Question

A sweet seller has $420$ Kaju burfis and $130$ Badam burfis. She wants to stack them in such a way that each stack has the same number and same type of burfis and they take up the least area of the tray. What is the number of burfis that can be placed in each stack for this purpose?

Answer 1

Hint: The number of burfis in each stack must be the same and be a factor of both $420$ and $130$. So, the number of burfis in each stack will be the H.C.F. of $420$ and $130$.
According to the data given in the question:
Number of Kaju barfis $ = 420$,
Number of Badam burfis $ = 130$.
The burfis need to be stacked in such a way that each of the stacks contains the same number and same type of burfis and they take the least area of the tray which means the number of stacks must be minimum.
$\therefore $So, the number of burfis in each stack must be such that it must be a factor of $420$ and $130$ and for taking the least area of the stack, this number must also be maximum. Therefore, the number of burfis in each stack must be the largest common factor (H.C.F.) of $420$ and $130$.
So, for calculating the H.C.F. , the numbers $420$ and $130$can be written as:
$
   \Rightarrow 420 = {2^4} \times 3 \times 5 \times 7, \\
   \Rightarrow 130 = 2 \times 5 \times 13. \\
$
The H.C.F. of them will be:
$ \Rightarrow $ H.C.F. $ = 2 \times 5 = 10.$
$\therefore $ H.C.F. of $420$ and $130$ is $10.$ Therefore, each stack must contain $10$ burfis.
Note: We can also calculate the total number of stacks thus formed. Since each stack is having $10$ burfis, the number of Kaju burfi stacks will be:
$ \Rightarrow \dfrac{{420}}{{10}} = 42.$
Whereas the number of Badam burfi stacks will be:
$ \Rightarrow \dfrac{{130}}{{10}} = 13.$
Therefore, the total number of stacks $ = 42 + 13 = 55.$