Six movie critics each gave the lastest Dan Smuthers film

Question

Six movie critics each gave the lastest Dan Smuthers film an integer score from 0-100. The mode and median of these scores was 82 and the range of scores was 50. How many possiblilities are there for the second lowest score if the mean of the six scores is 78 and the highest score is 99?

yezz · Answer

Six movie critics each gave the lastest Dan Smuthers film an integer score from 0-100. The mode and median of these scores was 82 and the range of scores was 50. How many possiblilities are there for the second lowest score if the mean of the six scores is 78 and the highest score is 99?

MEDIAN = MODE = 82 MEAN THAT AT LEAST THE MIDDLE TWO CRITICS GAVE 82......(IE IT EXISTS AS A VALUE)

RANGE DIFFERENCE BETWEEN THE EXTREEMS = 50

MEAN = 78 SUM = 78*6 = 468 - 164 = 304 REAMINING FOUR VALUES

IF HIGHEST SCORE IS 99 THUS THE REMAINING THREE = 205

LOWEST SCORE = 49

THE REMAINING TWO = 205 - 49= 156

THE SECONF LOWEST HAS TO BE > 44 AND THE REMAINING ONE IS LESS THAN 99 ( NOT EQUALL BECAUSE IF EQUAL 82 WONT BE THE MODE or more so it wont be the max value)

THUS IT COULD BE
156 - 99 = 57 < X ( OUR VALUE)

IF WE ASSUMED THAT THERE IS THREE 82 VALUES THEN WE CAN ASSUME THAT IT COULD BE EQUIVELANT TO 99

82*3 + 99*2 + 49= 493 BUT SUM IS 468 ....NOT A VALID ASSUMBTION

I BELIEVE IT CAN TAKE VALUE 58<=X<82 ( ie maximizing the third lowest to be 98)

total possibilities = 81 - 58+1 = 24

edited for wrong addition and subtraction

iced_tea · Answer

let the scores in ascending order be a1, a2, a3, a4, a5, a6

also, a1+a2+a3+a4+a5+a6 = 78*6 = 468

now a6 = 99 => a1 = 49 as range is 50.

since mode = 82, there need to be atleast 2 scores with values 82. For median to be 82 also, a3 = a4 = 82.

so we have

49, a2, 82, 82, a5, 99

so a2 + a5 = 156

now 82 <= a5 <= 99

so 57 <= a2 <= 74

hence number of values = 18 (edited to correct the subtraction)

mukeshnathani · Answer

Since median of six terms is 82.Hence 3 and 4 term add up to 164.Taking the last term as 99 and range as 50 gives first term as 49.Now since mean is 78 hence total sum is 468.Deducting from 468 the sumof(3 and 4 term) and sumof(1 and 6 term) gives the sumof(2 and 5 term) as 156.Again the 5 term has to be >=82 since the median is 82.So 5 term possibilities are >=82 and <=99.So the second term can be anywhere >= 57 and <=74.Hence a total of 18 possibilities

cicerone · Answer

Hey mukesh, when 5th term is 99 it cannot be treated as the second lowest term. So exclude 99 from ur count. So it must be 17. What do u say?

mailtheguru · Answer

You are confusing between second highest and lowest. The answer should be 18.

diegmat · Answer

Dang tough Q - where did u grab this one from kevincan?

anandsebastin · Answer

Any of 17 values from 68 to 74

We know
lowest score = 49
highest score = 99

Since mode = 82, there are atleast two 82s.
Since median = 82, sum of third and fourth scores/2 = 82.

Total of 6 scores = 468
Total of 4 known scores = 312

Sum of two unknown scores = 156.
The higher of the two scores is atleast 82 ( because if it is lower, the median will change)
The second lowest score can be anywhere between 68 and 74 (corresponding to the second highest score between 82 and 98).

Phew! took me nearly 7 minutes for this. :roll:

cicerone · Answer

Sorry i mean second highest term..........

I mean that 5th term cannot be 99........ because it cannot be treated as second highest ,infact it will be treated as highest .............

So only 17 are possible

kevincan · Answer

Suppose the scores are s1, s2,...,s6 in ascending order.
Then s1=49, a6=99, s3=s4=82 and s1+s2+...+s6=6*78
Also 82<=s5<99,  since the mode is 82, s5 cannot be 99 .

As (s1-78)+(s3-78)+(s4-78)+(s6-78)=-29+4+4+21=0

s2+s5=2(78) and s2=156-s5

Since 82<=s5<=98, s5 can have one of 17 different values, as can s2.

OA=17

AK47 · Answer

i agree,17 is the right choice. (49,58,82,82,98,99) to (49,74,82,82,82,99) where 57a5>81

yezz · Answer

:wall :wall i got the iniqulity the other wY ROUND......GREAT Kevin Keep them coming.

Six movie critics each gave the lastest Dan Smuthers film

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