question_answer

A probability of selecting a red ball at random from a jar that contains only red, blue and orange is \[\frac{1}{4}\]. The probability of selecting a blue ball at random from the same jar is \[\frac{1}{3}\]. If the jar contains 10 orange balls, find the total number of balls in the jar.

Answer:

Given, a jar contains red, blue and orange balls.

Let the number of red balls \[=x\]

and the number of blue balls \[=y\]

Number of orange bails \[=10\]

Then, total number of balls \[=x+y+10\]

Let P be the probability of selecting a red ball from the jar

\[P=\frac{x}{x+y+10}\]

But P (a red balls) \[=\frac{1}{4}\] (Given)

\[\therefore \frac{1}{4}=\frac{x}{x+y+10}\]

\[x+y+10=4x\]

\[3x-y=10\]                                            ?(i)

Similarly, P (a blue ball) \[=\frac{y}{x+y+10}\]

But      P (a blue ball) \[=\frac{1}{3}\]

\[\therefore \frac{1}{3}=\frac{y}{x+y+10}\]

\[x+y+10=3y\]

\[x-2y=-10\]                                           ?(ii)

On multiplying equation (ii) by 3, we get

\[3x-6y=-30\]                                           ...(iii)

On subtracting equation (iii) from (i)

\[_{\begin{smallmatrix}  3x\,\,-\,\,6y\,\,=\,\,-30 \\  -\,\,\,\,\,\,\,+\,\,\,\,\,\,\,\,\,\,\,\,\,\,+  \\  \overline{\,\,\,\,\,\,\,\,\,\,\,\,\,5y\,\,=\,\,40\,\,\,\,\,\,\,} \end{smallmatrix}}^{3x\,\,-\,\,y\,\,=\,\,10}\]

\[y=8\]

On putting the value of y in (iii), we get

\[3x-6\times 8=-30\]

\[3x=-30+48\]

\[x=\frac{18}{3}\]

\[x=6\]

Total number of balls \[=x+y+10\]

\[=6+8+10\]

\[=24\]

Hence, total number of balls in the jar is 24.