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Quantitative questions a day 51 to 100 - The discussions

I'm starting a new thread as the discussions will happen on this new thread for next 50 questions and the sectional test :-) Happy solving :smiley:
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Question:

The digits of a four digit number form an arithmetic progression, not necessarily in the same order. How many such four digit numbers are possible if the arithmetic mean of all the digits is an integer?

(a) 72 (c) 84 (c) 90 (d) 96

my solution-let the digits be a-3d,a-d,a+d,a+3d in any order a,d>0.where both a & d are integers.a.m of the four no.s is 'a' which is an integer.

hence clearly for a=9, 8 & 7 we do not have a single digit value of a+3d.

for a=2 & 1 we cannot have positive values of a-3d for any d.

for a=6,5,4 we get the four no.s which can be rearranged in 3*4!=72 ways.

for a=3 we get a-3d=0 for d=1. so the no. of ways of rearranging the no. =4!-3!=18

so total no.s=72+18=90 i..e ans option (c)

ASSUMPTION-an a.p cannot have 0 common difference i..e ignore no.s like
1111 & 22222 etc.

DISCLAIMER-answer could be wrong as i am still a novice for cat mathematics.

hi pagals,
After a long time:
if we take 1234 as one of the four digit no then there are 4! ways to rearrange and satisfy the given condition as per the question.
so in a sequence
123456789
5 possible ways(2345,3456,4567,5678,6789)
so in total =4!*5 ways =120
then taking 0123 = 8
then taking 2468= 24
then taking 3579= 24

in total=176 ways.:wow:
am i missing something or is the question lagging something...??:
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Question: The digits of a four digit number form an arithmetic progression, not necessarily in the same order. How many such four digit numbers are possible if the arithmetic mean of all the digits is an integer?
(a) 72 (c) 84 (c) 90 (d) 96

I am getting C - 90 as answer.

{0,2,4,6} - 18
{2,4,6,8} - 24
(1,3,5,7} - 24
(3,5,7,9} - 24

hence total 90
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36 hr - 720 Deg
So 1hr = 20 deg

120 min = 20 DEG (since 1 Hr = 120 Min)

1 Min = 20/120 = 1/6 DEG

So in I Min angle subtended by Min hand is 1/6 Deg. How come people getting 6 Deg??? Not able to understand!!

This is the angle subtended by hour hand for 1 minute.
For minute hand u will have 720/120=6 degrees.
(720 degrees = 120 minutes.)

And for todays question, the answer is 99,(as explained in above posts). I was wondering if the same question is asked for a harmonic progression instead of an AP!
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Sequence Permutation Total No of Numbers posssible
0246 3*3*2 18
1357 4! 24
2468 4! 24
3579 4! 24
Total 90

Ans Option c ie 90

Even I feel the options given should have included 99

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I'll also mark option (c) as answer. But if there was an option of none of the above i would have marked that bcoz i also think that common difference can be zero. In that case answer would have been 99.

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For odd common differences, the mean of the number cannot be an integer.So, common diffrence needs to be even.
common diffrence of 4 is not possible for 4 numbers.

So, the numbers can be and their rearrangement:
0246: 3*3*2
2468: 4!
1357: 4!
3579: 4!

they all add upto 90.so, my answer is 90.
So, the answer is (c)

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Let the nos be - > a-3d, a-d, a+d, a+3d
AM = a
Let d= 1 No a can take any value from 1- 9, a = {1- 9}
If a = 9,8,7 then a+3d no longer remains 1 digit
a = 4,5,6
Possible Valies = {1,3,5,8]
{2, 4, 6, 8}
{3, 5, 7, 9}
a-3, a-1, a+1 , a+3
a = 3
Possible Valies = {0, 2, 4, 6}
a = 1 ,2 ruled out as that makes the digits negative
So Possible Values are:
{1, 3, 5, 8] = 4!
{2, 4, 6, 8} = 4!
{3, 5, 7, 9} = 4!
{0, 2, 4, 6} = 4! - 3! = 3X3! (We remove the cases where 0 is in the Most Sig Digit)
Total = 3X4! + 3X3!
3X24 + 3X6 = 72 + 18 = 90

Hey that's option C)

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What I think is a AP cud have common difference 0.So according to me answer shud be 99.
A sequence (finite or infinite) is called an arithmetic progression (abbreviated A.P.) iff the difference of any term from its preceding term is constant. This constant is usually denoted by d and is called common difference.
Cheers Apple

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Quant Question # 76
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The digits of a four digit number form an arithmetic progression, not necessarily in the same order. How many such four digit numbers are possible if the arithmetic mean of all the digits is an integer?

(a) 72 (c) 84 (c) 90 (d) 96

My answer is (c).

Let numbers be a, a+d, a+2d, a+3d.

Given that mean of these is an integer.
=> (4a+6d)/4 is an integer
=> d is a multiple of 2.
=> d is 2 (because for d >= 4, a+3d will exceed 10)

Also we know that a+3d => a + 6 => a => a = 0, 1, 2, 3

For a = 1, 2, 3: number of combos is 3x4! = 72
For a = 0: number of combos is 3*3! = 18

Total 90.

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In an hour,
Hour hand makes -> 2o deg
min hand makes -> 720 deg => in one min, min hand makes -> 6 deg.

So in 11 hrs hr hand made -> 11 * 20 => 220 deg. further in 24 mins it covered ( since 24 is 1/5th of 120) 1/5 * 20 -> 4deg. so net coverage of 224 deg by hr hand.

Min hand covers 24 * 6 => 144 deg in 24 mins.

Hence the angle b/w two hands is 224 - 144 = 80deg.

This I guess was the easiest of all the ques that we have recieved so far with none of having any ambiguity about niehter the approach nor the answer.

Regards,
Contender

36 hr - 720 Deg
So 1hr = 20 deg

120 min = 20 DEG (since 1 Hr = 120 Min)

1 Min = 20/120 = 1/6 DEG

So in I Min angle subtended by Min hand is 1/6 Deg. How come people getting 6 Deg??? Not able to understand!!
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