To solve this question, let's start by understanding the given equation:
a:b = c:d
This equation represents a proportion, where the ratio of a to b is equal to the ratio of c to d.
Now, let's consider the expression (pa+qc)/(pb+qd). We can simplify this expression by multiplying both the numerator and denominator by the same value.
Multiplying the numerator by p and the denominator by q, we get:
(pa+qc)/(pb+qd) = (p*a + q*c)/(p*b + q*d)
Now, let's compare this expression to the given equation a:b = c:d.
The numerator of the expression (pa+qc)/(pb+qd) is p*a + q*c, which has the same ratio as a and c.
The denominator of the expression (pa+qc)/(pb+qd) is p*b + q*d, which has the same ratio as b and d.
Therefore, the expression (pa+qc)/(pb+qd) has the same ratio as a:b and c:d.
Hence, the correct answer is option D) a:b and c:d.