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 [tex]$\frac{3}{5}$[/tex]?
A. 11
B. 16
C. 20
D. 24
E. 32
Answer (1)
Set up the equation: 24 + x 8 + x = 5 3 .
Cross-multiply: 5 ( 8 + x ) = 3 ( 24 + x ) .
Expand and simplify: 40 + 5 x = 72 + 3 x ⇒ 2 x = 32 .
Solve for x : x = 16 .
Explanation
Analyze the problem We begin by analyzing the problem. We have a bag with 8 red marbles, 9 yellow marbles, and 7 green marbles. The total number of marbles is 8 + 9 + 7 = 24 . We want to add some red marbles, say x red marbles, such that the probability of drawing a red marble is 5 3 .
Set up the equation After adding x red marbles, the number of red marbles becomes 8 + x , and the total number of marbles becomes 24 + x . The probability of drawing a red marble is then 24 + x 8 + x . We want this probability to be equal to 5 3 . So we set up the equation: 24 + x 8 + x = 5 3
Cross-multiply and expand To solve the equation, we cross-multiply: 5 ( 8 + x ) = 3 ( 24 + x ) Expanding both sides, we get: 40 + 5 x = 72 + 3 x
Solve for x Now, we want to isolate x . Subtract 3 x from both sides: 40 + 5 x − 3 x = 72 + 3 x − 3 x 40 + 2 x = 72 Subtract 40 from both sides: 40 + 2 x − 40 = 72 − 40 2 x = 32 Divide both sides by 2: 2 2 x = 2 32 x = 16
State the answer Therefore, we need to add 16 red marbles to the bag so that the probability of randomly drawing a red marble is 5 3 .
Examples
Imagine you are designing a game with colored marbles. Initially, you have a bag with different colored marbles, and you want to adjust the number of red marbles to achieve a specific probability of drawing a red marble. This problem helps you determine how many red marbles you need to add to reach that desired probability, ensuring the game is balanced and fair.
