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# CAT 2020 Quant Question [Slot 1] with Solution 14

Question

The mean of all 4 -digit even natural numbers of the form 'aabb', where a >0, is

1. 5050
2. 4466
3. 5544
4. 4864
Option: 3
Solution:

The sum of possible even digit numbers in the form aabb is $1100 + 1122 + 1144 + 1166 + 1188 + 2200 + 2222 + 2288 + \ldots .9900 + 9922 + 9988$ i.e. $(45$ numbers)

$\Rightarrow 1100 + 1100 + 1100 + 1100 + 1100 + 22 + 44 + 66 + 88 + 2200 + 2200 + 2200 + 2200 + 2200 + 22 + 44 + 66 + 88 + \ldots \ldots + 9900 + 9900 + 9900 + 990 + 9900 + 22 + 44 + 66 + 88$

$\Rightarrow 5(1100 + 2200 + \ldots .9900) + 9(22 + 44 + 66 + 88)5 \times 1100(1 + 2 + \ldots \ldots 9) + 9 \times 22(1 + 2 + 3 + 4)$

$\Rightarrow 5500(45) + 45 \times 44 = 45(5544)$

Hence mean =5544

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