, the formula becomes 60 + 80 - (Both) = 90.
It follows that 140 - Both = 90.
Thus, Both must equal 50.
In light of this, 50 percent of young foxes were successful in catching at least one rabbit and at least one songbird.

Correct Answer: B

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