Out of the total pens, 10% of pens are black. 10% of remaining pens are blue. 20% of remaining pens are red. The remaining 648 pens are green. How many total pens are there?

Options

A. 1000
B. 1500
C. 2000
D. 2500

Answer

To solve the problem step by step, we can follow this approach:

1. **Define the Total Number of Pens**:
Let the total number of pens be \( N \).

2. **Calculate the Number of Black Pens**:
According to the problem, 10% of the total pens are black.
\[ \text{Number of black pens} = 10\% \text{ of } N = \frac{N}{10} \]

3. **Calculate the Remaining Pens After Black Pens**:
After removing the black pens, the remaining number of pens is:
\[ \text{Remaining pens} = N - \frac{N}{10} = \frac{9N}{10} \]

4. **Calculate the Number of Blue Pens**:
Now, 10% of the remaining pens are blue:
\[ \text{Number of blue pens} = 10\% \text{ of remaining} = 10\% \text{ of } \frac{9N}{10} = \frac{9N}{100} \]

5. **Calculate the Remaining Pens After Blue Pens**:
The remaining number of pens after removing blue pens is:
\[ \text{Remaining pens} = \frac{9N}{10} - \frac{9N}{100} \]
To simplify this, we need a common denominator:
\[ = \frac{90N}{100} - \frac{9N}{100} = \frac{81N}{100} \]

6. **Calculate the Number of Red Pens**:
Now, 20% of the remaining pens are red:
\[ \text{Number of red pens} = 20\% \text{ of remaining} = 20\% \text{ of } \frac{81N}{100} = \frac{81N}{500} \]

7. **Calculate the Remaining Pens After Red Pens**:
The remaining number of pens after removing red pens is:
\[ \text{Remaining pens} = \frac{81N}{100} - \frac{81N}{500} \]
Again, we need a common denominator:
\[ = \frac{405N}{500} - \frac{81N}{500} = \frac{324N}{500} \]

8. **Set the Remaining Pens Equal to 648**:
According to the problem, the remaining pens are 648:
\[ \frac{324N}{500} = 648 \]

9. **Solve for \( N \)**:
To find \( N \), we can cross-multiply:
\[ 324N = 648 \times 500 \]
\[ N = \frac{648 \times 500}{324} \]
Simplifying this:
\[ N = \frac{648 \times 500}{324} = 1000 \]

10. **Final Answer**:
Therefore, the total number of pens is \( N = 1000 \).