Question

How many Split Inverter ACs did D2 sell?

• A. 75%
• B. 20%
• C. 25%
• D. 4.33%
• A. D1 and D3 sold an equal number of Split ACs.
• B. D2 sold the highest number of ACs.
• C. D1 and D3 together sold more ACs as compared to D2 and D4 together.
• D. D4 sold more Split ACs as compared to D3.
• A. 5
• B. 7
• C. 6
• D. 4

Answer 1

The Correct Option is C

Let \( A \) represent the total quantity of AC units sold.

It is given that 25% of the ACs sold were of the Window variant.

The number of Window ACs sold is \( \frac{A}{4} \), and the number of Split ACs sold is \( \frac{3A}{4} \).

Let \( B \) denote the total number of inverter ACs sold.
Of the inverter ACs, 20% were of the Window variant.

The number of Window Inverter ACs is \( \frac{B}{5} \), and the number of Split Inverter ACs is \( \frac{4B}{5} \).

Based on condition (3):

\( \frac{A}{4} - \frac{B}{5} = 6 \) and \( \frac{4B}{5} = 36 \).

Solving these equations yields: \( B = 45 \) and \( A = 60 \).

Sales Distribution Summary:

Total Units Sold = 60
Total Split ACs = 45		Total Window ACs = 15
Total Inverter ACs = 36	Total Non-Inverter ACs = 9	Total Inverter ACs = 9	Total Non-Inverter ACs = 6

The following assumptions are made:

• Dealers D1 and D4 sold zero Window Non-Inverter ACs.
• Dealer D2 sold twice the quantity of Window Non-Inverter ACs as Dealer D3. Therefore, D2 sold 4 units and D3 sold 2 units.
• Dealer D1 sold \( x \) Window Inverter ACs. Consequently, D1 sold \( 13 - x \) Split Inverter ACs.
• Let \( y \) represent the total Window ACs sold by D3 and D4 combined. Then, D2 sold \( 3y \) Window ACs.
• Let \( z \) represent the Split Inverter ACs sold by D3 and D4 combined. Then, D2 sold \( 2z \) of these units.

The equation for total Window ACs sold is: \( x + 3y + y + y = 15 \), which simplifies to \( x + 5y = 15 \).
By testing values, we find \( x = 5 \) and \( y = 2 \).

Dealer-Specific Sales Data:

Dealer D1 Total Sales = 15
Split ACs = 10		Window ACs = 5
Inverter ACs = 8	Non-Inverter ACs = 2	Inverter ACs = 5	Non-Inverter ACs = 0

Dealer D2 Total Sales = 20
Split ACs = 14		Window ACs = 6
Inverter ACs = 14	Non-Inverter ACs = 0	Inverter ACs = 2	Non-Inverter ACs = 4

Dealer D3 Total Sales = 12
Split ACs = 10		Window ACs = 2
Inverter ACs = 7	Non-Inverter ACs = 3	Inverter ACs = 0	Non-Inverter ACs = 2

Dealer D4 Total Sales = 13
Split ACs = 11		Window ACs = 2
Inverter ACs = 7	Non-Inverter ACs = 4	Inverter ACs = 2	Non-Inverter ACs = 0

The total number of Non-Inverter ACs sold is \( 9 + 6 = 15 \).
The percentage of Non-Inverter ACs sold is \( \frac{15}{60} \times 100 = 25\% \).

Answer 2

Let \(A\) represent the total quantity of air conditioners (ACs) sold.

Provided information:

• Window variant ACs constituted 25% of the total: \( \frac{A}{4} \).
• Split variant ACs constituted 75% of the total: \( \frac{3A}{4} \).
• Let \(B\) denote the number of Inverter ACs.
• Of the Inverter ACs, 20% were Window types: \( \frac{B}{5} \).
• Consequently, Split Inverter ACs accounted for: \( \frac{4B}{5} \).
• Based on the given data, the following equations are established: \( \frac{A}{4} - \frac{B}{5} = 6 \) and \( \frac{4B}{5} = 36 \).

Calculations:

From the equation \( \frac{4B}{5} = 36 \), we derive \( B = 45 \). This signifies that there are 45 Inverter ACs.

Substituting this value into the first equation: \( \frac{A}{4} - \frac{45}{5} = 6 \). This simplifies to \( \frac{A}{4} - 9 = 6 \), leading to \( \frac{A}{4} = 15 \), and thus \( A = 60 \). Therefore, the total number of ACs sold is 60.

The number of Non-inverter ACs is calculated as Total ACs - Inverter ACs = \( 60 - 45 = 15 \).

The required percentage of Non-inverter ACs relative to the total ACs is:

\[ \frac{15}{60} \times 100 = 25\% \]

Result: 25%

Answer 3

Let \( A \) represent the total number of ACs sold. Of these, 25% were Window ACs, and 75% were Split ACs. Therefore, Window ACs = \( \frac{A}{4} \) and Split ACs = \( \frac{3A}{4} \).

Let \( B \) be the total number of Inverter ACs. It is given that 20% of Inverter ACs are Window ACs, which equals \( \frac{B}{5} \). The number of Window Non-inverter ACs is \( \frac{A}{4} - \frac{B}{5} \), and this is stated to be 6. Additionally, Split Inverter ACs are 36, so \( \frac{4B}{5} = 36 \), which implies \( B = 45 \). Substituting this value into the previous equation gives \( \frac{A}{4} - 9 = 6 \), leading to \( A = 60 \).

Total ACs Sold = 60
- Split ACs: 45
- Window ACs: 15
- Inverter ACs: 36 (composed of 9 Window and 27 Split)
- Non-inverter ACs: 24 (composed of 6 Window and 18 Split)

Based on the conditions provided:
Let \( x \) denote the number of Window Inverter ACs sold by D1. Then, Split Inverter ACs for D1 are \( 13 - x \).
Let \( y \) represent the number of Window ACs sold by D3 and D4. Then, D2 sold \( 3y \) Window ACs.
Let \( z \) be the number of Split Inverter ACs sold by D3 and D4. Then, D2 sold \( 2z \) Split Inverter ACs.

The total number of Window ACs sold is \( x + 3y + y + y = 15 \), which simplifies to \( x + 5y = 15 \). The only valid integer solution is \( x = 5 \) and \( y = 2 \).

Using the total Split Inverter ACs count of 36, we have \( 8 + 2z + z + z = 36 \), which means \( 4z = 28 \), so \( z = 7 \).

Dealer-wise AC Distribution:

D1:
- Window ACs: 5 (5 Inverter, 0 Non-inverter)
- Split ACs: 10 (2 Inverter, 8 Non-inverter)
- Total: 15

D2:
- Window ACs: 6 (2 Inverter, 4 Non-inverter)
- Split ACs: 21 (14 Inverter, 7 Non-inverter)
- Total: 27

D3:
- Window ACs: 2 (0 Inverter, 2 Non-inverter)
- Split ACs: 10 (7 Inverter, 3 Non-inverter)
- Total: 12

D4:
- Window ACs: 2 (2 Inverter, 0 Non-inverter)
- Split ACs: 4 (0 Inverter, 4 Non-inverter)
- Total: 6

Total ACs Sold by Dealers:
D1 + D3 = 15 + 12 = 27
D2 + D4 = 60 - 27 = 33

Answer 4

Problem Analysis:
Four dealers (D1, D2, D3, D4) sell two types of ACs: Window and Split. Each type can be either Inverter or Non-inverter.

Given Data:

• Window ACs constitute 25% of total sales; Split ACs constitute 75%.
• 20% of Inverter ACs are Window type.
• Sales figures: 6 Window Non-inverter ACs and 36 Split Inverter ACs.
• Dealer D1 sold 13 Inverter ACs.
• Dealer D3 sold 5 Non-inverter ACs.
• D1's Split AC sales equal twice its Window AC sales.
• D3 and D4 sold an equal number of Window ACs, which is one-third of D2's Window AC sales.
• Window Non-inverter ACs were sold exclusively by D2 and D3, with D2 selling twice as many as D3.
• D3 and D4 sold an equal number of Split Inverter ACs, each selling half the amount sold by D2.

Verification of Statements:

1. Statement: D1 and D3 sold an equal number of Split ACs.
   Analysis: D1's Split AC sales are known relative to its Window AC sales (D1 Split = 2 × D1 Window). D3's total Split AC sales are not directly calculable from the provided information.
   → Determination: Cannot be determined.
2. Statement: D2 sold the highest number of ACs.
   Analysis: D2 sold the most Window ACs (three times the amount sold by D3/D4 combined). D2 also sold the most Split Inverter ACs. This suggests D2 likely has the highest total sales.
   → Determination: Can be true.
3. Statement: D1 and D3 together sold more ACs than D2 and D4 together.
   Analysis: From the data, D2's sales significantly exceed those of D1 or D3 individually. As D4's sales are comparable to D3's in certain categories, the combined sales of D2 and D4 are likely greater than D1 and D3.
   → Determination: This is false.
4. Statement: D4 sold more Split ACs compared to D3.
   Analysis: D3 and D4 sold an equal quantity of Split Inverter ACs. Information regarding Split Non-inverter AC sales for D3 and D4 is missing.
   → Determination: Can be true.

Final Answer: The statement that is necessarily false is:
"D1 and D3 together sold more ACs as compared to D2 and D4 together."

Answer 5

To determine the quantity of Non-inverter ACs sold by D2, the provided data is analyzed.

1. As per the problem statement, D3 and D4 each sold identical quantities of Split Inverter and Window ACs. Let x represent the number of Window ACs sold by D3 and D4 individually.
2. D3 and D4's Window AC sales were equal and amounted to one-third of D2's sales. Consequently, D2 sold 3x Window ACs.
3. Window Non-inverter ACs were exclusively sold by D2 and D3, with D2 selling double D3's quantity. If D3 sold y Window Non-inverter ACs, then D2 sold 2y. The total for D2 and D3 is given as 3 ACs (2y + y = 3). Solving for y yields 1, thus D2 sold 2 ACs.
4. A total of six Window Non-inverter ACs were sold. D1 and D4 sold none. D3 sold 1, and D2 sold 2. The combined sales for D2 and D3 account for 3 of the total 6.
5. The total sales of Split Inverter ACs reached 36. D3 and D4 sold equal amounts, each selling half of D2's quantity. Let z be the quantity sold by D3 and D4 combined. Therefore, D2 sold 2z. With z = 9, D2's Split Inverter AC sales are confirmed at 18.

Conclusion: Considering that 5 Non-inverter ACs constitute the remaining sales after accounting for other categories, D2 sold a total of 5 Non-inverter ACs.