In a class the ratio of rural to urban students is 4 : 7. In an examination the average percentage marks of the rural and the urban studentsare respectively 65 and 63. Whatis the overall average percentage marks of the class (correct to two decimal places)?

The ratio of the number of rural students to urban students is $$4:7$$.
Hence, if we take a common multiple $$k$$, the numbers can be written as:

Number of rural students $$= 4k$$
Number of urban students $$= 7k$$

The average percentage marks are given as:
Rural students: $$65\%$$
Urban students: $$63\%$$

To find the overall average, use the weighted‐average formula:

Overall average $$= \dfrac{(\text{number of rural})(\text{average of rural}) + (\text{number of urban})(\text{average of urban})}{\text{total number of students}}$$

Substituting the expressions in terms of $$k$$:

Overall average $$= \dfrac{(4k)(65) + (7k)(63)}{4k + 7k}$$

Simplify numerator and denominator:

Numerator $$= 4k \times 65 + 7k \times 63 = 260k + 441k = 701k$$

Denominator $$= 4k + 7k = 11k$$

Therefore,

Overall average $$= \dfrac{701k}{11k} = \dfrac{701}{11}$$

Compute the division:

$$\dfrac{701}{11} = 63.727272\ldots\%$$

Correct to two decimal places, the overall average percentage marks $$= 63.73\%$$.

Option A which is: 63.73%