Max@Math Revolution wrote:[GMAT math practice question]

If a > b > c > d > 0, is d < 4?

1) 1/c + 1/d > 1/2
2) (1/a)+(1/b)+(1/c)+(1/d) = 1

$$a > b > c > d > 0\,\,\,\,\, \Rightarrow \,\,\,\,0 < {1 \over a} < {1 \over b} < {1 \over c} < {1 \over d}\,\,\,\,\left( * \right)$$
$$d\,\,\mathop < \limits^? \,\,4$$

$$\left( 1 \right)\,\,{1 \over c} + {1 \over d} > {1 \over 2}\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( {**} \right)\,\,\,d \ge 4\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,c > 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{1 \over c} + {1 \over d} < {1 \over 4} + {1 \over 4} = {1 \over 2}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{impossible}}$$

$$\left( 2 \right)\,\,\,1 = {1 \over a} + {1 \over b} + {1 \over c} + {1 \over d}\,\,\mathop < \limits^{\left( * \right)} \,\,4\left( {{1 \over d}} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,d\, > \,0} \,\,\,\,\,1 \cdot d < 4\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$


The correct answer is therefore (D).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
The original condition a>b>c>d>0 is equivalent to 0 < 1/a < 1/b < 1/c < 1/d. The question asks if d < 4. This is equivalent to asking if 1/d > 1/4.

By condition 1, 1/d > 1/4 since 1/c < 1/d. So, d < 4. Condition 1) is sufficient.

Condition 2)
Since 0 < 1/a < 1/b < 1/c < 1/d and (1/a)+(1/b)+(1/c)+(1/d)=1, we have 1/a<1/d, 1/b<1/d, 1/c<1/d and 1/a +1/b + 1/c + 1/d < 1/d +1/d +1/d + 1/d = 4/d. Therefore, 1 < 4/d and d < 4.
Condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

Note: This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is hard. For CMT4(B) questions, D is most likely to be the answer.