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the sequence is as follow: \(a_1=-81\) \(a_2=-78\) ... and this will reach 0 is 27 passages \(a_{28}=0\) and then will become positive and every term will balance its negative correspondent.

\(a_{27}=-3\) will be balanced by \(a_{29}=3\) => sum=0 and so on... This process continues for all terms, if the question were "What is the sum of the first 55 terms the balance will be perfect and the sum would be 0.

But here we are asked the sum of the 54 terms: the very first term will not be balanced! \(a_{1}=-81\), \(a_{2}=-78\), ..., \(a_{54}=78\)

The sum is -81.

Given \(a_1=-81\) , and \(a_n=a_{n-1}+3\) for \(n>1\),

Therefore, \(a_2=a_{1}+3\) & \(a_3=a_{2}+3\)

This means the Common increment of 3, & as we know that there are 54 terms in total but if we remove \(a_1\) , then we will have 53 terms each increasing with the common increment of 3. Therefore we have \(a_54\) as 3*53-81.

\(a_54\) =159 - 81 \(\Rightarrow\) \(a_54\) =78.

Now, we have first term & the last term & the common difference. so as per the properties.....

--> \(\frac{Last Term + First Term}{2} * Total # of Terms\)

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04 May 2013, 23:49

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Question :

Two cars move along a circular track 1.2 miles long at constant speeds. When they move in opposite directions, they meet every 15 seconds. However, when they move in same direction, once car overtakes the other car every 60 seconds. What is the speed of the faster car ?

Options :

A) 0.02 miles/s B) 0.03 miles/s C) 0.05 miles/s D) 0.08 miles/s E) 0.1 miles/s

- Let the speed of the 2 dots be "a" (faster dot) and "b" (slower dot) miles/s respectively. - When they move in opposite directions, \(\frac{1.2}{a+b}=15\) - When they move in same direction, \(\frac{1.2}{a-b}=60\) - Simplifying 2 equations, we get 15a + 15b = 1.2 and 60a - 60b = 1.2 - Solving 2 equations, 120a = 6 or a = 0.05 miles/s

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Last edited by dipen01 on 05 May 2013, 00:03, edited 1 time in total.

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05 May 2013, 00:02

Question :

How many chocolates did the two girls buy, if the sum of the cubes of the number of chocolates bought by them adds up to 189 and the result of subtracting the product of chocolates bought by them from the sum of the squares of the chocolates bought by them is 21 ?

- Let the number of chocolates bought by one of the toddlers be "x" and the number of chocolates bought by second toddler be "y" - Then, \(x^3 + y^3 = 189\)......... (I) - and \(x^2 + y^2 - xy = 21\)..........(II) - \(x^3 + y^3 = (x+y)(x^2 - xy + y^2)\) - Thus by dividing equation (I) by equation (II) we will get the value of x + y = 9

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16 May 2013, 13:26

dipen01 wrote:

Question :

How many chocolates did the two girls buy, if the sum of the cubes of the number of chocolates bought by them adds up to 189 and the result of subtracting the product of chocolates bought by them from the sum of the squares of the chocolates bought by them is 21 ?

- Let the number of chocolates bought by one of the toddlers be "x" and the number of chocolates bought by second toddler be "y" - Then, \(x^3 + y^3 = 189\)......... (I) - and \(x^2 + y^2 - xy = 21\)..........(II) - \(x^3 + y^3 = (x+y)(x^2 - xy + y^2)\) - Thus by dividing equation (I) by equation (II) we will get the value of x + y = 9

Another way to do this that involves only the first equation is to realize that x and y must be integers and that there is a max on x and y. Since \(6^3\) is 216, the max on x and y is 5. It is impossible for \(x^3 + y^3\) to equal 189 if \(x + y = 5\), so the only possible answer is 9. In fact, the only combination of positive integers that works for the first equation is 4 and 5.

The question difficulty could be improved if the answers were all between 5 and 10. It could also be improved by removing the second equation from the question (and then it would require the same trick as OG13 #64).

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17 May 2013, 04:50

A question paper had 40 questions. Every correct answer would fetch 4 marks and every wrong answer would deduct 1 mark from the total. The unanswered questions doesn't have any impact on the total score. How many distinct scores can be obtained by a student if he/she takes the test

The highest score can be obtained by any of the student is +160 and minimum score can be obtained is -40, Hence the total number of scores can be obtained is 160+40 + 1 (zero) = 201

Now some of the scores are not possible such as 159, 158, 157, 154, 153 and 149, hence 201 - 6 = D) 195

It can be solved using the AP of common difference of 1, that becomes lengthy.

Let's observe a pattern here.

3 can be written in consecutive terms as 1+2 - 1 way , 5 = 2+3 - 1 way, 15 = 8+7, 1+2+3+4+5, 4+5+6 - 3 ways, we can see it is number of odd factors of the number -1

Hence 5! can only two odd factors 3 and 5, hence the number of ways should be 4-1 = 3 ways C)

Let's say A+B+C = 23 As every body should get 1 chocolate, hence assigning 1 to each, the equation becomes A+B+C = 20, and number of ways of distributing chocolates = 22C2

Now none should get more than 10, lets assign 11 to A, then the equation becomes A+B+C = 9, hence number of ways of distribution being 11C2 -> That many cases is when A gets more than 10 and similarly for all the people, the number of cases being 3*11C2

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17 May 2013, 22:05

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My .02 cents

The Oxford press compiled a 2000 page dictionary but just before printing,it was found that page numbers are missing. How many times should typist press keys from 0-9 so as to number dictionary from 1-2000?

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17 May 2013, 22:50

X is divided by a divisor leaves 19 as a remainder, and when 5X is divided by the same divisor , it leaves 29 as the remainder. What remainder would be obtained if 6X is divided by the same divisor?

The explanation is as follows : The question doesnt state that the groups have to be equal. A group can have any number of girls between 1 and 9. So 12C3 x 9C3 x 6C3 wont be the answer. It can be solved using the following technique - - - - - - - - - - - - The 12 different girls can be arranged in a single line in 12! ways

To arrange them into 4 groups with atleast 1 girl in each lets consider markers which will divide the line into 4 parts. So we require 3 markers which can divide the line as follows

part1 Marker1 part2 marker2 part 3 marker3 part4

There are 11 spaces in the line of girls as shown above

-I-I-I-I-I-I-I-I-I-I-I- The markers can be places in the above 11 places in 11C3 ways

Therefore the ways of dividing the girls into 4 groups is 11C3 X 12! i.e C

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The explanation is as follows : The Balls are similar so the order or the arrangement will not matter. Only the ways the balls can be grouped is important. Also the balls have to be dropped into the bins so one scenario is that all the 14 balls are dropped into one bin and other 3 bins are empty The balls can be placed as such - - - - - - - - - - - - - -

To arrange them into 4 groups such that the no. of balls in any bin varies from 0-14.lets consider markers which will divide the line into 4 parts. So we require 3 markers which can divide the line as follows

part1 Marker1 part2 marker2 part 3 marker3 part4

So there are 14 balls and 3 markers

III and - - - - - - - - - - - - - -

In case the marker is placed as

III- - - - - - - - - - - - - - we have all the 14 balls in the 4th bin and none in the other 3

When its like II-I - - - - - - - - - - - - - There are 13 balls in the 4th bin and 1 ball in the 3rd bin

So the ways of arranging these 17 items is \(\frac{17!}{(14!*3!)}\) as 14 balls and 3 markers are the same

In other words the balls can be dropped in 4 different bins in 17C3 ways which is the same as \(\frac{17!}{(14!*3!)}\)

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Last edited by aceacharya on 18 May 2013, 02:08, edited 2 times in total.

The explanation is as follows : The Balls are similar so the order or the arrangement will not matter. Only the ways the balls can be grouped is important. Also the balls have to be dropped into the bins so one scenario is that all the 14 balls are dropped into one bin and other 3 bins are empty The balls can be placed as such - - - - - - - - - - - - - -

To arrange them into 4 groups such that the no. of balls in any bin varies from 0-14.lets consider markers which will divide the line into 4 parts. So we require 3 markers which can divide the line as follows

part1 Marker1 part2 marker2 part 3 marker3 part4

For this we have 16 places to place the markers

III-I-I-I-I-I-I-I-I-I-I-I-I-I-

In case the marker is placed as

III- - - - - - - - - - - - - - we have all the 14 balls in the 4th bin and none in the other 3

When its like II-I - - - - - - - - - - - - - There are 13 balls in the 4th bin and 1 ball in the 3rd bin

so the 3 markers can be placed in the 16 places in 16C3 ways

In other words the balls can be dropped in 4 different bins in 16C3 ways

I guess it should by 17C3, 14 balls + 3 markers.... Isn't it?
_________________

The explanation is as follows : The Balls are similar so the order or the arrangement will not matter. Only the ways the balls can be grouped is important. Also the balls have to be dropped into the bins so one scenario is that all the 14 balls are dropped into one bin and other 3 bins are empty The balls can be placed as such - - - - - - - - - - - - - -

To arrange them into 4 groups such that the no. of balls in any bin varies from 0-14.lets consider markers which will divide the line into 4 parts. So we require 3 markers which can divide the line as follows

part1 Marker1 part2 marker2 part 3 marker3 part4

For this we have 16 places to place the markers

III-I-I-I-I-I-I-I-I-I-I-I-I-I-

In case the marker is placed as

III- - - - - - - - - - - - - - we have all the 14 balls in the 4th bin and none in the other 3

When its like II-I - - - - - - - - - - - - - There are 13 balls in the 4th bin and 1 ball in the 3rd bin

so the 3 markers can be placed in the 16 places in 16C3 ways

In other words the balls can be dropped in 4 different bins in 16C3 ways

I guess it should by 17C3, 14 balls + 3 markers.... Isn't it?

Thanks for the heads up. I had got the solution wrong. Do check the corrected solution.
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19 May 2013, 00:57

One of my favorite question

A and B complete a piece of the task together in D days. A takes 9 days more than the days he will take to complete the task on his own and B takes 4 more days that he would take if he completes the task on his own. What is the value of D?

Lets say A takes a days and B takes b days to complete the work.

Together they will take ab/(a+b) days to complete the work.

The difference between the days taken by A alone and days taken when A and B worked together = a - ab/(a+b) = a^2/(a+b) Similarly difference of days in case of B = b^2/(a+b)

So if the difference is given, then the combined days can be obtained as sqrt (product of diffrence) = sqrt(a^2*b^2/(a+b)^2) = ab/(a+b)

Applying the formula, the combined number of days are sqrt(9*4) = 6 days

A number that can be expressed as a ratio of two numbers is rational. However, roots, \(\pi\), and endless non-repeating decimals are irrational numbers; they cannot be written in a ratio of two numbers.

(1) expression is not sufficient, need x (2) expression is not sufficient, need y

(1) (2) expression reduces to \(\frac{7\pi}{22}\). \(\frac{22}{7}\) is commonly used to approximate \(\pi\), yet \(\frac{22}{7}\neq\pi\). Thus, \(\frac{x}{y}\) is not an integer.

(1) If \(t^t-1\) is not positive, then \(t\leq0\). \(t\) can be any odd negative integer or 0. Not sufficient (2) If \(t! \neq 0\), then \(t\) can still be any positive integer. Not sufficient

(1) (2) If \(t^t -1\) is not negative and \(t! \neq 0\), then \(t=0\). \(0^0 - 1 = 0\) and \(0! = 1\). Sufficient

Therefore, the correct answer is C.

Last edited by mejia401 on 02 Jun 2013, 10:56, edited 1 time in total.

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