Multiple choice

When equal volume of two substances are mixed, the specific gravity of mixture is $4$. When equal weights of the same substances are mixed, the specific gravity of the mixture is $3$. The ratio of the specific gravity of two substances would be

  1. $2.0$
  2. $3.0$
  3. $3.5$
  4. $4.5$
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B Correct answer
Explanation

Let densities be d1, d2. Equal volume: (d1+d2)/2 = 4 => d1+d2 = 8. Equal weight: 2/(1/d1 + 1/d2) = 3 => 2d1d2/(d1+d2) = 3. 2d1d2/8 = 3 => d1d2 = 12. Roots of x^2 - 8x + 12 = 0 are 6 and 2. Ratio is 6/2 = 3.

AI explanation

When substances of specific gravities a and b are mixed in equal volumes, the mixture specific gravity of 4 equals the average of the two, meaning (a + b) / 2 = 4 and a + b = 8. When mixed in equal weights, the mixture specific gravity of 3 is given by the harmonic mean formula 2ab / (a + b) = 3. Substituting a + b = 8 into the second equation yields 2ab / 8 = 3, meaning the product ab = 12, and solving the system gives the ratio of the specific gravities as 3.0.

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