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, but the data in statement (2) alone is not sufficient.
• B. If the data in statement (2) alone is sufficient to answer the question, but the data in statement (1) alone is not sufficient.
• 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.
• E. If the data in neither statement (1) nor statement (2) is sufficient to answer the question, and more data is needed.

Hint:

Use volume = mass/density for both statements separately; check whether each equation alone has a unique valid (positive) solution for the density of A.

Answer 1

The Correct Option is D

Convert the mass to grams: 4.8 kg = 4800 g, and this mass is the same for both A and B. Let d be the density of A in g/cc, so density of B is d + 20. Since volume = mass/density, volume of A = 4800/d and volume of B = 4800/(d+20).

Statement 1 says volume of A minus volume of B is 12 cc: 4800/d - 4800/(d+20) = 12. Multiplying out, 4800 x 20 / [d(d+20)] = 12, which simplifies to d(d+20) = 8000, or d^2 + 20d - 8000 = 0. Factoring, (d+100)(d-80) = 0, so d = 80 or d = -100. A density cannot be negative, so d = 80 g/cc is the only valid value. This is a single, definite number, so statement 1 alone is sufficient.

Statement 2 says the ratio of volumes of A to B is 32:27. Since volume is inversely proportional to density for a fixed mass, this means density of B to density of A is 32:27, i.e. (d+20)/d = 32/27. Cross multiplying: 27(d+20) = 32d, so 27d + 540 = 32d, giving 5d = 540, so d = 108 g/cc. This is also a single, definite number, so statement 2 alone is sufficient too.

Since each statement independently lets us solve for a unique value of d (statement 1 gives 80, statement 2 gives 108), each statement alone is sufficient to answer the question on its own terms, which by the standard data-sufficiency rule means the answer is (d) - either statement alone suffices.

Note on the provided answer key: the key marks this as (c). However, verifying the algebra carefully shows statement 1 leads to a solvable quadratic with only one valid (positive) root, and statement 2 leads to a solvable linear equation, so each is genuinely independently sufficient by itself; our worked answer is (d) with this discrepancy flagged for review.