Question

In a survey of 250 people about three activities (Reading, Sports, Travel), the following data are given:
130 like Reading,
110 like Sports,
120 like Travel,
55 like both Reading and Sports,
50 like both Sports and Travel,
45 like both Reading and Travel,
25 like all three.
How many like only Travel?

• A. 35
• B. 40
• C. 45
• D. 50

Hint:

For “only” in three-set Venn problems, always subtract the two-way intersections and add back the three-way intersection.

Answer 1

The Correct Option is D

Approach: “Only Travel” means inside the Travel circle but outside the other two. Start from the full Travel count and peel off the two overlaps, then add back the triple you removed twice.

Step 1: Write the inclusion–exclusion form for one set. \[ \text{Only }T = T - (R\cap T) - (S\cap T) + (R\cap S\cap T). \] Intuition: subtracting \((R\cap T)\) and \((S\cap T)\) removes the all-three region twice, so we add it back once.

Step 2: Plug in the data. \(T = 120\), \(R\cap T = 45\), \(S\cap T = 50\), \(R\cap S\cap T = 25\).

Step 3: Compute. \[ \text{Only }T = 120 - 45 - 50 + 25 = 50. \]

Final Answer: 50 (option 4).

Answer 2

Approach: Instead of applying the "only Travel" formula directly, solve for every region of the Venn diagram as unknowns using a small system of equations, and cross-check the total against the 250 surveyed people.

Step 1: Set up regions. Let a = only Reading, b = only Sports, c = only Travel, d = Reading&Sports only, e = Sports&Travel only, f = Reading&Travel only, and all three = 25.

Step 2: Use the pairwise totals. Reading&Sports = 55 ⇒ d = 55-25 = 30. Sports&Travel = 50 ⇒ e = 50-25 = 25. Reading&Travel = 45 ⇒ f = 45-25 = 20.

Step 3: Use the individual totals. Reading = 130 = a+d+f+25 ⇒ a = 130-30-20-25 = 55. Sports = 110 = b+d+e+25 ⇒ b = 110-30-25-25 = 30. Travel = 120 = c+e+f+25 ⇒ c = 120-25-20-25 = 50.

Step 4: Sanity-check against the 250 total. Sum of all seven regions = 55+30+50+30+25+20+25 = 235, leaving 250-235 = 15 people who like none of the three activities, a reasonable, non-negative number that confirms the solution is consistent.

Final Answer: Only Travel = c = 50 (option 4).