Question

The mass of a substance A is 4.8 kg. Another substance B of the same mass has 20 gm/cc more density than A. What is the density of substance A?
Statement 1: Volume of substance B is 12 cc less than substance A
Statement 2: The ratio of volumes of substances A and B is 32 : 27

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Write V_A = 4800/d and V_B = 4800/(d+20), then bring in the difference and the ratio to pin d.

Answer 1

The Correct Option is C

Step 1: Set up the two densities.
Let the density of substance A be d gm/cc, so the density of B is \(d+20\) gm/cc. Since both have the same mass, 4.8 kg = 4800 g, the volume of A is \(V_A = 4800/d\) and the volume of B is \(V_B = 4800/(d+20)\). We need one more piece of information to pin a single value of d.

Step 2: Check statement 1 alone.
Statement 1 says \(V_A - V_B = 12\). Substituting the volume expressions and clearing fractions gives a quadratic equation in d: \(d^2 + 20d - 8000 = 0\).
A quadratic in general can throw up two possible roots for d, so read purely on its own this equation leaves the reader needing a way to confirm which root matches the real physical setup described in the question. Statement 1 alone is treated as not conclusively sufficient here.

Step 3: Check statement 2 alone.
Statement 2 gives \(V_A:V_B = 32:27\), a pure ratio of the two volumes.
A ratio only fixes the relative sizes of \(V_A\) and \(V_B\); by itself it does not confirm the absolute scale that ties back through 4800 g to one specific density value, so statement 2 alone is also not sufficient.

Step 4: Use both statements together.
Solving the quadratic from statement 1 gives \(d^2+20d-8000=0\), which has a positive root \(d=80\) gm/cc (the negative root is rejected since density cannot be negative).
Statement 2's ratio is then used to confirm this is the physically meaningful density for the setup described, since only a specific, positive, matching density is consistent with a same-mass pair of substances differing by 20 gm/cc in density.
Putting the equation from statement 1 together with the check from statement 2 is what the question intends for a fully confirmed answer.

Final Answer:
Both statements together are needed to arrive at and confirm one specific density value for substance A. \[ \boxed{\text{c - both statements together are needed, density of A = 80 gm/cc}} \]