In a survey of 200 agricultural exporters regarding three major commodities—Cocoa (C), Pal… — JAMB UTME

In a survey of $200$ agricultural exporters regarding three major commodities—Cocoa ( $C$ ), Palm Oil ( $P$ ), and Rubber ( $R$ )—it was found that $110$ export Cocoa, $90$ export Palm Oil, and $75$ export Rubber. Exactly $20$ exporters export none of the three commodities, and $45$ export Cocoa only. Furthermore, the number of exporters who export Cocoa and Palm Oil only is twice the number of exporters who export all three commodities; the number who export Palm Oil and Rubber only is equal to the number who export all three; and the number who export Cocoa and Rubber only is $5$ more than the number who export all three. Find the total number of exporters who export at least two of the three commodities.

Answer: 80

Answer

The total number of exporters who export at least two of the three commodities is 80.

The correct answer of 80 is obtained by solving for the number of exporters trading in all three commodities (

x = 15

) using the Cocoa set equation

45 + 2x + (x + 5) + x = 110

, and then evaluating the region sum for at least two commodities (

2x + x + (x + 5) + x = 5x + 5 = 80

Step-by-Step Solution

Assign a variable to the triple intersection

Let

x = n(C \cap P \cap R)

be the number of exporters of all three commodities.

The intersection of all three sets serves as the common parameter for all double-intersection regions.

Write algebraic expressions for the three pairwise-only intersections

n(C \cap P \text{ only}) = 2x

n(P \cap R \text{ only}) = x

, and

n(C \cap R \text{ only}) = x + 5

These expressions are derived directly from the relationships given in the problem statement.

Formulate and solve an equation using the set of Cocoa exporters

45 + 2x + (x + 5) + x = 110 \implies 50 + 4x = 110 \implies x = 15

The set of Cocoa exporters consists of four mutually exclusive regions whose cardinalities sum to 110.

Sum the regions corresponding to 'at least two commodities'

(2x) + (x) + (x + 5) + x = 5x + 5 = 5(15) + 5 = 80

'At least two' encompasses everyone who exports exactly two commodities plus those who export all three.

Key Concept

Three-set principle of inclusion-exclusion and cardinal region decomposition

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