The staff of a school planned a picnic. Among them, 45 suggested Godawari and 50 suggested Dhulikhel. 24 suggested other places than these and 15 agreed in any of these. If G and D denote the sets of staff who suggested Godawari and Dhulikhel respectively, answer the following questions. a. $$ n(G \cup D), n(G), n(D) $$ र $$ n(G \cap D) $$ विचको सम्बन्ध लेखुनुहोस्। Write a relation between $$ n(G \cup D), n(G), n(D) $$ and $$ n(G \cap D) $$. [1] b. उक्त विद्यालयमा कति कर्मचारी रहेछन् ? भेनचित्र वनाई पत्ता लगाउनुहोस् । How many staffs were there in the school? Calculate by drawing a Venn-diagram. FB-Tuition Class of Rajan Shrestha [3] c. एक ठाउंमात्र सुभाउने कति कर्मचारी रहेछन् ? How many of them were rigid in only one place? [1] d. गोदावरी छान्ने कर्मचारीहरूको सङ्य्या धुलिखेल छन्नेहरूको भन्दा कति प्रतिशत कम वा वढी रहेछ ? By what percent, the staff selecting Godawari is more or less than the staff selecting Dhulikhel? [1] वार्षिक $$ 12 \% $$ व्याजदरले 2 वर्षमा कुनै रकमको साधारण व्याज वार्षिक चक्रीय व्याज भन्दा रु 36 ले कम हुन आउंछ। The simple interest on a sum of money at the rate of $$ 12 \% $$ p.a. in 2 years is Rs. 36 less than the compound interest compounded annually. a. यदि SI र CI ले कमश: साधारण व्याज र चक्रीय व्याजलाई जनाउँछ भने, माधि दिइएको जानकारीको आधारमा SI र CI को सम्बन्ध लेलुहोस्। If SI and CI denote the simple interest and compound interest respectively, write the relation between CI and SI according to the Information given above. [1]

Question

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We are given the following information about the picnic suggestions:

• 45 staff suggested Godawari (set G: n(G) = 45).  
• 50 staff suggested Dhulikhel (set D: n(D) = 50).  
• 15 staff agreed on “any of these” (i.e. they chose both places, so n(G ∩ D) = 15).  
• 24 staff did not suggest either Godawari or Dhulikhel.

Below are the answers to the parts:

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(a) Write a relation between n(G ∪ D), n(G), n(D) and n(G ∩ D):

The basic relation between two sets is given by  
  n(G ∪ D) = n(G) + n(D) – n(G ∩ D).

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(b) Find the total number of staff (showing a Venn diagram approach):

Step 1. Draw the Venn Diagram with two overlapping circles labeled G and D.

• In the intersection (both G and D): 15 staff.  
  • In only G (Godawari only): 45 – 15 = 30 staff.  
  • In only D (Dhulikhel only): 50 – 15 = 35 staff.  
  • Outside both circles (neither): 24 staff.

Step 2. Total staff = (only G) + (only D) + (both) + (neither)  
  Total = 30 + 35 + 15 + 24 = 104.

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(c) How many staff suggested exactly one place?

Staff who suggested exactly one place = Staff in only G + Staff in only D  
  = 30 + 35 = 65.

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(d) By what percent is the number of staff choosing Godawari more or less than that choosing Dhulikhel?

We have:  
  n(G) = 45 and n(D) = 50.  
The difference = 50 – 45 = 5.  
Since Godawari choosers are fewer, we say they are (5/50) ×100 = 10% less than those choosing Dhulikhel.

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Now for the interest problem:

It is given that at an annual interest rate of 12% for 2 years, the simple interest (SI) on a sum is Rs.36 less than the compound interest (CI). We can express this as:

CI – SI = 36.

For a principal P:
  Simple Interest, SI = P × r × t = P × 0.12 × 2 = 0.24P.
  Compound Interest, CI = P[(1 + r)^2 – 1] = P[(1.12)^2 – 1] = P(1.2544 – 1) = 0.2544P.

Thus, the difference is:
  CI – SI = 0.2544P – 0.24P = 0.0144P = 36,  
which indeed reflects the given information.

Therefore, the relation between SI and CI is:  
  CI = SI + 36.

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Summary of Answers:
(a) n(G ∪ D) = n(G) + n(D) – n(G ∩ D).  
(b) Total number of staff = 104.  
(c) Staff suggesting exactly one place = 65.  
(d) Staff choosing Godawari are 10% less than those choosing Dhulikhel.  
(e) CI = SI + 36.

These are the required relations and results.
(a) $$ n(G \cup D) = n(G) + n(D) - n(G \cap D) $$ (b) Total staff = 104 (c) Staff suggesting exactly one place = 65 (d) Staff choosing Godawari are 10% less than those choosing Dhulikhel (e) $$ CI = SI + 36 $$

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