Question

A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint available to make the desired amount of the mixture?
(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

Hint:

For a Yes/No Data Sufficiency question, you do not need to find a specific value. You just need enough information to say "always yes" or "always no." If the information could lead to either a "yes" or a "no," it is insufficient.

Answer 1

Step 1: Understanding the Concept:
This is a Yes/No Data Sufficiency question based on ratios. To answer the question "is there enough of the yellow paint," we need to compare the amount of yellow paint required with the amount available.
Amount Required \(\le\) Amount Available \(\rightarrow\) Yes
Amount Required \(>\) Amount Available \(\rightarrow\) No
The ratio of Blue:Yellow:Red is 2:3:1. The total number of parts in the ratio is \(2+3+1=6\). Therefore, yellow paint makes up \(3/6 = 1/2\) of the total mixture.
Step 2: Detailed Explanation:
Evaluating Statement (1) Alone:
"Exactly 20 quarts of the mixture are needed."
From this, we can calculate the amount of yellow paint required.
Yellow Paint Required = (Fraction of Yellow) \(\times\) (Total Mixture)
\[ \text{Yellow Paint Required} = \frac{1}{2} \times 20 = 10 \text{ quarts} \] However, we do not know how much yellow paint is available. We cannot answer the question. This statement is not sufficient.
Evaluating Statement (2) Alone:
"Exactly 10 quarts of the yellow paint are available."
This tells us the amount of yellow paint available. However, we do not know the "desired amount of the mixture," so we cannot calculate how much yellow paint is required. This statement is not sufficient.
Evaluating Statements (1) and (2) Together:
From statement (1), we know that 10 quarts of yellow paint are required.
From statement (2), we know that 10 quarts of yellow paint are available.
Since the amount required (10 quarts) is equal to the amount available (10 quarts), we can definitively answer "Yes," there is enough yellow paint.
Because we can provide a definitive Yes/No answer, the statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient. This corresponds to option (C).