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There are 108 positive integers. 50% of the integers have...

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There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post Updated on: 10 Nov 2018, 03:57
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There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tens digit, 1/3 of integers have 6 as tens digit and 1/3 of integers have 8 as tens digit.
What is the tens digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0

Originally posted by Kaczet on 05 Nov 2018, 13:22.
Last edited by Kaczet on 10 Nov 2018, 03:57, edited 1 time in total.
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 05 Nov 2018, 13:46
Kaczet wrote:
There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 6 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0



so we have numbers

60 and 65

80 and 85

20 and 25

sum them up and you get 330

So, tenth digit is 3

:)
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 05 Nov 2018, 13:58
1
dave13 wrote:
Kaczet wrote:
There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 6 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0



so we have numbers

60 and 65

80 and 85

20 and 25

sum them up and you get 330

So, tenth digit is 3

:)


Hello. Imagine it were

There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 0 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

then it is

0 + 5

80 and 85

20 and 25

and it's 215 but the unit digit should be 7...

Thx.
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 05 Nov 2018, 14:03
Kaczet wrote:
dave13 wrote:
Kaczet wrote:
There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 6 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0



so we have numbers

60 and 65

80 and 85

20 and 25

sum them up and you get 330

So, tenth digit is 3

:)


Hello. Imagine it were

There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 0 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

then it is

0 + 5

80 and 85

20 and 25

and it's 215 but the unit digit should be 7...

Thx.



Kaczet hi there, thanks for your comments :) but i dont get what you mean:) is it a question ? or was my answer wrong ? did you find any flaw in my reasoning ?
cheers
D :)
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 05 Nov 2018, 14:06
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dave13,

If we were to change the question a bit (i.e. change one of the tenth digits), your reasoning no longer yields to the correct answer.
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 05 Nov 2018, 14:22
Kaczet wrote:
dave13 wrote:
Kaczet wrote:
There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 6 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0



so we have numbers

60 and 65

80 and 85

20 and 25

sum them up and you get 330

So, tenth digit is 3

:)


Hello. Imagine it were

There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 0 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

then it is

0 + 5

80 and 85

20 and 25

and it's 215 but the unit digit should be 7...

Thx.


Kaczet well

if we have changed numbers in question stem, then we gonna have

25 20
05 00
85 80

subtotal =215 :) if numbers changed answer choices should be changed too, on the other hand 00 doesnt look like valid number :)

and why you think the unit digit should be 7 in number 215 :?

am i missing something ? :)
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 05 Nov 2018, 14:29
1
dave13,

108 integers. 54 have 0 as unit digit and 54 have 5 as unit digit. 36 have 2 as tenth digit, 36 have 0 as tenth digit and 36 have 8 as tenth digit.

Let's suppose that they all are of 1XY form (it doesn't matter if it's 1XY, 2XY or 3XY, etc.). X = tenth digit, Y = unit digit.
If we were to add them 36*125 + 18*105 + 18*100 + 36*180 (which is impossible to do on the exam), we would get 14670. 7 is the tenth digit.
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 06 Nov 2018, 12:01
Bunuel,

any ideas for this question?
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 10 Nov 2018, 00:42
Kaczet wrote:
There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tenth digit, 1/3 of integers have 6 as tenth digit and 1/3 of integers have 8 as tenth digit.
What is the tenth digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0


Hi,

In general, we can write two-digit numbers as follows:

\(n_i = a_i + 10b_i\) .

In the above question \(a_i \in \{0,5\}\) and \(b_i \in \{2,6,8\}\).

From question we have the following information:

54 numbers with 5 as unit digit
54 numbers with 0 as unit digit
36 numbers with 2 as the tens digit
36 numbers with 6 as the tens digit
36 numbers with 8 as the tens digit

Sum of all the numbers = 36*10 (2 + 6 + 8) + 54*1*(5 + 0) = 6030.

Hence, at tens digit of the sum of 108 integers = 3. Answer (B).

Note: In the question prompt "tenth" should be replaced by "tens".

Thanks.
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 10 Nov 2018, 00:46
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Hi Folks.

Break the question into 2 sections.
1. Unit Digit

54*5 ending
54*0 ending
add this value together, we will get total of 270.
Then we keep this number and move to tenth digits. we note 7 to add up in the next process.

2. tenth digits
As the question ask about the tenth digits
we will have to find digits in these formula.
36*2= unit digit is 2
36*6= unit digit is 6
36*8= unit digit is 8
Add all the value including 7.
Thus 2+6+8+7 = 23

the tenth digits is 3.

So Answer =B
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 10 Nov 2018, 05:50
1
1
Kaczet wrote:
There are 108 integers. 50% of the integers have 5 as unit digit and 50% have 0 as unit digit. 1/3 of integers have 2 as tens digit, 1/3 of integers have 6 as tens digit and 1/3 of integers have 8 as tens digit.
What is the tens digit of the sum of those 108 integers?

(A) 8

(B) 3

(C) 6

(D) 2

(E) 0


54 numbers have 0 in units digit and 54 numbers have 5 in units digit. When you add all units digits, you will get 54*5 = 270. So 0 will come in units digit and 27 will be carry forward.

36 numbers have 2 in tens digit, 36 have 6 in tens digit and 36 have 8 in tens digit. The 36 numbers with 2 and 36 numbers with 8 will add up to give 0 so we can ignore them.
36 numbers with 6 in tens digit give 36*6 = 216.
Add 27 of carry from before to get 243. the tens digit will be 3.

Answer (B)
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Re: There are 108 positive integers. 50% of the integers have...  [#permalink]

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New post 15 Jan 2019, 19:40
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Since the units digit has two options and the tens digit has 3 options you can have 6 numbers of equal proportion:
    20
    25
    60
    65
    80
    85

Each of these numbers could be multiplied by 18 in order to represent the 108 numbers in the problem.

Adding these numbers together yields 335 and this number accounts for 1/18 of the numbers. Simply multiply 335 by 18 and you will yield 5,930 whereby 3 is the tens digit and is the correct answer.
Coincidentally, '3' is the same number in the first group of 6, but don't let that misguide you for if this question stem replaced the tens digit of 6 with 0 (as someone suggested below in the comments), that would yield 215 or '1' for the first six and after you multiple by 18 you get 3,870 whereby 7 is the correct units digit.
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Re: There are 108 positive integers. 50% of the integers have...   [#permalink] 15 Jan 2019, 19:40
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