In a certain examination 52 candidate offered biology 60 candidate offered history and 96 offered mathematics.if 21candidate offered biology and history 22 candidate offered history and mathematics 17 candidate offered all three subject. Draw a vanies diagram to illustrate these information and (a) how many candidate sat for the examination ?(b) how many candidate offered only one subject? (c) how many candidate offered two subject?
The Venn diagram has three big circles. In the very center of the circles is the 17 that took all three.
Remember to subtract the 17 when you make the numbers inside the arcs of where the circle hits. For example... inside the section where 22 were offered history and mathematics. The two numbers would be the 17, then only 5 that have both history and mathematics (as 22 - 17 = 5)
Continue in that manner around the circles : )
The History only section of the circle is 60 - 4 - 17 - 5 which is 34
The 60 minus the pieces that are shared with other subjects : )
To draw the Venn diagram, we start by labeling the three sets - Biology (B), History (H), and Mathematics (M).
Based on the information given, we have:
Number of candidates offering Biology (B) = 52
Number of candidates offering History (H) = 60
Number of candidates offering Mathematics (M) = 96
Number of candidates offering Biology and History (B ∩ H) = 21
Number of candidates offering History and Mathematics (H ∩ M) = 22
Number of candidates offering all three subjects (B ∩ H ∩ M) = 17
Now, let's construct the Venn diagram step by step:
Step 1: Draw three overlapping circles to represent Biology (B), History (H), and Mathematics (M).
Step 2: Label the regions where the circles overlap. Starting from the center, the first overlap is B ∩ H (Biology and History) with 21 candidates.
Step 3: The second overlap is H ∩ M (History and Mathematics) with 22 candidates.
Step 4: The third overlap is B ∩ M (Biology and Mathematics). However, we don't have this information directly. To find the number of candidates in this region, we can use the formula:
B ∩ M = B + M - (B ∩ H) - (H ∩ M) - (B ∩ H ∩ M)
B ∩ M = 52 + 96 - 21 - 22 - 17
B ∩ M = 88
Step 5: Finally, place the 17 candidates in the center region where all three subjects overlap (B ∩ H ∩ M).
Now, let's answer the questions:
(a) To find the total number of candidates, we add up the number of candidates in each set:
Candidates in Biology (B) = 52
Candidates in History (H) = 60
Candidates in Mathematics (M) = 96
Total candidates = B + H + M = 52 + 60 + 96 = 208
Therefore, there were 208 candidates who sat for the examination.
(b) To find the number of candidates who offered only one subject, we sum the candidates in each set excluding the overlaps:
Candidates offering only Biology (B - B ∩ H - B ∩ M) = 52 - 21 - 88 = -57 (ignore negative values)
Candidates offering only History (H - B ∩ H - H ∩ M) = 60 - 21 - 22 = 17
Candidates offering only Mathematics (M - H ∩ M - B ∩ M) = 96 - 22 - 88 = -14 (ignore negative values)
So, 17 candidates offered only one subject.
(c) To find the number of candidates who offered two subjects, we sum the candidates in each overlap region:
Candidates offering Biology and History (B ∩ H) = 21
Candidates offering History and Mathematics (H ∩ M) = 22
Candidates offering Biology and Mathematics (B ∩ M) = 88
Candidates offering two subjects = B ∩ H + H ∩ M + B ∩ M = 21 + 22 + 88 = 131
Therefore, 131 candidates offered two subjects.
I hope this step-by-step explanation helps!
To draw a Venn diagram illustrating the given information, follow these steps:
Step 1: Draw three overlapping circles to represent Biology (B), History (H), and Mathematics (M).
Step 2: Label the intersection of all three circles as "B, H, M" since 17 candidates offered all three subjects.
Step 3: Label the section where Biology and History overlap as "B, H" since 21 candidates offered both subjects but not Mathematics.
Step 4: Label the section where History and Mathematics overlap as "H, M" since 22 candidates offered both subjects but not Biology.
Step 5: Label the section where Biology and Mathematics overlap as "B, M" since no value is given.
Step 6: Finally, outside of the overlapping regions, label the sections representing only one subject as "B only," "H only," and "M only."
Now, let's answer the questions using the Venn diagram:
(a) To calculate the total number of candidates who sat for the examination, we need to find the sum of all the regions in the Venn diagram. The diagram shows that there are 17 candidates who offered all three subjects, 21 candidates who offered Biology and History only, 22 candidates who offered History and Mathematics only, and no information is given about the Biology and Mathematics overlap. However, we know that the total number of candidates who offered Biology alone is 52, History alone is 60, and Mathematics alone is 96. Therefore, to get the total number of candidates, we add all these values:
Total = (B only) + (H only) + (M only) + (B, H) + (H, M) + (B, M) + (B, H, M)
Total = 52 + 60 + 96 + 21 + 22 + unknown + 17
(b) To find the number of candidates who offered only one subject, we add up the values in the "B only," "H only," and "M only" regions:
Candidates offered only one subject = (B only) + (H only) + (M only)
(c) To find the number of candidates who offered two subjects, we add up the values in the "B, H," "H, M," and "B, M" regions:
Candidates offered two subjects = (B, H) + (H, M) + (B, M)
Unfortunately, without knowing the value of the "B, M" region, we cannot give an exact answer to parts (b) and (c).