Data Sufficiency

The figure shows triangle ABC and points D, E, and F, which are on line AB, BC, and CA, respectively. The area of triangle ABC is 15. What is the area of triangle BCF?

1) (BE) : ( EC) = 3 : 4
2) The area of \(\triangle\ BCF\) is equal to the area of Quadrilateral ECFD.

Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem.

Remember that equal numbers of variables and independent equations ensure a solution.

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Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the area of triangle BCF, and since the area of triangle ABC is 15, we should know the ratio of AF: FC to determine the area of triangle BCF.

Follow the second and the third step: From the original condition, we have many variables (many triangles and each triangle has 3 variables). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since we have \(\triangle\ BCF\ =\ \triangle\ BEF\ \ +\ \triangle\ ECF\ \) ,
Quadrilateral ECFD = \(\triangle DEF+\triangle ECF\) and Triangle BCF = Quadrilateral ECFD

Then, we have \(\triangle BEF\ =\ \triangle DEF\)

Since triangles BEF and DEF have a common base of EF, their heights are equal to each other. Thus, AB and EF are parallel to each other.

Then triangles CEF and CBA are similar.

Since CF:AC = EC:BC = 4:7, we have △BCF = ( \(\frac{4}{7}\) ) △ABC = ( \(\frac{4}{7}\) ) * 15 = \(\frac{60}{7}\) .

The answer is unique, so both conditions are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.