A student obtained equal marks in History and Sociology. The ratioof marks is Sociology and Geography is 2:3 and the ratio fo the marks in History and Philosophy is 1:2. If he has scored an aggregate of 55% marks. The maximum marks in each subject is same. In how many subjects did he score equal to or greager than 60% marks?

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the maximum marks in each subject be \( M \).
Let the marks obtained in History and Sociology be \( H \) (since they are equal, \( H = S \)).
Let the marks in Geography be \( G \) and in Philosophy be \( P \).

### Step 2: Set Up Ratios
From the problem, we have the following ratios:
1. The ratio of marks in Sociology to Geography is \( S:G = 2:3 \). This can be expressed as:
\[ S = \frac{2}{3}G \quad \text{(1)} \]
2. The ratio of marks in History to Philosophy is \( H:P = 1:2 \). This can be expressed as:
\[ H = \frac{1}{2}P \quad \text{(2)} \]

### Step 3: Express All Marks in Terms of One Variable
From equation (1), substituting \( S \) with \( H \) (since \( H = S \)):
\[ H = \frac{2}{3}G \implies G = \frac{3}{2}H \quad \text{(3)} \]
From equation (2):
\[ H = \frac{1}{2}P \implies P = 2H \quad \text{(4)} \]

### Step 4: Calculate Total Marks
The total marks obtained by the student in all subjects can be expressed as:
\[ \text{Total Marks} = H + S + G + P = H + H + \frac{3}{2}H + 2H \]
Combining these:
\[ \text{Total Marks} = 2H + \frac{3}{2}H + 2H = 2H + 2H + \frac{3}{2}H = 4H + \frac{3}{2}H = \frac{8H}{2} + \frac{3H}{2} = \frac{11H}{2} \]

### Step 5: Calculate Aggregate Percentage
The aggregate percentage is given as 55%. Therefore:
\[ \frac{\text{Total Marks}}{4M} \times 100 = 55 \]
Substituting the total marks:
\[ \frac{\frac{11H}{2}}{4M} \times 100 = 55 \]
Simplifying this:
\[ \frac{11H \times 100}{8M} = 55 \implies 11H \times 100 = 55 \times 8M \implies 1100H = 440M \implies H = \frac{440M}{1100} = \frac{44M}{110} = \frac{4M}{10} = 0.4M \]

### Step 6: Calculate Marks in Each Subject
Using \( H = 0.4M \):
- Marks in History \( H = 0.4M \)
- Marks in Sociology \( S = 0.4M \)
- Marks in Geography \( G = \frac{3}{2}H = \frac{3}{2} \times 0.4M = 0.6M \)
- Marks in Philosophy \( P = 2H = 2 \times 0.4M = 0.8M \)

### Step 7: Determine Subjects with Marks ≥ 60%
Now we check each subject:
- History: \( 0.4M \) (not ≥ 60%)
- Sociology: \( 0.4M \) (not ≥ 60%)
- Geography: \( 0.6M \) (≥ 60%)
- Philosophy: \( 0.8M \) (≥ 60%)

### Conclusion
The student scored equal to or greater than 60% in 2 subjects (Geography and Philosophy).

### Final Answer
**The student scored equal to or greater than 60% in 2 subjects.**

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