A salt and water mixture, of which 12.5% is salt, costs ₹22 per litre. Another salt and water mixture, of which 27.5% is salt, costs ₹38.50 per litre. How many litres of a salt and water mixture, of which 30% is salt, can be bought for ₹165?

Correct option is C

Given:

First mixture: 12.5% salt, costs ₹22 per litre.

Second mixture: 27.5% salt, costs ₹38.50 per litre.

Required: How many litres of a mixture, of which 30% is salt, can be bought for ₹165?

Formula Used:

Using the allegation rule to find the required number of litres.

Solution:

Mixture      Salt               Price

P              12.5% →       Rs. 22

Q             27.5% →       Rs. 38.50

R             30% →          Rs. 165

Mixture Q can be formed by mixing P and R

1             :                     6

Cost should be in same ratio:

Let, salt of last mixture be R. x /litres

Cost →  22        :          x

Ratio → 1         :           6

Average = Rs. 38.50 = $$\frac{77}{2}$$​​

Average = $$\frac{22\times 1 + 6 \times x}{7}$$

$$\frac{77}{2}$$ = $$\frac{22 + 6 x}{7}$$ 

12x = 539 - 44

12x = 495

x = $$\frac{495}{12}$$ = $$\frac{165}{4}$$

Then amount = $$\frac{\frac{165}{1}}{\frac{165}{4}}$$ = 4 litres

Therefore, 4 litres can be bought for ₹165.

Thus the correct option is (c) 4

Alternate Method:

Cost difference for the increase in salt percentage:

First mixture has 12.5% salt, costs ₹22 per litre.

Second mixture has 27.5% salt, costs ₹38.50 per litre.

The difference in salt percentage between the two mixtures is:

27.5% − 12.5% = 15%

The difference in cost for this 15% increase in salt concentration is:

₹38.50 - ₹22 = ₹16.50

So, the cost difference for a 15% increase in salt content is ₹16.50.

The cost increases for each 1% increase in salt concentration.

​$$\frac{₹16.50}{15} = ₹1.10$$​​

This means that for each 1% increase in salt content, the cost increases by ₹1.10.

The first mixture (12.5% salt) costs ₹22 per litre.

The cost increase for moving from 12.5% salt to 30% salt is:

​$$(30 - 12.5) \times ₹1.10 = 17.5 \times ₹1.10 = ₹19.25$$​​

So, the cost of the mixture with 30% salt will be:

₹22 + ₹19.25 = ₹41.25

Thus, the cost of 1 litre of the new mixture is ₹41.25.

Calculating the number of litres that can be bought for ₹165:

​$$\frac{₹165}{₹41.25} = 4 \text{ litres}$$​​

So, you can buy 4 litres of the mixture with 30% salt for ₹165.