Question

A sweet – shop makes gift packets of sweets by combining two special types of sweets A and B which are 7 kg. At least 3 kg of A and no more than 5 kg of B should be used. The shop makes a profit of Rs. 15 on A and Rs. 20 on B per kg. Determine the product mix so as to obtain maximum profit.

Answer 1

Hint:In this question we are given to obtain a maximum profit of product with the combination of sweets A and B. To calculate this first we need to find the profit of a product mix with different quantities by taking few points into consideration which is given in the question, i.e. sweet A should be equal or greater than 3 $\left( A\ge 3 \right)$ and sweet B should be equal or less than 5 $\left( B\le 5 \right)$ then compare the profit of both products as to ascertain the answer.

Complete step-by-step answer:
From the question we get to know that a gift packed by a sweet shop is a combination of two sweets such as A and B which is of 7 kg i.e.\[A+B=7kg\] .......... (1)
To make this product mix we have to take at least 3 kg of A and not more than 5 kg of B i.e. $A\ge 3$ and $B\le 5$ .
It is given that a shop makes a profit of Rs 15/- on A and Rs 20/- on B per kg.
Let us consider that the product 1 contains: \[A=3kg\] and \[B=4kg.\]
We know that the profit on sweet A is Rs 15/- per kg and we have 3kg. So, we can take the total profit on A as $\left( 15\times 3 \right)$. And the profit on sweet B is Rs 20/- per kg and we have 4kg. So, we can take the total profit on sweet B as $\left( 4\times 20 \right)$ .
Thus, we will get the profit earned on product 1 by taking the total profit earned on A and B in equation (1) we get –
 $\left( 3\times 15 \right)+\left( 4\times 20 \right)$
$=45+80$
$=125.$
Let us consider that the product 2 contains: \[A=4kg\] and \[B=3kg\]
We know that the profit on sweet A is Rs 15/- per kg and we have 4kg. So, we can take the total profit on A as $\left( 15\times 4 \right)$. And the profit on sweet B is Rs 20/- per kg and we have 3kg. So, we can take the total profit on sweet B as $\left( 3\times 20 \right)$ .
Thus, we will get the profit earned on product 2 by taking the total profit earned on A and B in equation (1) we get –
$\left( 4\times 15 \right)+\left( 3\times 20 \right)$ .
= 60+60
= 120.
Therefore, the profit earned on product 1 > product 2.
Hence, the product mix of \[A\text{ }=\text{ }3kg\] and \[B\text{ }=\text{ }4kg\] can obtain maximum profit.

Note: Students can solve this problem directly on a logical basis.
For example: The shop makes a maximum profit of Rs 20/- on B than A per kg, we will take the maximum amount of B in the product mix.
As the maximum limit of B is 5 kg in a product for which we have to take \[A\text{ }=\text{ }2\text{ }kg\] , that can’t be possible because, the minimum limit of A in a product mix is 3 kg.So, we will take 4 kg of B and 3 kg of A to obtain maximum profit.