Question

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

Accepted Answer

B. Let the strengths of A, B, C be $a, b, c$ respectively. From first mixture (ratio 1 : 2 : 3, total 6 parts): $$\frac{a + 2b + 3c}{6} = 20 \implies a + 2b + 3c = 120$$ From second mixture (ratio 3 : 2 : 1, total 6 parts): $$\frac{3a + 2b + c}{6} = 30 \implies 3a + 2b + c = 180$$ Subtracting first equation from second: $$2a - 2c = 60 \implies a - c = 30 \implies c = a - 30$$ Substitute $c = a - 30$ into first equation: $$a + 2b + 3(a - 30) = 120$$ $$4a + 2b - 90 = 120 \implies 4a + 2b = 210 \implies 2b = 210 - 4a$$ Strength of solution D (mixing B and C in ratio 2 : 7): $$d = \frac{2b + 7c}{9} = \frac{(210 - 4a) + 7(a - 30)}{9} = \frac{210 - 4a + 7a - 210}{9} = \frac{3a}{9} = \frac{a}{3}$$ Ratio of strength of D to A is $\frac{d}{a} = \frac{a/3}{a} = \frac{1}{3} = 1 : 3$.

Ratio of salt solution strengths

Explanation

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