A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5 : 6 : 7 : 8 : 9 : 10. Overall he scored 60%. In how many number of papers did he score less than 60% of the maximum marks?

Correct Answer: The correct option is 3.

Step-by-Step Explanation:

Step 1: Understand the given ratio and parameters

Let the maximum marks for each of the 6 papers be $M$.

The total maximum marks for all 6 papers combined is:

$\text{Total Maximum Marks}=6\times M$

The marks scored by the student in the 6 papers are in the proportion of $5:6:7:8:9:10$.

Let the common ratio multiplier be $x$.

Therefore, the marks obtained in each paper are:

$5x,6x,7x,8x,9x,\text{and}10x$

Step 2: Calculate the total marks obtained

Summing the marks across all 6 papers gives:

$\text{Total Marks Obtained}=5x+6x+7x+8x+9x+10x=45x$

Step 3: Relate total marks obtained to the maximum marks using overall percentage

It is given that the overall percentage scored by the student is 60%.

$\text{Total Marks Obtained}=60\%\text{of Total Maximum Marks}$

$45x=\frac{60}{100}\times 6M$

$45x=3.6M$

Solving for $M$ in terms of $x$:

$M=\frac{45x}{3.6}=12.5x$

Step 4: Find the threshold mark for 60% in a single paper

The maximum marks for a single paper is $M=12.5x$.

60% of the maximum marks for one paper is:

$60\%\text{of}M=0.60\times 12.5x=7.5x$

Step 5: Determine in how many papers he scored less than 60% of maximum marks

We compare the marks in each paper with $7.5x$:

1. Paper 1: $5x<7.5x$ (Less than 60%)

2. Paper 2: $6x<7.5x$ (Less than 60%)

3. Paper 3: $7x<7.5x$ (Less than 60%)

4. Paper 4: $8x>7.5x$ (More than 60%)

5. Paper 5: $9x>7.5x$ (More than 60%)

6. Paper 6: $10x>7.5x$ (More than 60%)

Thus, the student scored less than 60% of the maximum marks in exactly 3 papers.