A certain jar contains only \(b\) black marbles, \(w\) white marbles, and \(r\) red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater then the probability that the marble chosen will be white?
(1) \(\dfrac{r}{b+w}>\dfrac{w}{b+r}\)
(2) \(b-w>r\)
Answer: A
Source: GMAT Prep
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A certain jar contains only \(b\) black marbles, \(w\) white marbles and \(r\) red marbles. If one marble is to be chose
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Gmat_mission
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Target question: Is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white?Gmat_mission wrote: ↑Thu Oct 01, 2020 7:11 amA certain jar contains only \(b\) black marbles, \(w\) white marbles, and \(r\) red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater then the probability that the marble chosen will be white?
(1) \(\dfrac{r}{b+w}>\dfrac{w}{b+r}\)
(2) \(b-w>r\)
Answer: A
Source: GMAT Prep
We can rephrase the target question as...
REPHRASED target question: Is r > w?
Statement 1: r/(b + w) > w/(b + r)
Let's let T = the TOTAL number of marbles in the jar.
This means that b + w + r = T
This also means that b + w = T - r
And it means that b + r = T - w
So, we can take statement 1, r/(b + w) > w/(b + r), and rewrite it as...
r/(T - r) > w/(T - w)
Multiply both sides by (T - r) to get: r > w(T - r)/(T - w)
Multiply both sides by (T - w) to get: r(T - w) > w(T - r)
Expand both sides: rT - rw > wT - rw
Add rw to both sides: rT > wT
Divide both sides by T to get: r > w
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: b - w > r
Add w to both sides to get: b > w + r
All this means is that there are more black marbles than there are white and red marbles combined.
Given this information, there's no way to determine whether or not r is greater than w
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
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Brent
Brent Hanneson - Creator of GMATPrepNow.com
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Is \(r>w\)?Gmat_mission wrote: ↑Thu Oct 01, 2020 7:11 amA certain jar contains only \(b\) black marbles, \(w\) white marbles, and \(r\) red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater then the probability that the marble chosen will be white?
(1) \(\dfrac{r}{b+w}>\dfrac{w}{b+r}\)
(2) \(b-w>r\)
Answer: A
Source: GMAT Prep
\(\dfrac{r}{b+w} > w(b+r)\)
\(rb + r^2\) ? \(wb + w^2\)
\((r+w)(r-w) > b(w-r)\)
Now if \(r-w > 0\) then \(w-r < 0\) and inequality holds true.
Other way around if \(r-w < 0\), LHS is negative and RHS is positive and inequality does NOT hold true.
So only valid scenario is \(r-w > 0\) and thus \(r > w\)
Sufficient \(\Large{\color{green}\checkmark}\)
2) Not enough Info \(\Large{\color{red}\chi}\)
Therefore, A
