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 papers did he receive less than $60\%$ marks?

Let the marks scored in the 6 papers be $5k,6k,7k,8k,9k,$ and $10k$.
Total marks scored = $5k+6k+7k+8k+9k+10k=45k$

Let $M$ be the maximum marks of each paper. Total maximum marks = $6M$.
Since he scored $60\%$ overall:

$$\frac{45k}{6M}=\frac{60}{100}=\frac{3}{5}$$

$$\frac{45k}{6M}=\frac{3}{5}⟹M=\frac{45k\times 5}{18}=12.5k$$

The threshold for $60\%$ marks in an individual paper is:

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

Comparing the scored marks with $7.5k$:

* Paper 1: $5k<7.5k$ (Less)
* Paper 2: $6k<7.5k$ (Less)
* Paper 3: $7k<7.5k$ (Less)
* Paper 4: $8k>7.5k$ (More)
* Paper 5: $9k>7.5k$ (More)
* Paper 6: $10k>7.5k$ (More)

Hence, he received less than $60\%$ marks in exactly $3$ papers.