In a school 156 passed Geography or history,75 students passed both subjects.If nine more passed Geography as passed history, how many students passed each subjects?
Let x be the number of students who passed only Geography, y be the number of students who passed only History, and z be the number of students who passed both subjects (75 students).
We know that:
x + y + z = 156 (Equation 1)
And we also know that:
x + z = (y + 9) (Equation 2)
This is because nine more students passed Geography than History, which means that the number of students who passed both subjects is equal to the number of students who passed History plus nine.
We can use equation 2 to solve for x in terms of y and z:
x = y + 9 - z
We substitute this expression for x into equation 1:
(y + 9 - z) + y + z = 156
Simplifying:
2y + 9 = 156
2y = 147
y = 73.5
This doesn't make sense since y must be a whole number. However, we do know that y + z = 75, which means that z = (75 - y). We can substitute this expression for z into equation 2:
x + (75 - y) = (y + 9)
Simplifying:
x = y + 9 - 75 + y
x = 2y - 66
Now we can substitute this expression for x into equation 1:
(2y - 66) + y + 75 = 156
Simplifying:
3y = 147
y = 49
Now that we know y, we can use y + z = 75 to solve for z:
49 + z = 75
z = 26
Finally, we can use x = 2y - 66 to solve for x:
x = 2(49) - 66
x = 32
Therefore, 32 students passed only Geography, 49 students passed only History, and 26 students passed both subjects.
Let's solve this step by step:
Step 1: We know that a total of 156 students passed either Geography or History.
Step 2: We also know that 75 students passed both subjects.
Step 3: If we assume "x" as the number of students who passed History, then "156 - x" would represent the number of students who passed Geography but not History.
Step 4: If "9 more" students passed Geography than History, then we can say that "x + 9" represents the number of students who passed Geography.
Step 5: Now, we can add the students who passed both subjects and the students who passed only Geography to find the total number of students who passed Geography: (x + 9) + (156 - x) = (156 - x) + x + 9
Step 6: Simplifying the equation, we get: (156 - x) + x + 9 = 156 + 9
Step 7: By combining like terms, we get: 165 = 165
Step 8: Since this equation is true, it means that the value of "x" doesn't matter.
Step 9: Therefore, the number of students who passed History can be any value, and the number of students who passed Geography would be x + 9.
In summary, the number of students who passed History can have any value, while the number of students who passed Geography would be 9 more than the number who passed History.
To solve this problem, we can use a visual representation called a Venn diagram. A Venn diagram consists of overlapping circles, where each circle represents a subject, and the overlapping region represents the students who passed both subjects.
Given:
- Total number of students who passed Geography or history is 156.
- Number of students who passed both subjects is 75.
Let's assume the number of students who passed only Geography is 'x', the number of students who passed only history is 'y', and the number of students who passed both subjects is 75.
According to the given information, the total number of students who passed either Geography or history is 156. We can write this as the sum of the number of students who passed only Geography (x), the number of students who passed only history (y), and the number of students who passed both subjects (75):
x + y + 75 = 156
Now, let's consider the second part of the question: "If nine more passed Geography than passed history." This means that the number of students who passed only Geography (x) is 9 more than the number of students who passed only history (y):
x = y + 9
Now we have two equations:
x + y + 75 = 156
x = y + 9
We can solve these equations simultaneously to find the values of x and y. Substituting the second equation into the first equation, we get:
(y + 9) + y + 75 = 156
2y + 84 = 156
2y = 156 - 84
2y = 72
y = 72/2
y = 36
Now, substitute the value of y back into the second equation to find x:
x = y + 9
x = 36 + 9
x = 45
Therefore, the number of students who passed Geography is 45, and the number of students who passed history is 36.