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# Each of the 25 balls in a certain box is either red, blue or

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04 May 2015, 11:14
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Why is the following logic wrong?

1) P(W and E) = P(W)*P(E) = 0 --> P(W) or P(E) or both are zero. Insufficient.
2) P(W) - P(E) = .2. Insufficient.
1+2) P(W)=.2, P(E)=0. Therefore P(W) + P(E) - P(W and E) = .2

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Re: Each of the 25 balls in a certain box is either red, blue or [#permalink]

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04 May 2015, 21:18
swaggerer wrote:
Why is the following logic wrong?

1) P(W and E) = P(W)*P(E) = 0 --> P(W) or P(E) or both are zero. Insufficient.
2) P(W) - P(E) = .2. Insufficient.
1+2) P(W)=.2, P(E)=0. Therefore P(W) + P(E) - P(W and E) = .2

Under what conditions is P(W and E) = P(W)*P(E)? Is it always true?
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17819 [0], given: 235 e-GMAT Representative Joined: 04 Jan 2015 Posts: 746 Kudos [?]: 2164 [1], given: 123 Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 05 May 2015, 04:38 1 This post received KUDOS Expert's post swaggerer wrote: Why is the following logic wrong? 1) P(W and E) = P(W)*P(E) = 0 --> P(W) or P(E) or both are zero. Insufficient. Think about it like this. (Represent the given info in a tree structure as shown below). We are given that every ball is either Red or Blue or White. We are also given that ball of each color can have either an even number or an odd number written on them. Now from statement 1, we can see that the number of white balls that have an even number written on them is 0. But this doesn't mean that the total number of white balls is 0. Similarly, this doesn't also mean that there are no even numbers written on other colored balls. (Referring to your inference that P(E) is 0). The correct way to write the above formula would be: P(choosing a white ball with an even number) = P(choosing a white ball out of all balls) * P(choosing an even numbered ball out of all white balls). Notice that the second term is not P(E). Since you chose the short notation, you incorrectly inferred that the P(E) is 0. In fact, the correct inference would be P(choosing Even numbered ball from white balls) = 0) Hope this helps. Regards, Krishna _________________ | '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com Kudos [?]: 2164 [1], given: 123 Manager Joined: 18 Aug 2014 Posts: 129 Kudos [?]: 78 [0], given: 36 Location: Hong Kong Schools: Mannheim Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 23 Aug 2015, 07:14 Bunuel wrote: Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? Probability ball: white - $$P(W)$$; Probability ball: even - $$P(E)$$; Probability ball: white and even - $$P(W&E)$$. Probability ball picked being white or even: $$P(WorE)=P(W)+P(E)-P(W&E)$$. (1) The probability that the ball will both be white and have an even number painted on it is 0 --> $$P(W&E)=0$$ (no white ball with even number) --> $$P(WorE)=P(W)+P(E)-0$$. Not sufficient (2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2 --> $$P(W)-P(E)=0.2$$, multiple values are possible for $$P(W)$$ and $$P(E)$$ (0.6 and 0.4 OR 0.4 and 0.2). Can not determine $$P(WorE)$$. (1)+(2) $$P(W&E)=0$$ and $$P(W)-P(E)=0.2$$ --> $$P(WorE)=2P(E)+0.2$$ --> multiple answers are possible, for instance: if $$P(E)=0.4$$ (10 even balls) then $$P(WorE)=1$$ BUT if $$P(E)=0.2$$ (5 even balls) then $$P(WorE)=0.6$$. Not sufficient. Answer: E. Hope it's clear. Normally in probability problems we say "and" means multiply. Hence why are we not saying P(W) * P(EVEN) = 0 --> Either P(W) or P(EVEN) must be 0 ? Kudos [?]: 78 [0], given: 36 Intern Joined: 16 Jun 2015 Posts: 1 Kudos [?]: [0], given: 1 Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 23 Aug 2015, 22:54 Bunuel wrote: jananijayakumar wrote: But how can this be solved in less than 2 mins??? You can solve this problem in another way. Transform probability into actual numbers and draw the table. Given: Attachment: 1.JPG So we are asked to calculate $$\frac{a+b-c}{25}$$ (we are subtracting $$c$$ not to count twice even balls which are white). (1) The probability that the ball will both be white and have an even number painted on it is 0 --> $$c=0$$ --> $$\frac{a+b}{25}=?$$. Not sufficient. Attachment: 4.JPG (2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2 --> $$\frac{white}{25}-\frac{even}{25}=0.2$$ --> $$white-even=25*0.2=5$$ --> $$a-b=5$$ --> $$b=a-5$$ --> $$\frac{a+a-5-c}{25}=?$$. Not sufficient. Attachment: 2.JPG (1)+(2) $$c=0$$ and $$b=a-5$$ --> $$\frac{a+a-5+0}{25}=\frac{2a-5}{25}$$. Not sufficient. Attachment: 3.JPG Answer: E. I would like one clarification, have we assumed that there is a possibility that a each ball will have a unique number on it, so if we translate the statement the probability of a ball being whit and even is 0, that mean only 5 possible combination of white balls are present (1,3,5,7,9) there fore 10 blue and 10 red balls would be present. then A would be sufficient. would it be explicitly written in the question that each ball has a unique number? Kudos [?]: [0], given: 1 Intern Joined: 06 Dec 2015 Posts: 1 Kudos [?]: [0], given: 0 Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 12 Dec 2015, 03:45 bkk145 wrote: lexis wrote: Each of the 25 balls in a certain box is either red, blue, or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? 1). The probability that the ball will both be white and have an even number painted on it is 0. 2). The probability that the ball will be white minus the probability that have an eve number painted on it is 0.2 I got E. The question is asking for: P(white) + P(even) - P(white&even) = ? (1) is saying: P(white&even) = 0 Still cannot find the answer INSUFFICIENT (2) is saying: P(white) - P(even) = 0.2 We don't know P(white&even), INSUFFICIENT Together, you have P(white) - P(even) = 0.2 and want to find: P(white) + P(even)=? cannot complete the calculation with information given. INSUFFICIENT I have a query related this explanation. Doesn't statement 1 tell us that either P(white) or P(even) is ZERO? If it does, then from statement 2, we can conclude that P(white) + P(even) is also 0.2, and hence the overall answer becomes 0.2; both statements combined become sufficient (C). Explain please! Kudos [?]: [0], given: 0 Intern Joined: 23 May 2016 Posts: 11 Kudos [?]: [0], given: 2 Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 23 May 2016, 13:45 Bunuel wrote: Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? Probability ball: white - $$P(W)$$; Probability ball: even - $$P(E)$$; Probability ball: white and even - $$P(W&E)$$. Probability ball picked being white or even: $$P(WorE)=P(W)+P(E)-P(W&E)$$. (1) The probability that the ball will both be white and have an even number painted on it is 0 --> $$P(W&E)=0$$ (no white ball with even number) --> $$P(WorE)=P(W)+P(E)-0$$. Not sufficient (2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2 --> $$P(W)-P(E)=0.2$$, multiple values are possible for $$P(W)$$ and $$P(E)$$ (0.6 and 0.4 OR 0.4 and 0.2). Can not determine $$P(WorE)$$. (1)+(2) $$P(W&E)=0$$ and $$P(W)-P(E)=0.2$$ --> $$P(WorE)=2P(E)+0.2$$ --> multiple answers are possible, for instance: if $$P(E)=0.4$$ (10 even balls) then $$P(WorE)=1$$ BUT if $$P(E)=0.2$$ (5 even balls) then $$P(WorE)=0.6$$. Not sufficient. Answer: E. Hope it's clear. --- Isnt P(E) = 12 / 25 (each ball has numbers 1- 10, assuming 5 from first 10 balls, another 5 from second 10 balls, and 2 from the remaining 5 balls) which will let us calculate P(W), and then calculate P(W) + P(E) - (PWnE = 0) Kudos [?]: [0], given: 2 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17819 [0], given: 235 Location: Pune, India Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 23 May 2016, 19:31 alizainulabedeen wrote: bkk145 wrote: lexis wrote: Each of the 25 balls in a certain box is either red, blue, or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? 1). The probability that the ball will both be white and have an even number painted on it is 0. 2). The probability that the ball will be white minus the probability that have an eve number painted on it is 0.2 I got E. The question is asking for: P(white) + P(even) - P(white&even) = ? (1) is saying: P(white&even) = 0 Still cannot find the answer INSUFFICIENT (2) is saying: P(white) - P(even) = 0.2 We don't know P(white&even), INSUFFICIENT Together, you have P(white) - P(even) = 0.2 and want to find: P(white) + P(even)=? cannot complete the calculation with information given. INSUFFICIENT I have a query related this explanation. Doesn't statement 1 tell us that either P(white) or P(even) is ZERO? If it does, then from statement 2, we can conclude that P(white) + P(even) is also 0.2, and hence the overall answer becomes 0.2; both statements combined become sufficient (C). Explain please! Statement 1 tells us that P(White AND Even) = 0 That is, there is no ball such that it is white and has an even number painted on it. Obviously, there could be balls that are white and there could be other balls with even numbers painted on them. So P(White) or P(Even) is not given 0. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Each of the 25 balls in a certain box is either red, blue or [#permalink]

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23 May 2016, 19:34
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Expert's post
rjivani wrote:
Bunuel wrote:
Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it?

Probability ball: white - $$P(W)$$;
Probability ball: even - $$P(E)$$;
Probability ball: white and even - $$P(W&E)$$.

Probability ball picked being white or even: $$P(WorE)=P(W)+P(E)-P(W&E)$$.

(1) The probability that the ball will both be white and have an even number painted on it is 0 --> $$P(W&E)=0$$ (no white ball with even number) --> $$P(WorE)=P(W)+P(E)-0$$. Not sufficient

(2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2 --> $$P(W)-P(E)=0.2$$, multiple values are possible for $$P(W)$$ and $$P(E)$$ (0.6 and 0.4 OR 0.4 and 0.2). Can not determine $$P(WorE)$$.

(1)+(2) $$P(W&E)=0$$ and $$P(W)-P(E)=0.2$$ --> $$P(WorE)=2P(E)+0.2$$ --> multiple answers are possible, for instance: if $$P(E)=0.4$$ (10 even balls) then $$P(WorE)=1$$ BUT if $$P(E)=0.2$$ (5 even balls) then $$P(WorE)=0.6$$. Not sufficient.

Hope it's clear.

---

Isnt P(E) = 12 / 25 (each ball has numbers 1- 10, assuming 5 from first 10 balls, another 5 from second 10 balls, and 2 from the remaining 5 balls) which will let us calculate P(W), and then calculate P(W) + P(E) - (PWnE = 0)

What says that numbers are allotted in sequence? There could be 5 balls with number 1 on them, 10 balls with number 2 on them, 3 balls with number 3 on them and then rest of the 7 balls with numbers 4 to 10 on them. Or any other combination.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17819 [2], given: 235 Intern Joined: 02 Oct 2013 Posts: 6 Kudos [?]: [0], given: 238 Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 09 Oct 2016, 10:34 This is my approach. Please correct me where I went wrong. I chose A for this one..... - Label from 1 to 10 >> maximum number of each balls = 10 My curious is, from statement 1, can't we conclude that white balls are only marked with odd numbers? Then, the total number of balls is 25 thus there are 5 white balls, 10 blue balls, and 10 red balls. Blue balls and red balls have 5 even balls each. >>> Even balls = 10 White balls = 5 P(white)+P(Even) = 10/25+5/25 = 3/5 Kudos [?]: [0], given: 238 Board of Directors Status: Aiming MBA Joined: 18 Jul 2015 Posts: 2764 Kudos [?]: 916 [1], given: 67 Location: India GPA: 3.65 WE: Information Technology (Health Care) Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 10 Oct 2016, 09:36 1 This post received KUDOS ariiet wrote: This is my approach. Please correct me where I went wrong. I chose A for this one..... - Label from 1 to 10 >> maximum number of each balls = 10 My curious is, from statement 1, can't we conclude that white balls are only marked with odd numbers? Then, the total number of balls is 25 thus there are 5 white balls, 10 blue balls, and 10 red balls. Blue balls and red balls have 5 even balls each. >>> Even balls = 10 White balls = 5 P(white)+P(Even) = 10/25+5/25 = 3/5 Yes, we can conclude that white balls are only marked with odd numbers. But we have numbers on the balls from 1 to 10 only and we have total 25 balls. SO, it is obvious that numbers are repeated. Now, We don't have any information whether we do have only 5 ODD numbered white balls or more than 5 ODD numbered white balls. So, we cannot conclude the number of white balls. Hence, Statement 1 is also insufficient. I hope it is clear now. Let me know if you have more doubts. _________________ How I improved from V21 to V40! ? Kudos [?]: 916 [1], given: 67 Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 1821 Kudos [?]: 933 [2], given: 3 Location: United States (CA) Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 11 Oct 2016, 16:55 2 This post received KUDOS Expert's post lexis wrote: Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? (1) The probability that the ball will both be white and have an even number painted on it is 0 (2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2 We are given that there are 25 balls in a box and each one is either red, blue, or white and has a number from 1 to 10 painted on it. We need to determine the probability of selecting a white ball or an even-numbered ball. Since each ball is colored and has a number painted on it, selecting a ball with a number or color is not mutually exclusive. Thus, to determine the probability of selecting a white or even ball we use the following formula: P(white or even ball) = P(white ball) + P(even ball) – P(white and even ball) Statement One Alone: The probability that the ball will both be white and have an even number painted on it is 0. Using the information in statement one, we know that P(white and even ball) = 0. However, we still cannot determine the probability of selecting a white or even ball. We can eliminate answer choices A and D. Statement Two Alone: The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2. Statement two does not provide enough information to determine the probability of selecting a white or even ball. We can eliminate answer choice B. Statements One and Two Together: Using the information from statements one and two we still only know the following: P(white or even ball) = P(white ball) + P(even ball) – P(white and even ball) P(white or even ball) = P(white ball) + P(even ball) – 0 Without knowing the sum of P(white ball) and P(even ball), we cannot answer the question. Answer: E _________________ Scott Woodbury-Stewart Founder and CEO GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Kudos [?]: 933 [2], given: 3 Manager Joined: 08 Feb 2016 Posts: 75 Kudos [?]: 6 [0], given: 25 Location: India Concentration: Technology Schools: AGSM '20 (A) GMAT 1: 650 Q49 V30 GPA: 4 Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 04 Feb 2017, 05:49 How do you get to this formula.: $$P(WorE)=P(W)+P(E)-P(W&E)$$ Any underlying concept ? Bunuel wrote: Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? Probability ball: white - $$P(W)$$; Probability ball: even - $$P(E)$$; Probability ball: white and even - $$P(W&E)$$. Probability ball picked being white or even: $$P(WorE)=P(W)+P(E)-P(W&E)$$. (1) The probability that the ball will both be white and have an even number painted on it is 0 --> $$P(W&E)=0$$ (no white ball with even number) --> $$P(WorE)=P(W)+P(E)-0$$. Not sufficient (2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2 --> $$P(W)-P(E)=0.2$$, multiple values are possible for $$P(W)$$ and $$P(E)$$ (0.6 and 0.4 OR 0.4 and 0.2). Can not determine $$P(WorE)$$. (1)+(2) $$P(W&E)=0$$ and $$P(W)-P(E)=0.2$$ --> $$P(WorE)=2P(E)+0.2$$ --> multiple answers are possible, for instance: if $$P(E)=0.4$$ (10 even balls) then $$P(WorE)=1$$ BUT if $$P(E)=0.2$$ (5 even balls) then $$P(WorE)=0.6$$. Not sufficient. Answer: E. Hope it's clear. Kudos [?]: 6 [0], given: 25 Board of Directors Status: Aiming MBA Joined: 18 Jul 2015 Posts: 2764 Kudos [?]: 916 [1], given: 67 Location: India GPA: 3.65 WE: Information Technology (Health Care) Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 04 Feb 2017, 05:55 1 This post received KUDOS ajay2121988 wrote: How do you get to this formula.: $$P(WorE)=P(W)+P(E)-P(W&E)$$ Any underlying concept ? Yes, it is a general formula to determine the probability of Events A or B when both A and B are NOT mutually exclusive. For example, say I have the numbers from 1 to 10. Event A : All even numbers Event B : All multiples of 3. Probability of A or B will be All even + All multiple of 3 - Those that have been repeated(only 6 in this case). I hope it is clear now. _________________ How I improved from V21 to V40! ? Kudos [?]: 916 [1], given: 67 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17819 [1], given: 235 Location: Pune, India Re: Each of the 25 balls in a certain box is either red, blue or [#permalink] ### Show Tags 07 Feb 2017, 09:49 1 This post received KUDOS Expert's post ajay2121988 wrote: How do you get to this formula.: $$P(WorE)=P(W)+P(E)-P(W&E)$$ Any underlying concept ? Exactly same as that of two overlapping sets. n(A or B) = n(A) + n(B) - n(A and B) Why do you subtract n(A and B) out of it? Because it is double counted - once in n(A) and another time in n(B). Similarly, P(A and B) is double counted - once in probability of A and another time in probability of B. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Each of the 25 balls in a certain box is either red, blue or [#permalink]

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25 Jun 2017, 18:34
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lexis wrote:
Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it?

(1) The probability that the ball will both be white and have an even number painted on it is 0

(2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2

The goal is to find the probability of picking a ball that is either white or even-numbered. In other words, P(W) + P(E) - P(W&E).

Statement 1) P(W&E) = 0.

We don't know P(W) or P(E), so this statement is insufficient.

Statement 2) P(W) - P(E) = 0.2

We don't know the value of P(W) + P(E) or P(W&E). Insufficient.

Statements 1+2) P(W&E) = 0 and P(W) - P(E) = 0.2.

We still don't know the value of P(W) + P(E). Insufficient.

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Re: Each of the 25 balls in a certain box is either red, blue or [#permalink]

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09 Oct 2017, 01:16
guys correct me where am i going wrong and lets not get into the grammar details of the question,lets solve it the pure mathematical way:
we need to find =P(w) + P(e) - [P(w)*P(e)]

1) tells us that P(w)* P(e)=0 ,, so either P(w)=0 or P(e)=0 or both =0 {basically w=0 or e=0 or both 0}
2)tells us that P(w)-P(e)=0 or w-e=5

clearly 1) and 2) alone are insufficient

but when we combine both we get w=5. and e=0,hence we can get the answer....

i dont get the flaw in reasoning here ..anyone please?

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Re: Each of the 25 balls in a certain box is either red, blue or [#permalink]

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10 Oct 2017, 09:55
lexis wrote:
Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it?

(1) The probability that the ball will both be white and have an even number painted on it is 0

(2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

...........| White | Not White
------------------------------
.....Even.| a | b
---------------------------------
Not Even.| c | d

=> Q: ( a + b + c ) / 25 ?

a + b + c + d = 25

Since we have 4 variables and 1 equation, E could be the answer most likely.

1) & 2)
From the condition 1), we have a = 0.
From the condition 2), ( a + c ) / 25 - ( a + b ) / 25 = ( c - b ) / 25 = 0.2
c - b = 5

Since we have only 2 equations a = 0 and c - b = 5, we can't figure out b + c and solve this question.

Therefore, the answer is E as expected.

For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80 % chance that E is the answer, while C has 15% chance and A, B or D has 5% chance. Since E is most likely to be the answer using 1) and 2) together according to DS definition. Obviously there may be cases where the answer is A, B, C or D.
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.
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Re: Each of the 25 balls in a certain box is either red, blue or   [#permalink] 10 Oct 2017, 09:55

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