A bag contain 20 balls which x are red balls and y are blue balls. When 5 red balls were added and 10 blue balls were removed, the probability of getting a blue ball is ⅓. Find x and y.

We are given the following problem:

- A bag contains 20 balls in total, where \( x \) are red balls and \( y \) are blue balls. So, the initial total number of balls is:
  \[
  x + y = 20
  \]
- Then, 5 red balls are added and 10 blue balls are removed. The new number of red balls is \( x + 5 \), and the new number of blue balls is \( y - 10 \).
- The probability of drawing a blue ball after these changes is \( \frac{1}{3} \). The probability of drawing a blue ball is calculated as:
  \[
  \frac{y - 10}{(x + 5) + (y - 10)} = \frac{1}{3}
  \]

Now, we can set up our system of equations.

### Step 1: Set up equations
From the total number of balls initially, we have:
\[
x + y = 20 \tag{1}
\]
From the new probability condition, we have:
\[
\frac{y - 10}{(x + 5) + (y - 10)} = \frac{1}{3}
\]
Simplifying the denominator:
\[
\frac{y - 10}{x + y - 5} = \frac{1}{3} \tag{2}
\]

### Step 2: Solve the system of equations
From equation \((2)\), cross-multiply to eliminate the fraction:
\[
3(y - 10) = x + y - 5
\]
Expanding and simplifying:
\[
3y - 30 = x + y - 5
\]
\[
3y - y = x - 5 + 30
\]
\[
2y = x + 25 \tag{3}
\]

### Step 3: Substitute into the first equation
Substitute equation \((3)\) into equation \((1)\):
\[
x + y = 20
\]
\[
(2y - 25) + y = 20
\]
\[
3y - 25 = 20
\]
\[
3y = 45
\]
\[
y = 15
\]

### Step 4: Find \( x \)
Substitute \( y = 15 \) into equation \((1)\):
\[
x + 15 = 20
\]
\[
x = 5
\]

### Final Answer:
The number of red balls \( x \) is 5, and the number of blue balls \( y \) is 15.

Let me know if you'd like more details or have any questions!

Here are 5 related questions:
1. How is probability affected if we add or remove balls?
2. What if 10 red balls were added instead of 5?
3. What happens to the total number of balls when adding or removing?
4. How do we calculate probability for other ratios?
5. Can this same method apply to different objects, not just balls?

**Tip:** Always double-check the total number of items after changes to avoid calculation errors!

Explore how to solve probability problems involving red and blue balls. Learn step-by-step how to determine the number of red and blue balls in a bag after adding and removing balls. Suitable for advanced high school students.

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