In a container, the quantities of red, blue, and green balls are in the ratio 2 : 3 : 5. If 15 red balls, 10 blue balls, and an unknown number of green balls are added to the container, the new ratio of red, blue, and green balls becomes 3 : 4 : 6. Find the number of green balls added.

The correct answer is 0.

Step 1: Define initial quantities using a common multiplier

The initial ratio of red, blue, and green balls is given as 2 : 3 : 5.

Let the common ratio multiplier be $x$.

Therefore, the initial number of balls can be represented as:

Red balls = $2x$

Blue balls = $3x$

Green balls = $5x$

Step 2: Represent the new quantities after adding balls

We are given that 15 red balls, 10 blue balls, and an unknown number of green balls (let this be $g$) are added to the container.

The new quantities of each color become:

New Red balls = $2x+15$

New Blue balls = $3x+10$

New Green balls = $5x+g$

Step 3: Solve for $x$ using the red and blue ratio

The new ratio of Red : Blue : Green is 3 : 4 : 6.

Using the ratio of Red to Blue balls:

$\frac{2x+15}{3x+10}=\frac{3}{4}$

Cross-multiply to solve for $x$:

$4(2x+15)=3(3x+10)$

$8x+60=9x+30$

$9x-8x=60-30$

$x=30$

Step 4: Calculate the number of green balls added ($g$)

Now, set up the ratio for Blue to Green balls, which is 4 : 6 (or simplified as 2 : 3):

$\frac{3x+10}{5x+g}=\frac{4}{6}$

Substitute $x=30$ into the equation:

$\frac{3\left(30\right)+10}{5\left(30\right)+g}=\frac{2}{3}$

$\frac{90+10}{150+g}=\frac{2}{3}$

$\frac{100}{150+g}=\frac{2}{3}$

Cross-multiplying gives:

$2(150+g)=3\times 100$

$300+2g=300$

$2g=300-300$

$2g=0$

$g=0$

Therefore, the number of green balls added to the container is 0.