If the sum of five consecutive positive integers is A, then the sum of the next

If the sum of five consecutive positive integers is A, then the sum of the next five consecutive integers in terms of A is:

a) A+1
b) A+5
c) A+25
d) 2A
e) 5A 







n + n+1 + n+2 + n+3 + n+4 = 5n + 10 = A

n+5 + n+6 + n+7 + n+8 + n+9 = 5n + 35 = A+25

The sum of n consecutive positive integers is 45

The sum of n consecutive positive integers is 45. What is the value of n?

(1) n is even

(2) n < 9

(1) n is even. Now try to choose n such that average i.e, 45/n gives you .5 in decimal. Why? Because we would like to quickly identify the middle two.

So, choose n=6. Average=7.5. So, the case is 5, 6, 7, 8, 9, 10.
Now choose n=2. The case is 22, 23. Insufficient.

(2) No need for further calculation. Just look at (1). Insufficient.

Combining: Just look (1) & (2). E

The sum of n consecutive positive integers is 45

The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is odd
(2) n >= 9

(1)   The sum of consecutive integers will always be divisible by the number of integers only if the number of integers is odd.
n can be 3/5/9. Insufficient.

(2) We can have n=9/10 but since we are talking of +ve int only. Hence it would be 9. This is sufficient.

Ans : B

The sum of the even numbers between 1 and k is 79*80

The sum of the even numbers between 1 and k is 79*80, where k is an odd number, then k=?

(A) 79
(B) 80
(C) 81
(D) 157
(E) 159 






Sn=nA1+[n(n-1)d]/2

n=(k-1)/2, A1=2, d=2 ===>

2n+n(n-1)=n(n+1) ===>

[(k-1)/2]*[(k-1+2)/2]=(k-1)*(k+1)/4 ====>

(k-1)*(k+1)=79*2*80*2 ===>

k=159

At a certain theater, the cost of each adult's ticket is $5 and the cost of each child's ticket is $2. What was the average cost of all the adult's and children's tickets sold at the theater yesterday?
(1) Yesterday ratio of # of children's ticket sold to the # of adult's ticketr sold was 3 to 2
(2) Yesterday 80 adult's tickets were sold at the theater.









Av. cost=(2*C+5*A)/(C+A)
(1) 3A=2C A=2C/3 --> Av.cost C(2+5*2/3)/C(1+2/3) --> (2+5*2/3)/(1+2/3) Sufficient
(2) A=80 know nothing about C Not sufficient.

Answer: A

Are all of the numbers in a certain list of 15 numbers equal

Are all of the numbers in a certain list of 15 numbers equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.








(1) S=60 list can contain numerous combination of 15 numbers totaling 60. Not sufficient.
(2) If the sum of ANY 3 numbers=12 all numbers=12/3=4. Sufficient.

Answer: B.

a certain meter records voltage between 0 and 10 volts

a certain meter records voltage between 0 and 10 volts, inclusive. if the average value of 3 recordings from the meter was 8 volts, what was the smallest possible recording in volts?

A) 2
B) 3
C) 4
D) 5
E) 6

(v1+v2+v3)/3=8

v1+v2+v3=24

For any one to be min, the other two must be max. ****

max for v =10, If v1 is min and v2,v3 are max

v1= 24-20=4

Ans : C

****TAKEAWAY:

when you have numbers with a fixed sum or product:

if you want to MINIMIZE A QUANTITY, then you must MAXIMIZE ALL OTHER QUANTITIES in the sum or product.

if you want to MAXIMIZE A QUANTITY, then you must MINIMIZE ALL OTHER QUANTITIES in the sum or product.

What is the sum of all integers from 132 to 531, inclusive

What is the sum of all integers from 132 to 531, inclusive?

in this problem, you're looking for the SUM of a large SET OF NUMBERS.

when it comes to large SUMS, you should use the SUM FORMULA:
SUM = AVERAGE x NUMBER OF DATA POINTS

in this case, the NUMBER OF DATA POINTS is 400 (i.e., there are 500 integers being summed).
there are two ways to see this:
(1) use the "add one before you're done" rule (see the number properties guide):
number of integers = 531 - 132 + 1 = 532 - 132 = 400.
(2) if you subtract 131 from all of them, then 132, 133, 134, ..., 531 becomes 1, 2, 3, ..., 400. therefore, there are 400 integers. (this "matching technique" is only used for finding the NUMBER of integers in the list; obviously you can't do this when it comes to finding the average, or the sum, of the numbers.)

also, the AVERAGE is 331.5.
since this is a list of consecutive integers, you can just take the average of the first and last numbers, and that's the same as the average of the entire list.
this average is (132 + 531) / 2, or 331.5.

therefore, the sum is
400 x 331.5
= 132,600

The lifetimes of all the batteries produced by a certain company

The lifetimes of all the batteries produced by a certain company in a year have a distribution that is symmetric about the mean m. If the distribution has a standard deviation of d, what percent of the distribution is greater than m+d?

(1) 68 percent of the distribution lies in the interval from m-d to m+d, inclusive.
(2) 16 percent of the distribution is less than m-d.

The question stem tells you that Distribution is symmetrical around mean m so 50% above m and 50% below.

St-1 68% is between m-d and m+d, this tells you that on the side which is higher 34% is between m and m+d so, remaining 16% has to be above m+d. SUFFICIENT

St-2 tells you that 16% is below m-d, so on the other side 16% will be above m+d. SUFFICIENT as well.

This is actually a normal distribution, where 68% is between 1 Standard deviation (SD), 96% between 2 SD and rest within 3 SD.

Ans :D

The sum of first N-1 terms of AP is either zero or positive. but sum

The sum of first N-1 terms of AP is either zero or positive. but sum of first N terms is negative. What is the value of N?

A) Common difference is -4 and 7th term is last positive term
B) 25 is first term and there are 7 positive terms and all are integers

In an arithmetic sequence, if we know one term and we know the common difference, we know everything else. So we can rephrase this question: what is one term in the sequence and the common difference?

(1) Just by looking at the statement, you might guess this is insufficient because although we have the common difference, we don't have any terms in the sequence. Let's process this a bit just to make sure.

If the common difference is -4, and the 7th term is the last positive term, that means that the 7th term is either 1, 2, 3 or 4. Because the common difference is -4, the first term with be (term 7) + 24.

If the 7th term is 4,
- the 8th term is 0,
- the 9th through 15th terms all cancel out the 1st through 7th terms.
- the 16th term is when the sum becomes negative.
In this case, N must be 16.

However, if the 7th term is 2,
- the 8th term is -2
- so the 8th through 14th terms cancel out the 1st through 7th terms.
- the 15th term is when the sum becomes negative.
In this case, N must be 15.

(If you work it out for the others, N is 15 when the 7th term is 3, and N is 14 when the 7th term is 1.)

So (1) is insufficient.

(2) If there are 7 positive terms and they are all integers, then the common difference must be -4 (it cannot be -3 or -5). Since we know the first term and we know the common difference, then we can conclude that this statement is sufficient.

If 58 is 2 standart deviation below the mean and 98 - 3 standart deviation above the mean, what is mean?



(let us consider m as mean and s as S.D.)
58=m-2s
next if 98 is 3 S.D above the mean,this can be re-written as:
98=m+3s.
Now since we have two equations,solving them we can find out the value of mean and the S.D. The answer that i got is mean is 74 and S.D is 8.

notice one important theme at work here: in problems in which you're given a numerical value of the standard deviation, it's actually completely unimportant that it's called "the standard deviation".

Last year the avg. salary of the 10 employees of Company X was $42,800. What is the avg. salary of the same 10 employees this year?
1) For 8 of the 10 employees, this year's salary is 15 percent greater than last year's salary.
2) For 2 of the 10 employees, this year's salary is the same as last year's salary








Sum = Avg * data points
Note - the SUM of the salaries will answer the data sufficiency problem just as well as the AVERAGE that's explicitly requested.

I believe it is E because given (1) and (2) together, we have no way of knowing the individual salaries and we cannot assume that the individual salaries are each equal to the previous year's average. For example, perhaps the highest 8 salaries increased by 15% and the lowest 2 did not change at all, the average would surely be different than if the highest 2 salaries did not change and the other 8 increased by 15%.

Ans - E

If 58 is 2 standart deviation below the mean and 98 - 3 standart deviation above the mean, what is mean?





(let us consider m as mean and s as S.D.)
58=m-2s
next if 98 is 3 S.D above the mean,this can be re-written as:
98=m+3s.
Now since we have two equations,solving them we can find out the value of mean and the S.D. The answer that i got is mean is 74 and S.D is 8.

notice one important theme at work here:
in problems in which you're given a numerical value of the standard deviation, it's actually completely unimportant that it's called "the standard deviation".

Committee X and Committee Y , which have no common members, will combine to form Committee Z . Does Committee X have more members than Committee Y ?
(1) The average (arithmetic mean) age of the members of Committee X is 25.7 years and the average age of the members of Committee Y is 29.3 years.
(2) The average (arithmetic mean) age of the members of Committee Z will be 26.6 years






general comment:
in a WEIGHTED AVERAGE, if you have the 2 distances between the mean and each of the individual component averages, that's good enough to figure out the RATIO of the two components. here's how:
distance between committee A and average = 29.3 - 26.6 = 2.7
distance between committee B and average = 26.6 - 25.7 = 0.9
ratio = 3:1
committee B must be bigger (because the average is closer to committee B's average), so the ratio of committee B : committee A is 3:1 (not 1:3).
that's how it works. the same process applies to any other weighted average - good stuff.
WARNING:
this procedure will NOT allow you to determine the actual SIZE of committee A or committee B, without any additional information; all it will give you is the RATIO of the two sizes. that's plenty sufficient to answer this problem, but, if the problem had asked for the NUMBER of members of any of the committees, the answer would be (e) as you had expected.

During an experiment some water was removed from each of the 6 tanks .If the Std deviation of the volumes of the water at the beginning of the experiment was 10 gallons what was the standard deviation of the volumes of the water after the experiment ?

1) For each tank 30% of the volume of the water that was in the tank before the beginning of the experiment were removed during the experiment

2) The average ( mean ) volume of water in the at the end of the experiment was 63 gallons






(1) tells us that both the mean and the standard deviation of the set will decrease by 30%. Therefore, the new standard deviation will decrease to 7 gallons. SUFFICIENT.


Statement (2) tells us nothing about standard deviation, which measures SPREAD of numbers. If we achieved the 63 gallons by taking most of the water out of the tanks that were already lowest, then the standard deviation will be huge (because you'll have some tanks almost full and some almost empty). If we got there by taking most of the water out of the fullest tanks, then the standard deviation will be a lot smaller. INSUFFICIENT.


Ans : A


Note ----- Make sure you know that, when ALL numbers in a set are multiplied or divided by some number,** the mean and standard deviation are multiplied/divided by the same number


**This includes increasing or decreasing all the numbers in the set by some percentage (which can be accomplished by multiplication: e.g., 30% increase = multiplication by 1.3).

What is the average (aritmetic mean) hight of the n people in a certain group?

1 the average hight of the n/3 talles people in the group is 6 feet and 2 1/2 inches and the average hight of the rest of the people in the group is 5 feet and 10 inches.

2 The sum of the lenghts of the people is 178 feet and 9 inches.









Note - if you have the averages of all the FRACTIONS or PERCENTAGES of a group, then you'll be able to calculate the overall average of the group. This is a worthwhile fact to memorize for the data sufficiency problems.


stmt 1) Convert feet to inches..

[(n/3)(74.5) + (2n/3)(70)] / (n)
Suff

Stmt 2) Suff


Ans - D

List K consists of 12 consecutive integers. If -4 is the least integer in the list K, what is the range of the positive integers in list K.

A.3

B.6

C.7

D.11

E.5.5



the positive integers in the list are 1, 2, 3, 4, 5, 6, and 7, so the range is biggest - smallest = 7 - 1 = 6.
--
lesson one: READ PROBLEMS CAREFULLY. don't miss words like 'positive'. this is not a hard problem, but it's easy to miss if you are inattentive.
lesson two: KNOW YOUR TERMINOLOGY / CLASSIFICATIONS. zero is not a positive number.
Ans : 6

Ann $450,000Bob $360,000Cal $190,000Dot $210,000Ed $680,000
The table above shows the total sales recorded in July for the 5 salespeople. It was discovered that one of Cal’s sales was incorrectly recorded as one of Ann’s sales. After this error was corrected, Ann’s total sales were still higher than Cal’s total sales, and the median of 5 sales totals was $330,000. What was the value of the incorrectly recorded sale?


fact one: there are an odd number of data points, so the median is actually one of the numbers in the problem. this is cool here, because you know that exactly one of two things has happened: (a) cal's new total is $330,000, (b) ann's new total is $330,000. since none of the other salespeople's totals are going to change, it's impossible for the new median to be anyone's other than ann's or cal's.
fact two: however much you take away from ann's sales, you have to add to cal's sales. you're basically just exchanging money between the two of them, so whatever dollars you take away from ann must go to cal.
fact three: in any problem like this one, you should put the salary figures in order. this goes without saying in any problem that has to do with the median of a set

if x and y are points on the number line, what is the value of x + y?

1. 6 is the halfway between x and y.

2. y=2x









Ans : A

If M is a positive odd integrer, what is the average of a certain set of M integers?

1) The integers in th set are consecutive multiples of 3

2) The median of the set of integers is 33





(1) Insufficient. No way to know the mean. Consider one set {3, 6, 9} and another {6, 9, 12}.
(2) Insufficient. Knowing that there is an odd number of terms in the set and that the median is 33 does not tell us what the mean is.
(1&2) Sufficient. In an ordered set with an odd number of terms, the median is equal to the "middle" term. Moreover, in an equally distributed set of integers (like this one... consecutive multiples of 3) the median will equal the mean itself