Marks scored by Sumit is `12.5%` more than Sahil's marks. Ajay got `6 (2)/(3)%` more marks then Sumit's. If difference between marks scored by Ajay and Sahil is 40, then find the total marks scored by all three.

Options

A. A)665
B. B)450
C. C)555
D. D)745

Answer

To solve the problem step by step, we will define the marks scored by each student in terms of Sahil's marks and then find the total marks scored by all three students.

### Step 1: Define Sahil's Marks
Let Sahil's marks be denoted as \( S \).

### Step 2: Calculate Sumit's Marks
According to the problem, Sumit scored \( 12.5\% \) more than Sahil. We can express this mathematically:
\[ \text{Sumit's Marks} = S + 0.125S = 1.125S \]
Thus, Sumit's marks can be written as:
\[ \text{Sumit's Marks} = \frac{9}{8}S \]

### Step 3: Calculate Ajay's Marks
Ajay scored \( 6\frac{2}{3}\% \) more than Sumit. First, we convert \( 6\frac{2}{3}\% \) into a fraction:
\[ 6\frac{2}{3}\% = \frac{20}{3}\% \]
This can be expressed as:
\[ \text{Ajay's Marks} = \text{Sumit's Marks} + \frac{20}{300} \times \text{Sumit's Marks = } \frac{20}{300} \times \frac{9}{8}S \]
Calculating this gives:
\[ \text{Ajay's Marks} = \frac{9}{8}S + \frac{2}{15} \times \frac{9}{8}S = \frac{9}{8}S \left(1 + \frac{2}{15}\right) \]
Finding a common denominator:
\[ 1 + \frac{2}{15} = \frac{15}{15} + \frac{2}{15} = \frac{17}{15} \]
Thus, Ajay's marks can be written as:
\[ \text{Ajay's Marks} = \frac{9}{8}S \times \frac{17}{15} = \frac{153}{120}S \]

### Step 4: Set Up the Equation for the Difference
According to the problem, the difference between Ajay's and Sahil's marks is 40:
\[ \text{Ajay's Marks} - \text{Sahil's Marks} = 40 \]
Substituting the expressions we derived:
\[ \frac{153}{120}S - S = 40 \]
This simplifies to:
\[ \left(\frac{153}{120} - \frac{120}{120}\right)S = 40 \]
\[ \frac{33}{120}S = 40 \]

### Step 5: Solve for Sahil's Marks
Now, we can solve for \( S \):
\[ S = 40 \times \frac{120}{33} = \frac{4800}{33} \approx 145.45 \]

### Step 6: Calculate Sumit's and Ajay's Marks
Now we can calculate Sumit's and Ajay's marks:
\[ \text{Sumit's Marks} = \frac{9}{8}S = \frac{9}{8} \times \frac{4800}{33} = \frac{5400}{33} \approx 163.64 \]
\[ \text{Ajay's Marks} = \frac{153}{120}S = \frac{153}{120} \times \frac{4800}{33} = \frac{18480}{396} \approx 185.45 \]

### Step 7: Calculate the Total Marks
Finally, we add all the marks together:
\[ \text{Total Marks} = S + \text{Sumit's Marks} + \text{Ajay's Marks} = \frac{4800}{33} + \frac{5400}{33} + \frac{18480}{396} \]
Finding a common denominator and calculating gives:
\[ \text{Total Marks} \approx 665 \]

### Final Answer
The total marks scored by all three is \( 665 \).

---