In a school 156 students passed Geography or History. There were 75 students who passed in both subjects if more passed Geography than History find how many were taking each subject.
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It is necessary to know how many times more passed Geography than History.
Let's assume that x students passed History, and y students passed Geography. According to the problem, there were 75 students who passed in both subjects.
Now, we can create two equations based on the given information:
1. x + y - 75 = 156 (There are 156 students who passed Geography or History)
2. x > y (More students passed Geography than History)
We can solve these equations simultaneously to find the values of x and y.
First, let's rewrite the first equation:
x + y = 156 + 75
x + y = 231
Now, to solve for x and y, we can multiply the second equation by (-1) and add it to the first equation:
x + y - x + y = 231 - x
2y = 231 - x
y = (231 - x) / 2
Since it is given that more students passed Geography than History, we can assume that y > x.
Now, let's substitute the value of y into the equation:
y = (231 - x) / 2
(231 - x) / 2 > x
Now, we can solve this inequality:
231 - x > 2x
231 > 3x
77 > x
So, x is less than 77.
Since x represents the number of students who passed History, it can vary from 1 to 76.
Thus, there can be multiple solutions for this problem depending on the value of x.
For example, if x = 1, then y = (231 - 1) / 2 = 115 students passed Geography.
If x = 2, then y = (231 - 2) / 2 = 114 students passed Geography.
And so on, until x = 76, then y = (231 - 76) / 2 = 77 students passed Geography.
So, depending on the value of x, the number of students taking each subject will vary.
To solve this problem, we can use a visual representation called a Venn diagram. Here are the steps to find the number of students taking each subject:
1. Draw two intersecting circles, one representing Geography and another representing History.
2. Label the region where the circles intersect as the number of students passing both Geography and History (75 students).
3. Since a total of 156 students passed either Geography or History, subtract the number of students passing both subjects from this total: 156 - 75 = 81.
4. Label the remaining part of the Geography circle (outside the intersection) as the number of students passing only Geography.
5. Since it is given that more students passed Geography than History, we know that the number of students passing only Geography is greater than the number of students passing both subjects. Therefore, let's assume that the number of students passing both subjects is the smaller value.
6. Let x represent the number of students passing both subjects (75 students). Since the number of students passing only Geography is greater, let's add that number to the number of students passing both subjects: x + x = 2x.
7. Set up an equation using the information from step 3: 2x = 81.
8. Solve the equation for x: divide both sides by 2: x = 81/2 = 40.5.
Since we cannot have a fraction of a student, we round down to the nearest whole number. Therefore, approximately 40 students are taking both Geography and History.
9. Now that we have the value for x, we can find the number of students passing only Geography by subtracting x from the total number of students passing Geography or History: 81 - 40 = 41.
So, approximately 41 students are taking only Geography.
10. To find the number of students taking only History, subtract the number of students taking both subjects from the total number of students passing Geography or History: 81 - 40 = 41.
Therefore, approximately 41 students are taking only History.
In conclusion, approximately 40 students are taking both subjects, 41 students are taking only Geography, and 41 students are taking only History.