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
What was the total number of ACs sold by D2 and D4? (This Question was asked as TITA)
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
What was the total number of ACs sold by D2 and D4? (This Question was asked as TITA)
- 33 AC's
- 39 AC's
- 42 AC's
- 28 AC's
The Correct Option is A
Solution and Explanation
The problem is solved by systematically analyzing the given conditions:
1. Let $W$ represent the count of Window ACs and $S$ represent the count of Split ACs. It is given that 25% of the total ACs sold ($T$) are Window type, thus $W = 0.25T$ and $S = 0.75T$.
2. Out of the total Inverter ACs ($I$), 20% are Window variants. Therefore, the number of Window Inverter ACs is $0.2I$.
3. Dealer D1 sold 13 Inverter ACs. The number of Split ACs sold by D1 was double the number of Window ACs sold by D1. Let $x$ be the number of Window ACs sold by D1. Then, Split ACs sold by D1 $= 2x$. The total ACs sold by D1 $= x + 2x = 3x$.
4. Dealers D3 and D4 sold an equal quantity of Window ACs, and each sold one-third of the quantity sold by D2. Let $y$ be the number of Window ACs sold by each of D3 and D4. Consequently, D2 sold $3y$ Window ACs. The total Window ACs from D2, D3, and D4 $= 3y + y + y = 5y$. Since D1 sold $x$ Window ACs and the total Window ACs is $W = 7x$, we have: $7x = 5y \Rightarrow \dfrac{x}{y} = \dfrac{5}{7}$.
5. Dealers D2 and D3 collectively sold all the Window Non-Inverter ACs. D2 sold twice the number of Window Non-Inverter ACs sold by D3. The total Window Non-Inverter ACs is 6. Let D3 have sold $z$ Window Non-Inverter ACs, then D2 sold $2z$. Thus, $z + 2z = 6 \Rightarrow z = 2$, and $2z = 4$.
6. The total number of Split Inverter ACs sold is 36. D3 and D4 sold equal numbers of Split Inverter ACs, each selling half of what D2 sold. Let D3 and D4 each sell $p$ Split Inverter ACs. Then D2 sold $2p$. The total is $2p + p + p = 4p = 36 \Rightarrow p = 9$. Therefore, D2 sold 18 Split Inverter ACs, and D3 and D4 each sold 9 Split Inverter ACs.
The total ACs sold by D2 and D4 are calculated as follows:
D2: Window ACs $= 3y$, Split Inverter ACs $= 18$, Window Non-Inverter ACs $= 4$. Total for D2 $= 3y + 18 + 4 = 3y + 22$.
D4: Window ACs $= y$, Split Inverter ACs $= 9$. Total for D4 $= y + 9$.
The combined total for D2 and D4 $= 3y + 22 + y + 9 = 4y + 31$.
From the earlier derivation, $7x = 5y \Rightarrow x = \dfrac{5y}{7}$, allowing all values to be expressed in terms of $y$.
Consequently, the total number of ACs sold by D2 and D4 is 33.
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