a bag contains 12 red marbles, 5 yellow marbl...
a bag contains 12 red marbles, 5 yellow marbles, and 15 green marbles. how many additional red marbles must be added to the 32 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5? f. 13 g. 18 h. 28 j. 32 k. 40 10. 1 point the coordinates of the endpoints of cd, in the standard (x,y) coordinate plane, are (-4,-2) and (14,2). what is the x - coordinate of the midpoint of cd? a. 0 b. 2 c. 5 d. 9 e. 10
Answer
### Explanation:
#### Step1: Set up probability equation
Let $x$ be the number of additional red - marbles. The total number of marbles originally is $12 + 5+15=32$. After adding $x$ red marbles, the number of red marbles is $12 + x$ and the total number of marbles is $32 + x$. The probability of drawing a red marble is $\frac{12 + x}{32 + x}$, and we want this probability to be $\frac{3}{5}$. So we have the equation $\frac{12 + x}{32 + x}=\frac{3}{5}$.
#### Step2: Cross - multiply
Cross - multiplying the equation $\frac{12 + x}{32 + x}=\frac{3}{5}$ gives us $5(12 + x)=3(32 + x)$.
Expanding both sides: $60+5x = 96+3x$.
#### Step3: Solve for $x$
Subtract $3x$ from both sides: $60 + 5x-3x=96+3x - 3x$, which simplifies to $60 + 2x=96$.
Subtract 60 from both sides: $2x=96 - 60$, so $2x = 36$.
Divide both sides by 2: $x = 18$.
#### Step4: Find mid - point $x$ - coordinate
The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For the points $(-4,-2)$ and $(14,2)$, the $x$ - coordinate of the mid - point is $\frac{-4 + 14}{2}=\frac{10}{2}=5$.
### Answer:
G. 18
C. 5