An air conditioner (AC) company has four dealers - D1, D2, D3 and D4 in a city. It is evaluating sales performances of these dealers. The company sells two variants of ACs Window and Split. Both these variants can be either Inverter type or Non-inverter type. It is known that of the total number of ACs sold in the city, 25% were of Window variant, while the rest were of Split variant. Among the Inverter ACs sold, 20% were of Window variant.
The following information is also known:
1. Every dealer sold at least two window ACs.
2. D1 sold 13 inverter ACs, while D3 sold 5 Non-inverter ACs.
3. A total of six Window Non-inverter ACs and 36 Split Inverter ACs were sold in the city. 4. The number of Split ACs sold by D1 was twice the number of Window ACs sold by it. 5. D3 and D4 sold an equal number of Window ACs and this number was one-third of the number of similar ACs sold by D2.
4. D2 and D3 were the only ones who sold Window Non-inverter ACs. The number of these ACs sold by D2 was twice the number of these ACs sold by D3.
5. D3 and D4 sold an equal number of Split Inverter ACs. This number was half the number of similar ACs sold by D2
How many Split Inverter ACs did D2 sell?
The following information is also known:
1. Every dealer sold at least two window ACs.
2. D1 sold 13 inverter ACs, while D3 sold 5 Non-inverter ACs.
3. A total of six Window Non-inverter ACs and 36 Split Inverter ACs were sold in the city. 4. The number of Split ACs sold by D1 was twice the number of Window ACs sold by it. 5. D3 and D4 sold an equal number of Window ACs and this number was one-third of the number of similar ACs sold by D2.
4. D2 and D3 were the only ones who sold Window Non-inverter ACs. The number of these ACs sold by D2 was twice the number of these ACs sold by D3.
5. D3 and D4 sold an equal number of Split Inverter ACs. This number was half the number of similar ACs sold by D2
How many Split Inverter ACs did D2 sell?
- 27 AC's
- 22 AC's
- 20 AC's
- 14 AC's
The Correct Option is D
Solution and Explanation
To address this problem, we will analyze the given data and formulate equations to calculate the sales figures.
Analysis Steps:
1. AC Type Distribution:
Let \( T \) represent the total number of ACs. Based on the 25% proportion of Window ACs:
- Window ACs: \( 0.25T \)
- Split ACs: \( 0.75T \)
Let \( I \) denote the total number of Inverter ACs. Given that 20% of Inverter ACs are Window types:
- Window Inverter ACs: \( 0.2I \)
- Split Inverter ACs: \( 0.8I \)
2. Problem Data:
(a) 6 Window Non-inverter ACs were sold.
(b) 36 Split Inverter ACs were sold.
Dealer-Specific Data:
(i) Dealer D1 sold 13 Inverter ACs, with Split ACs being twice the number of Window ACs.
(ii) Dealer D3 sold 5 Non-inverter ACs and the same quantity of Window ACs as Dealer D4.
(iii) Dealer D2 sold twice the quantity of Window Non-inverter ACs as Dealer D3. This implies D3 sold 1 unit and D2 sold 2 units.
(iv) Split Inverter ACs: Dealers D3 and D4 each sold half the number of units sold by Dealer D2 (D3 = D4 = \( \frac{1}{2} \) × D2).
3. Calculations:
(a) From point 2(iii), the total Window Non-inverter ACs sold by D3 and D2 is 1 + 2 = 3. Since the overall total is 6, Dealer D1 must have sold the remaining 3 units.
(b) Let Dealer D2 sell \( x \) Split Inverter ACs. Consequently, Dealers D3 and D4 each sell \( \frac{x}{2} \) ACs, totaling \( x + \frac{x}{2} + \frac{x}{2} = 2x \) units among them. However, this approach leads to a contradiction with prior assumptions. We will utilize an alternative consistent equation.
Let Dealer D2 sell \( x \) Split Inverter ACs. Then D3 and D4 combined sell \( 2x \) Split Inverter ACs. The total Split Inverter ACs sold by D2, D3, and D4 is 36, so \( x + 2x = 36 \), which simplifies to \( 3x = 36 \), yielding \( x = 12 \).
4. Conclusion:
Dealer D2 sold 12 ACs that were Split Inverter models.
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