A bag contains 20 balls . These are of three different colours : green red and blue .A ball is chosen at random from the bag . The probability of a green ball is `(1)/(4)`. The probability of a red ball is `(2)/(5)` .
How many balls are blue ?

Answer

To solve the problem step by step, we will first summarize the information given and then calculate the number of blue balls in the bag.

### Step 1: Understand the total number of balls
We know that there are a total of 20 balls in the bag. These balls are of three different colors: green, red, and blue.

### Step 2: Write down the probabilities
The probabilities given in the problem are:
- Probability of choosing a green ball, \( P(G) = \frac{1}{4} \)
- Probability of choosing a red ball, \( P(R) = \frac{2}{5} \)

### Step 3: Set up the equation for total probability
The total probability of all outcomes must equal 1. Therefore, we can express this as:
\[ P(G) + P(R) + P(B) = 1 \]
where \( P(B) \) is the probability of choosing a blue ball.

### Step 4: Substitute the known probabilities
Substituting the known values into the equation gives us:
\[ \frac{1}{4} + \frac{2}{5} + P(B) = 1 \]

### Step 5: Find a common denominator
To combine the fractions, we need a common denominator. The least common multiple of 4 and 5 is 20. We can rewrite the fractions:
\[ \frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{2}{5} = \frac{8}{20} \]
Now substituting these values into the equation:
\[ \frac{5}{20} + \frac{8}{20} + P(B) = 1 \]

### Step 6: Combine the fractions
Now we can combine the fractions:
\[ \frac{5 + 8}{20} + P(B) = 1 \]
This simplifies to:
\[ \frac{13}{20} + P(B) = 1 \]

### Step 7: Solve for \( P(B) \)
To find \( P(B) \), we subtract \( \frac{13}{20} \) from 1:
\[ P(B) = 1 - \frac{13}{20} = \frac{20}{20} - \frac{13}{20} = \frac{7}{20} \]

### Step 8: Relate probability to the number of blue balls
The probability of choosing a blue ball can also be expressed in terms of the number of blue balls. Let \( n_B \) be the number of blue balls. Then:
\[ P(B) = \frac{n_B}{\text{Total number of balls}} = \frac{n_B}{20} \]
We already found that \( P(B) = \frac{7}{20} \). Therefore:
\[ \frac{n_B}{20} = \frac{7}{20} \]

### Step 9: Solve for \( n_B \)
Multiplying both sides by 20 gives:
\[ n_B = 7 \]

### Conclusion
Thus, the number of blue balls in the bag is \( 7 \).

---