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

Let the marks obtained in the $6$ papers be $5x,6x,7x,8x,9x,$ and $10x$.
Let the maximum marks for each paper be $M$. Total maximum marks for $6$ papers $=6M$.

The total marks obtained by the student is:

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

Given that his overall score is $60\%$:

$$45x=60\%\text{ of }6M$$

$$45x=\frac{60}{100}\times 6M=3.6M⟹M=\frac{45x}{3.6}=12.5x$$

Now, we calculate what value corresponds to $60\%$ of individual paper's maximum marks ($M$):

$$60\%\text{ of }M=0.6\times 12.5x=7.5x$$

Any paper where marks are less than $7.5x$ is below $60\%$.
The marks obtained are: $5x,6x,7x,8x,9x,10x$.
Comparing with $7.5x$, the papers with scores less than $7.5x$ are $5x,6x,$ and $7x$.

Thus, there are exactly $3$ papers where he scored less than $60\%$.