  Question 32

# The strength of a salt solution is p% if 100 ml of the solution contains p grams of salt. If three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%. If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%. A fourth solution, D, is produced by mixing B and C in the ratio 2 : 7. The ratio of the strength of D to that of A is

Solution

Let 'a', 'b' and 'c' be the concentration of salt in solutions A, B and C respectively.

It is given that three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%.

$$\Rightarrow$$ $$\dfrac{a+2b+3c}{1+2+3} = 20$$

$$\Rightarrow$$ $$a+2b+3c = 120$$ ... (1)

If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%.

$$\Rightarrow$$ $$\dfrac{3a+2b+c}{1+2+3} = 30$$

$$\Rightarrow$$ $$3a+2b+c = 180$$ ... (2)

From equation (1) and (2), we can say that

$$\Rightarrow$$ $$b+2c = 45$$

$$\Rightarrow$$ $$b = 45 - 2c$$

Also, on subtracting (1) from (2), we get

$$a - c = 30$$

$$\Rightarrow$$ $$a = 30 + c$$

In solution D, B and C are mixed in the ratio 2 : 7

So, the concentration of salt in D = $$\dfrac{2b + 7c}{9}$$ = $$\dfrac{90 - 4c + 7c}{9}$$ = $$\dfrac{90 + 3c}{9}$$

Required ratio = $$\dfrac{90 + 3c}{9a}$$ = $$\dfrac{90 + 3c}{9 (30 + c)}$$ = $$1 : 3$$

Hence, option B is the correct answer.

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