A crate contains green and red apples in the ratio 7 : 11 . When ten green apples and ten red apples are removed from the crate , the ratio becomes 9 : 17. How many red apples were originally in the bag ?

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Define Variables
Let the number of green apples be \( 7X \) and the number of red apples be \( 11X \), where \( X \) is a positive constant.

### Step 2: Set Up the Initial Ratio
The initial ratio of green apples to red apples is given as \( 7:11 \). This means:
- Green apples = \( 7X \)
- Red apples = \( 11X \)

### Step 3: Account for Removed Apples
When 10 green apples and 10 red apples are removed, the new quantities become:
- Remaining green apples = \( 7X - 10 \)
- Remaining red apples = \( 11X - 10 \)

### Step 4: Set Up the New Ratio
After the removal, the new ratio of green apples to red apples is \( 9:17 \). Therefore, we can set up the equation:
\[ \frac{7X - 10}{11X - 10} = \frac{9}{17} \]

### Step 5: Cross-Multiply to Eliminate the Fraction
Cross-multiplying gives:
\[ 17(7X - 10) = 9(11X - 10) \]

### Step 6: Expand Both Sides
Expanding both sides results in:
\[ 119X - 170 = 99X - 90 \]

### Step 7: Rearrange the Equation
Rearranging the equation to isolate \( X \):
\[ 119X - 99X = 170 - 90 \]
\[ 20X = 80 \]

### Step 8: Solve for \( X \)
Dividing both sides by 20 gives:
\[ X = 4 \]

### Step 9: Calculate the Number of Red Apples
Now that we have \( X \), we can find the number of red apples:
\[ \text{Number of red apples} = 11X = 11 \times 4 = 44 \]

### Conclusion
Thus, the number of red apples originally in the bag is **44**.

---