In a class of 70 students,6 offer Economics only,18 offer Economics but not mathematics,36 offer Economics and Geography,53 offer Economics,50 offer geography and 34 offer mathematics and geography.All students offer at least one subject.
Using a Venn diagram, we can visualize the information given in the problem:
[asy]
label("Economics", (2,85));
label("Mathematics", (-55,0));
label("Geography", (65,0));
draw(Circle((0,45), 29.5));
draw(Circle((-35, -20), 29.5));
draw(Circle((35, -20), 29.5));
label("$6$", (0, 75));
label("$12$", (-13, 45));
label("$6$", (13, 45));
label("$6$", (-45, -10));
label("$12$", (0, -10));
label("$24$", (45, -10));
label("$18$", (0, 18));
label("$15$", (-23.5, 22));
label("$21$", (23.5, 22));
label("$19$", (-17.5, -10));
label("$15$", (0, -32.5));
label("$15$", (17.5, -10));
label("$15$", (17.5, -32.5));
label("$4$", (-27.5, -32.5));
label("$15$", (27.5, -32.5));
label("$19$", (13.5, -45));
label("$15$", (-13.5, -45));
filldraw(circle((0,45),29.5),white);
filldraw(circle((-35, -20), 29.5),white);
filldraw(circle((35, -20), 29.5),white);
[/asy]
Let:
- $x$ be the number of students who offer Economics, Mathematics, and Geography
- $y$ be the number of students who offer Mathematics and Geography only
- $z$ be the number of students who offer Mathematics only
- $w$ be the number of students who offer Geography only
Using these variables, we can create a system of equations based on the given information:
- $6 + 12 + x + 15 + 19 = 70$ (all students offer at least one subject)
- $6 + 12 + x + 18 + y = 70$ (all students offer at least one subject)
- $6 + 12 + x + 15 + y + z = 70$ (all students offer at least one subject)
- $6 + x + 15 + 19 + y + w = 70$ (all students offer at least one subject)
- $x + y + z + w = 0$ (no students offer no subjects)
Simplifying and solving the system of equations, we get:
- $x = 4$
- $y = 15$
- $z = 10$
- $w = 5$
Therefore, there are 10 students who offer Mathematics only, 4 students who offer Economics, Mathematics, and Geography, and 15 students who offer Mathematics and Geography only. The remaining numbers for each category can be found by adding the relevant numbers from the diagram:
- Economics only: 6
- Economics and Mathematics only: 12
- Economics and Geography only: 18
- Mathematics and Geography only: 15 + 5 = 20
- Economics, Mathematics, and Geography: 4
Checking that the sum of these numbers is equal to 70, we can conclude that our solution is correct.
To solve this problem, let's break it down step-by-step:
Step 1: Identify the number of students who offer both Economics and Mathematics.
Given that 34 students offer Mathematics and Geography, and 36 students offer Economics and Geography, we can infer that the number of students who offer both Economics and Mathematics is the same as the number of students who offer both Economics, Mathematics, and Geography.
Therefore, the number of students who offer both Economics and Mathematics is 36.
Step 2: Calculate the number of students who offer Economics only.
We know that 6 students offer Economics only, and there are 36 students who offer Economics and Mathematics.
To calculate the number of students who offer Economics only, we subtract the number of students who offer both Economics and Mathematics (36) from the total number of students who offer Economics (53).
Therefore, the number of students who offer Economics only is 53 - 36 = 17.
Step 3: Determine the number of students who offer Economics and Geography only.
We know that 18 students offer Economics but not Mathematics, and there are 36 students who offer both Economics and Geography.
To calculate the number of students who offer Economics and Geography only, we subtract the number of students who offer both Economics, Mathematics, and Geography (36) from the total number of students who offer Economics and Geography (36).
Therefore, the number of students who offer Economics and Geography only is 36 - 36 = 0.
Step 4: Find the number of students who offer Geography only.
We know that 50 students offer Geography, and there are 36 students who offer both Geography and Mathematics.
To determine the number of students who offer Geography only, we subtract the number of students who offer both Economics and Geography (36) from the total number of students who offer Geography (50).
Therefore, the number of students who offer Geography only is 50 - 36 = 14.
Step 5: Calculate the number of students who offer Mathematics only.
To find the number of students who offer Mathematics only, we need to subtract the number of students who offer both Mathematics and Geography (34) from the total number of students who offer Mathematics (34).
Therefore, the number of students who offer Mathematics only is 34 - 34 = 0.
Step 6: Determine the number of students who offer all three subjects (Economics, Mathematics, and Geography).
We know that there are 34 students who offer both Mathematics and Geography, and there are 36 students who offer both Economics, Mathematics, and Geography.
To calculate the number of students who offer all three subjects, we take the smaller value, which is 34.
Step 7: Summarize the number of students offering each subject.
- The number of students offering Economics only is 17.
- The number of students offering Economics and Mathematics is 36.
- The number of students offering Economics and Geography only is 0.
- The number of students offering Geography only is 14.
- The number of students offering Mathematics only is 0.
- The number of students offering all three subjects is 34.
Note: To verify our calculations, we can add up the number of students offering each category:
17 (Economics only) + 36 (Economics and Mathematics) + 0 (Economics and Geography only) + 14 (Geography only) + 0 (Mathematics only) + 34 (all three subjects) = 101, which is greater than the total number of students (70). If the calculations are correct, it indicates that some students take more than one subject.