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There are $50$ students in an auditorium, of which $2x$ are boys and $y$ are girls. After $(y-6)$ boys leave the auditorium and $(2x -5)$ girls enter the auditorium, the probability of selecting a girl at random becomes $\frac{9}{13}$ . Find the value of $x$ and $y$ .

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Identify independent events

Full solution

Q. There are

50

students in an auditorium, of which

2x

are boys and

y

are girls. After

(y-6)

boys leave the auditorium and

(2x -5)

girls enter the auditorium, the probability of selecting a girl at random becomes

\frac{9}{13}

. Find the value of

x

and

y

.

Initial Setup: Initial number of boys is $2x$ and girls is $y$ , so total students is $50$ . $\newline$ $2x + y = 50$
Boys Leaving: After $y-6$ boys leave, the number of boys becomes $2x - (y-6)$ . Simplify to get $2x - y + 6$ .
Girls Entering: After $2x-5$ girls enter, the number of girls becomes $y + (2x-5)$ . Simplify to get $y + 2x - 5$ .
New Probability Ratio: The new probability of selecting a girl is $\frac{9}{13}$ , so the ratio of girls to total students is: $\newline$ $\frac{y + 2x - 5}{2x - y + 6 + y + 2x - 5} = \frac{9}{13}$ $\newline$ Simplify the denominator to get $4x + 1$ .
Probability Equation: Set up the equation from the probability: $\newline$ $(y + 2x - 5) / (4x + 1) = \frac{9}{13}$ $\newline$ Cross multiply to solve for $x$ and $y$ . $\newline$ $13(y + 2x - 5) = 9(4x + 1)$
Equation Simplification: Distribute both sides: $13y + 26x - 65 = 36x + 9$
Isolating Variables: Rearrange the equation to isolate terms with $x$ and $y$ on one side: $13y - 65 = 36x + 9 - 26x$ Simplify to get $13y - 65 = 10x + 9$
Solving for y: Add $65$ to both sides to isolate $y$ : $\newline$ $13y = 10x + 74$ $\newline$ Divide by $13$ to solve for $y$ in terms of $x$ : $\newline$ $y = \frac{10x + 74}{13}$
Substitute $y$ into First Equation: Substitute $y$ back into the first equation: $\newline$ $2x + \frac{10x + 74}{13} = 50$ $\newline$ Multiply through by $13$ to clear the fraction: $\newline$ $26x + 10x + 74 = 650$
Calculate $x$ : Combine like terms: $\newline$ $36x + 74 = 650$ $\newline$ Subtract $74$ from both sides: $\newline$ $36x = 576$ $\newline$ Divide by $36$ to solve for $x$ : $\newline$ $x = \frac{576}{36}$
Calculate $y$ : Calculate $x$ : $x = 16$
Calculate $y$ : Calculate $x$ : $x = 16$ Substitute $x$ back into the equation for $y$ : $y = \frac{(10(16) + 74)}{13}$ Calculate $y$ : $y = \frac{(160 + 74)}{13}$ $y = \frac{234}{13}$ $y = 18$

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