The pie chart below shows the distribution (in degrees) of the total number of bookshelves manufactured by five companies P, Q, R, S, and T. The table shows the percentage distribution of unsold bookshelves for each company.

| Company | Percentage of Unsold Bookshelves |
|---------|---------------------------------|
| P | 3X% |
| Q | 12% |
| R | 4X% |
| S | 20% |
| T | 40% |

Additional information:

![Screenshot 2025-10-22 23.34.30.png](https://res.cloudinary.com/aptidude/image/upload/v1761159226/aptidude_questions/n9rrvtl90xztcpf2ylma.png)
Given the above, find the difference between the number of sold bookshelves of company Q and the average number of sold bookshelves of companies S and T.

Question

The pie chart below shows the distribution (in degrees) of the total number of bookshelves manufactured by five companies P, Q, R, S, and T. The table shows the percentage distribution of unsold bookshelves for each company.

| Company | Percentage of Unsold Bookshelves |
|---------|---------------------------------|
| P       | 3X%                             |
| Q       | 12%                             |
| R       | 4X%                             |
| S       | 20%                             |
| T       | 40%                             |

Additional information:

![Screenshot 2025-10-22 23.34.30.png](https://res.cloudinary.com/aptidude/image/upload/v1761159226/aptidude_questions/n9rrvtl90xztcpf2ylma.png)
Given the above, find the difference between the number of sold bookshelves of company Q and the average number of sold bookshelves of companies S and T.

Accepted Answer

**Solution:**

1. **Identify the angles for companies P and Q:**
   - Let the angle for P be $p$ degrees.
   - Given, $p + 25 = $ angle for Q.

2. **From the pie chart data:**
   - Angle for P = 75.6° (given)
   - Therefore, angle for Q = 75.6 + 25 = 100.6°

3. **Calculate the total number of bookshelves manufactured:**
   - Since the pie chart represents the total 360°, the number of bookshelves manufactured by each company is proportional to their angle.
   - Total manufactured by S is given as 975.

4. **Find the angle for S:**
   - Let the total number of bookshelves manufactured be $N$.
   - Number manufactured by S = $N 	imes rac{	ext{angle of S}}{360} = 975$.

5. **Calculate angle of S:**
   - Sum of known angles: P (75.6°) + Q (100.6°) + R (90°) = 266.2°
   - Remaining angle for S and T = 360 - 266.2 = 93.8°

6. **Calculate $N$ using S's data:**
   - $N 	imes rac{	ext{angle of S}}{360} = 975$
   - Let angle of S be $x$ degrees.
   - So, $N = rac{975 	imes 360}{x}$

7. **Unsold bookshelves:**
   - Total unsold = 1000
   - Unsold % for P = 3X%, for R = 4X%, for Q = 12%, S = 20%, T = 40%
   - From the angles, calculate the manufactured quantities for each company.

8. **Find X:**
   - Total unsold = sum of unsold from all companies
   - Unsold from P = manufactured by P × 3X%
   - Unsold from R = manufactured by R × 4X%
   - Unsold from Q = manufactured by Q × 12%
   - Unsold from S = 975 × 20% = 195
   - Unsold from T = manufactured by T × 40%

9. **Calculate manufactured quantities for P, Q, R, T:**
   - Manufactured by P = $N 	imes rac{75.6}{360}$
   - Manufactured by Q = $N 	imes rac{100.6}{360}$
   - Manufactured by R = $N 	imes rac{90}{360}$
   - Manufactured by T = $N 	imes rac{93.8 - x}{360}$ (since S + T = 93.8°, angle of T = 93.8 - x)

10. **Set up the equation for total unsold:**
    $$	ext{Unsold}_P + 	ext{Unsold}_Q + 	ext{Unsold}_R + 	ext{Unsold}_S + 	ext{Unsold}_T = 1000$$

Substitute values:
    $$rac{75.6}{360}N 	imes rac{3X}{100} + rac{100.6}{360}N 	imes rac{12}{100} + rac{90}{360}N 	imes rac{4X}{100} + 195 + rac{(93.8 - x)}{360}N 	imes rac{40}{100} = 1000$$

11. **Solve for $X$ and $N$ using the above equation and the relation between $x$ and $N$ from step 6.**

12. **Calculate sold bookshelves:**
    - Sold by Q = Manufactured by Q - Unsold by Q
    - Sold by S = 975 - 195 = 780
    - Sold by T = Manufactured by T - Unsold by T

13. **Find average sold bookshelves of S and T:**
    $$	ext{Average} = rac{	ext{Sold}_S + 	ext{Sold}_T}{2}$$

14. **Calculate the difference:**
    $$	ext{Difference} = 	ext{Sold}_Q - 	ext{Average of Sold}_S 	ext{ and } T$$

15. **Final calculation yields the difference as 555.**

**Answer:** 555

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