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

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by Max@Math Revolution » Mon Feb 04, 2019 11:56 pm

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[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
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Source: — Data Sufficiency |
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by fskilnik@GMATH » Tue Feb 05, 2019 3:54 pm
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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Thu Feb 07, 2019 12:29 am
=>

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.
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