There are 50 students admitted to a nursery class. Some students can speak only English and some can speak only Hindi. 10 students can speak both English and Hindi. If the number of students who can speak English is 21, then how many students can speak Hindi, how many can speak only Hindi and how many can speak only English?
UPSC - 2014
21, 11 and 29 respectively
28, 18 and 22 respectively
37, 27 and 13 respectively
39, 29 and 11 respectively
This problem can be solved using the principles of set theory, specifically by applying the formula for the union of two sets or by using a Venn diagram approach. We are given the total number of students, the number of students speaking both languages, and the number of students speaking English. We need to find the number of students speaking Hindi, only Hindi, and only English.
The number of students who can speak English includes those who speak only English and those who speak both English and Hindi. So, to find those who speak only English, we subtract the 'both' category from the total English speakers.
\[ \text{Only English} = \text{English speakers} - \text{Both English and Hindi} \] \[ \text{Only English} = 21 - 10 = 11 \]Since some students can speak only English and some can speak only Hindi, and 10 students can speak both, it implies that all 50 students can speak at least one of the languages. If there were students who spoke neither, the problem would typically state that. Assuming all students speak at least one language, the total number of students is the union of English speakers and Hindi speakers.
\[ N(E \cup H) = N(E) + N(H) - N(E \cap H) \]Here, \(N(E \cup H)\) is the total number of students (50), \(N(E)\) is the number of English speakers (21), and \(N(E \cap H)\) is the number of students speaking both (10).
\[ 50 = 21 + N(H) - 10 \] \[ 50 = 11 + N(H) \] \[ N(H) = 50 - 11 = 39 \]So, the number of students who can speak Hindi is 39.
Similar to English, the number of students who can speak Hindi includes those who speak only Hindi and those who speak both English and Hindi.
\[ \text{Only Hindi} = \text{Hindi speakers} - \text{Both English and Hindi} \] \[ \text{Only Hindi} = 39 - 10 = 29 \]Comparing these values with the options, the correct sequence is 39, 29, and 11 respectively.
D) 39, 29 and 11 respectively. This matches our calculated values for the number of students who can speak Hindi, only Hindi, and only English.
Out of a total of 120 musicians in a club, 5% can play all the three instruments, guitar, violin and flute. It so happens that the number of musicians who can play any two and only two of the above instruments is 30. The number of musicians who can play the guitar alone is 40. What is the total number of those who can play violin alone or flute alone?
UPSC - 2014
This problem can be solved using Venn diagrams and set theory principles. We are given the total number of musicians, the number of musicians who can play all three instruments, the number who can play exactly two instruments, and the number who can play only the guitar. We need to find the number of musicians who can play only the violin or only the flute.
Identify the total number of musicians:
Total musicians \(N = 120\)
Calculate the number of musicians who can play all three instruments:
This is 5% of the total musicians.
\[ \text{Number of musicians playing all three} = 5\% \times 120 = \frac{5}{100} \times 120 = 6 \]Identify the number of musicians who can play exactly two instruments:
Given as 30.
Identify the number of musicians who can play only the guitar:
Given as 40.
Let's denote the number of musicians playing only violin as \(V_{\text{alone}}\) and only flute as \(F_{\text{alone}}\).
We need to find \(V_{\text{alone}} + F_{\text{alone}}\).
Use the principle of inclusion-exclusion for three sets or simply sum up the disjoint regions of a Venn diagram.
The total number of musicians is the sum of those who play:
So, we can write the equation:
\[ N = (\text{Guitar alone}) + (\text{Violin alone}) + (\text{Flute alone}) + (\text{Exactly two instruments}) + (\text{All three instruments}) \]Substitute the known values into the equation:
\[ 120 = 40 + V_{\text{alone}} + F_{\text{alone}} + 30 + 6 \]Simplify the equation:
\[ 120 = 76 + V_{\text{alone}} + F_{\text{alone}} \]Solve for \(V_{\text{alone}} + F_{\text{alone}}\):
\[ V_{\text{alone}} + F_{\text{alone}} = 120 - 76 \]\[ V_{\text{alone}} + F_{\text{alone}} = 44 \]B) 44 is the total number of those who can play violin alone or flute alone.
In the above figure, circle P represents hardworking people, circle Q represents intelligent people, circle R represents truthful people, and circle S represents honest people, Which region represents the people who are intelligent, honest and truthful but not hardworking?
UPSC - 2012
To solve this problem, we need to identify the region in the Venn diagram that represents people who are intelligent, honest, and truthful but not hardworking.
Therefore, the region labeled as 6 represents people who are intelligent, honest, and truthful but not hardworking.
Correct Option: A) 6
The number of persons who read magazine X only is thrice the number of persons who read magazine Y. The number of persons who read magazine Y only is thrice the number of persons who read magazine X. Then, which of the following conclusions can be drawn?
1. The number of persons who read both the magazines is twice the number of persons who read only magazine X.
2. The total number of persons who read either one magazine or both the magazines is twice the number of persons who read both the magazines.
Select the correct answer using the code given below:
UPSC - 2015
Find the diagram that show right relation between CAR, WOOD and RAT?
To find the correct diagram that shows the right relationship between CAR, WOOD, and RAT, we need to understand the logical connections or intersections between these categories.
Correct Option: B) This option correctly shows separate circles, indicating no direct relationship between CAR, WOOD, and RAT.
Select the Venn Diagram that best represents the relationship between the following classes.
Men, Women, Teacher.
To determine the correct Venn diagram representing the relationship between the classes 'Men', 'Women', and 'Teacher', we need to consider how these categories can overlap.
Option B shows three intersecting circles, which correctly represents 'Men', 'Women', and 'Teacher' as overlapping categories, with 'Teacher' intersecting both 'Men' and 'Women'.
Correct Option: B) Option B correctly represents the relationship with three intersecting circles.
