Question

a bag contains 8 red marbles, 9 yellow marbles, and 7 green marbles. how many additional red marbles must be added to the 24 marbles already in the bag so that the probability of randomly drawing a red marble is \\(\frac{3}{5}\\)\
11\
16\
20\
24\
32

Explanation:

Step1: Define variables

Let \( x \) be the number of additional red marbles added.
The initial number of red marbles is \( 8 \), so the new number of red marbles is \( 8 + x \).
The initial total number of marbles is \( 24 \), so the new total number of marbles is \( 24 + x \).

Step2: Set up the probability equation

The probability of drawing a red marble is the number of red marbles divided by the total number of marbles. We want this probability to be \( \frac{3}{5} \), so:
\[
\frac{8 + x}{24 + x} = \frac{3}{5}
\]

Step3: Cross - multiply to solve for \( x \)

Cross - multiplying gives \( 5(8 + x)=3(24 + x) \).
Expand both sides: \( 40+5x = 72 + 3x \).
Subtract \( 3x \) from both sides: \( 40 + 2x=72 \).
Subtract \( 40 \) from both sides: \( 2x=72 - 40=32 \).
Divide both sides by \( 2 \): \( x = 16 \).

Answer:

16 (corresponding to the option "16")