Mindyard | Comprehension: 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?

Comprehension:

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?

Correct Answer :

Solution :

Let the total number of ACs sold in the city be $T$ .
According to the problem:
- Window ACs sold = 25% of $T = 0.25 T$
- Split ACs sold = 75% of $T = 0.75 T$
- Among the Inverter ACs sold, 20% were Window variant and 80% were Split variant.

We are given that a total of 36 Split Inverter ACs were sold.
If the total number of Inverter ACs sold is $I$ , then:
$0.80 I = 36 \Rightarrow I = 45$
Thus, the number of Window Inverter ACs sold = $0.20 \times 45 = 9$ .

We are also given that a total of 6 Window Non-inverter ACs were sold.
Therefore, the total number of Window ACs sold is:
$Total Window ACs = 9 (Inverter) + 6 (Non-inverter) = 15$

Since Window ACs account for 25% of the total ACs sold:
$0.25 T = 15 \Rightarrow T = 60$
So, the total number of ACs sold in the city is 60.

Now, let's distribute the Window ACs sold among the dealers D1, D2, D3, and D4:
- Every dealer sold at least 2 Window ACs.
- D3 and D4 sold an equal number of Window ACs, which was one-third of the number of Window ACs sold by D2.
Let the number of Window ACs sold by D3 and D4 be $x$ . Then D2 sold $3 x$ Window ACs.
Let the number of Window ACs sold by D1 be $W_{1}$ .
$W_{1} + 3 x + x + x = 15 \Rightarrow W_{1} + 5 x = 15$
Since every dealer sold at least 2 Window ACs, we must have $x \geq 2$ and $W_{1} \geq 2$ .
If $x = 2$ , then $W_{1} = 5$ (valid).
If $x \geq 3$ , then $W_{1} \leq 0$ (invalid).
So, $x = 2$ .
- Window ACs sold by D1 = 5
- Window ACs sold by D2 = 6
- Window ACs sold by D3 = 2
- Window ACs sold by D4 = 2

Next, we determine the Split ACs sold by D1:
- The number of Split ACs sold by D1 was twice the number of Window ACs sold by it:
$Split ACs (D1) = 2 \times 5 = 10$
So, the total number of ACs sold by D1 is $5 + 10 = 15$ .

Now we analyze the Window Non-inverter ACs:
- 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 sold by D3.
Since the total number of Window Non-inverter ACs is 6:
- Window Non-inverter ACs sold by D3 = 2
- Window Non-inverter ACs sold by D2 = 4
For D3, the total Window ACs sold is 2, which means both are Non-inverter. Thus, Window Inverter ACs sold by D3 = 0.

Now we analyze the Split Inverter ACs:
- D1 sold 13 inverter ACs in total. Since Window Inverter(D1) = 5 (as Window Non-inverter for D1 is 0), Split Inverter(D1) = $13 - 5 = 8$ .
- The remaining Split Inverter ACs (out of 36) are distributed among D2, D3, and D4:
$Split Inverter (D2 + D3 + D4) = 36 - 8 = 28$
- D3 and D4 sold an equal number of Split Inverter ACs, and this number was half the number sold by D2.
Let Split Inverter ACs sold by D3 and D4 be $y$ . Then D2 sold $2 y$ .
$y + y + 2 y = 28 \Rightarrow 4 y = 28 \Rightarrow y = 7$
Thus, Split Inverter(D3) = 7, Split Inverter(D4) = 7, and Split Inverter(D2) = 14.

For D3:
- We are given that D3 sold 5 Non-inverter ACs.
- Since Window Non-inverter(D3) = 2, we have Split Non-inverter(D3) = $5 - 2 = 3$ .
- Thus, the total number of ACs sold by D3 = Window ACs + Split ACs = $2 + (7 + 3) = 12$ .

Finally, we can find the total number of ACs sold by D2 and D4 combined:
$Total ACs (D2 + D4) = Total ACs - Total ACs (D1) - Total ACs (D3)$
$Total ACs (D2 + D4) = 60 - 15 - 12 = 33$

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