There are three subjects i.e. History, Economics and Geography in 1 semester. 120 students choose History while 150 students choose exactly one subject. A student who chooses Geography does not choose either History or Economics. 20 students don't choose any subject.
Quantity 1: Number of students who like only History are 2 times the number of students who choose Geography. If ratio of number of students who choose only Economics and the number of students who choose Geography is 3:4, then find total number of students in 1" semester.
Quantity II: If number of students who choose multiple subjects is equal to number of students who choose Geography, then find the number of students who choose only Economics.
Quantity III: Minimum number of students in 1 semester.
A. Quantity 1> Quantity II <Quantity III
B. Quantity II > Quantity III <Quantity I
C. Quantity II > Quantity III <Quantity I
D. Quantity II <Quantity I < Quantity III
E. Quantity I < Quantity II > Quantity III

Let's break down the problem step by step:

### Given Information:
- **Total number of students**: \( N \)
- Students who choose **only History**: \( H_1 \)
- Students who choose **only Economics**: \( E_1 \)
- Students who choose **only Geography**: \( G_1 \)
- Students who choose **exactly one subject**: 150
- Students who don't choose any subject: 20
- Students who choose **History**: 120
- Students who choose **multiple subjects**: \( M \)
- Students who choose **Geography** and neither of the other subjects: \( G_1 \)

### Equations:
1. **Total students equation**:
   \[
   H_1 + E_1 + G_1 + M + 20 = N
   \]
   But since \( H_1 + E_1 + G_1 = 150 \) and \( 20 \) students don't choose any subject:
   \[
   N = 150 + M + 20 = 170 + M
   \]

2. **Number of students who choose only History**:
   \[
   H_1 = 2 \times G_1
   \]

3. **Ratio of students choosing only Economics to Geography**:
   \[
   \frac{E_1}{G_1} = \frac{3}{4}
   \]
   \[
   E_1 = \frac{3}{4}G_1
   \]

4. **Total students choosing only one subject**:
   \[
   H_1 + E_1 + G_1 = 150
   \]
   Substituting \( H_1 \) and \( E_1 \) in terms of \( G_1 \):
   \[
   2G_1 + \frac{3}{4}G_1 + G_1 = 150
   \]
   \[
   2G_1 + 1.75G_1 = 150
   \]
   \[
   3.75G_1 = 150 \implies G_1 = \frac{150}{3.75} = 40
   \]
   Therefore:
   \[
   H_1 = 2 \times 40 = 80
   \]
   \[
   E_1 = \frac{3}{4} \times 40 = 30
   \]

### Substituting in the total:
\[
N = 170 + M
\]

### Quantity I: 
- **Number of students who like only History**: 80.

### Quantity II:
- **Number of students who choose multiple subjects** is equal to **number of students who choose Geography**:
   \[
   M = G_1 = 40
   \]
- Therefore, \( N = 170 + 40 = 210 \).

### Quantity III:
- **Minimum number of students in the semester**:
   Since multiple students can choose multiple subjects, and \( M \) is equal to \( G_1 \), the minimum number is when \( M = 0 \).
   So:
   \[
   N_{\text{min}} = 170
   \]

### Final comparison:
- **Quantity I**: Number of students who like only History = \( 80 \).
- **Quantity II**: Number of students who choose only Economics = \( 30 \).
- **Quantity III**: Minimum number of students in the semester = \( 170 \).

From this, we have:
- **Quantity I**: 80
- **Quantity II**: 30
- **Quantity III**: 170

So the comparison is:

**Answer: D. Quantity II < Quantity I < Quantity III**

---

Would you like more details or have any questions?

Here are some related questions:
1. What if the ratio of students who choose only Economics to Geography was different?
2. How does the presence of students who don't choose any subject affect the total count?
3. What is the impact of increasing the number of students choosing multiple subjects on the total?
4. How would the equation change if more subjects were introduced?
5. What happens if the ratio of students choosing multiple subjects to those choosing Geography changes?

**Tip:** Always start by defining variables clearly and translating the word problem into mathematical equations. This helps in systematic problem-solving.

Explore a complex math problem involving student subject choices in a semester. Learn how to calculate the number of students choosing only History, only Economics, and those choosing multiple subjects. Understand the impact of students who choose Geography and those who don't choose any subject. Suitable for Grades 9-12.

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