In a bag the ratio of red balls to green balls is `15 : 26`. If 12 more green balls are added to the bag the ratio of red balls to green balls will become `1: 2`. How" many red balls are there in the bag ?

Options

A. 30
B. 45
C. 15
D. 60

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the Variables
Let the number of red balls be represented as \( R \) and the number of green balls be represented as \( G \). According to the problem, the initial ratio of red balls to green balls is given as \( 15:26 \).

### Step 2: Express the Number of Balls in Terms of a Variable
From the ratio \( 15:26 \), we can express the number of red and green balls in terms of a variable \( x \):
- Red balls \( R = 15x \)
- Green balls \( G = 26x \)

### Step 3: Update the Number of Green Balls
According to the problem, 12 more green balls are added. Therefore, the new number of green balls becomes:
\[ G' = G + 12 = 26x + 12 \]

### Step 4: Set Up the New Ratio
After adding the green balls, the new ratio of red balls to green balls is \( 1:2 \). We can express this as:
\[ \frac{R}{G'} = \frac{1}{2} \]

Substituting the expressions for \( R \) and \( G' \):
\[ \frac{15x}{26x + 12} = \frac{1}{2} \]

### Step 5: Cross Multiply to Solve for \( x \)
Cross-multiplying gives us:
\[ 2 \cdot 15x = 1 \cdot (26x + 12) \]
\[ 30x = 26x + 12 \]

### Step 6: Simplify the Equation
Now, we will simplify the equation:
\[ 30x - 26x = 12 \]
\[ 4x = 12 \]

### Step 7: Solve for \( x \)
Dividing both sides by 4:
\[ x = 3 \]

### Step 8: Calculate the Number of Red Balls
Now that we have the value of \( x \), we can find the number of red balls:
\[ R = 15x = 15 \cdot 3 = 45 \]

### Conclusion
The number of red balls in the bag is \( 45 \).