Understanding the Provision Problem

The question asks for the minimum and maximum cost of purchasing a total of 5 kg of provisions, consisting of two varieties of rice (A and B) and two varieties of pulses (C and D). There are specific cost per kg for each variety and also minimum quantity constraints for purchasing each variety.

Let's list the given information:

The total quantity to be purchased is 5 kg.

Let the quantities (in kg) of varieties A, B, C, and D purchased be denoted by \(a\), \(b\), \(c\), and \(d\), respectively.

The conditions are:

• Total quantity: \(a + b + c + d = 5\)
• Minimum quantity constraints: \(a \ge 1\), \(b \ge 1\), \(c \ge 0.5\), \(d \ge 0.5\)
• Quantities must be non-negative: \(a, b, c, d \ge 0\) (The minimum constraints already ensure this)

The total cost is given by the formula:

\(\text{Cost} = 30a + 90b + 100c + 120d\)

Cost of Minimum Required Quantities

First, let's calculate the total quantity and cost if we only buy the minimum required amount of each variety:

• Minimum quantity of A = 1 kg
• Minimum quantity of B = 1 kg
• Minimum quantity of C = 0.5 kg
• Minimum quantity of D = 0.5 kg

Total minimum quantity = \(1 + 1 + 0.5 + 0.5 = 3\) kg.

Cost of minimum quantities = \(30(1) + 90(1) + 100(0.5) + 120(0.5)\)

= \(30 + 90 + 50 + 60 = 230\) Rs.

We need a total of 5 kg. Since the minimum required quantity is 3 kg, the remaining quantity to purchase is \(5 - 3 = 2\) kg.

This remaining 2 kg must be added to the quantities already purchased, such that the total quantities still satisfy the minimum constraints (which they will, as we are only adding to the minimums) and the total quantity is 5 kg.

Minimum Cost Calculation

To find the minimum total cost, we should add the remaining 2 kg to the variety with the lowest cost per kg. Looking at the costs (A=30, B=90, C=100, D=120), variety A has the lowest cost (Rs. 30 per kg).

So, we add the remaining 2 kg to variety A.

• Quantity of A: \(1 \text{ kg (minimum)} + 2 \text{ kg (remaining)} = 3\) kg
• Quantity of B: 1 kg (minimum)
• Quantity of C: 0.5 kg (minimum)
• Quantity of D: 0.5 kg (minimum)

Total quantity = \(3 + 1 + 0.5 + 0.5 = 5\) kg (Correct total).

Minimum Cost = \(30(\text{Quantity of A}) + 90(\text{Quantity of B}) + 100(\text{Quantity of C}) + 120(\text{Quantity of D})\)

= \(30(3) + 90(1) + 100(0.5) + 120(0.5)\)

= \(90 + 90 + 50 + 60\)

= \(290\) Rs.

Maximum Cost Calculation

To find the maximum total cost, we should add the remaining 2 kg to the variety with the highest cost per kg. Looking at the costs (A=30, B=90, C=100, D=120), variety D has the highest cost (Rs. 120 per kg).

So, we add the remaining 2 kg to variety D.

• Quantity of A: 1 kg (minimum)
• Quantity of B: 1 kg (minimum)
• Quantity of C: 0.5 kg (minimum)
• Quantity of D: \(0.5 \text{ kg (minimum)} + 2 \text{ kg (remaining)} = 2.5\) kg

Total quantity = \(1 + 1 + 0.5 + 2.5 = 5\) kg (Correct total).

Maximum Cost = \(30(\text{Quantity of A}) + 90(\text{Quantity of B}) + 100(\text{Quantity of C}) + 120(\text{Quantity of D})\)

= \(30(1) + 90(1) + 100(0.5) + 120(2.5)\)

= \(30 + 90 + 50 + 300\)

= \(470\) Rs.

Summary of Costs

• Minimum Cost = Rs. 290
• Maximum Cost = Rs. 470

The minimum and maximum costs of a total of 5 kg of these provisions are Rs. 290 and Rs. 470, respectively.