Cost of \[5kg\] of wheat is \[Rs.30.50\].
\[\left( a \right)\] What will be the cost of \[8kg\] of wheat?
\[\left( b \right)\] What quantity of wheat can be purchased in \[Rs.61\].

Hint: In both of the cases, we will apply the proportion method so as to find the cost and quantity respectively. We can see that the cost is directly proportional to quantity. So we will be applying the formula of direct proportion by assigning a variable to the unknown value in both of the cases. Upon solving them, we obtain the required values.

Complete step-by-step solution:
Now let us briefly discuss proportions. Proportion is nothing but saying that two ratios are equal. Two ratios can be written in proportion in the following ways- \[\dfrac{a}{b}=\dfrac{c}{d}\] or \[a:b=c:d\]. From the second way of notation, the values on the extreme end are called as extremes and the inner ones as means. Proportions are of two types: direct proportions and indirect or inverse proportions. In the direct proportion, there would be direct relation between the quantities. In the case of indirect proportion, there exists indirect relation between the quantities.
Let us consider the first case given.
\[\left( a \right)\] let us find the cost of \[8kg\] of wheat.
Given that, the cost of \[5kg\] of wheat is \[Rs.30.50\].
By applying the proportion, we get
\[\begin{align}
  & \Rightarrow 5kg:8kg::30.50:x \\
 & \Rightarrow x=\dfrac{8\times 30.50}{5} \\
 & \Rightarrow x=48.8 \\
\end{align}\]
\[\therefore \] The cost of \[8kg\] of wheat is \[Rs.48.8\].
\[\left( b \right)\]now let us find the quantity of wheat that can be purchased for \[Rs.61\]
By applying proportion, we get
\[\begin{align}
  & \Rightarrow 5kg:x::30.50:61 \\
 & \Rightarrow x=\dfrac{5\times 61}{30.50} \\
 & \Rightarrow x=10 \\
\end{align}\]
\[\therefore \] The wheat that can be purchased for \[Rs.61\] is \[10kg\].

Note: We must always assign the variable to the value to be found. We can use ratios and proportions in our daily life. We can apply a ratio for adding the quantity of milk to water or water to milk. We can apply proportions for finding the height of the buildings and trees and many more.