A set of five positive integers has an arithmetic mean of 150. A parti

VERITAS PREP OFFICIAL SOLUTION:

Let’s assume that the 5 natural numbers in increasing order are: a, b, b, b, a+100

We are given that a < b < a+100.

Also, we are given that a and b are positive integers. This information is critical – we will see later why.

The average of the 5 numbers is (a+b+b+b+a+100)/5 = 150

(a+b+b+b+a+100) = 5*150

2a+3b = 650

We need to find the number of distinct values that a can take because a+100 will also take the same number of distinct values.

Now there are two methods to proceed. Let’s discuss both of them.

Method 1: Pure Algebra – Write b in terms of a and plug it in the inequality

b = (650 – 2a)/3

a < (650 – 2a)/3 < a+100

3a < 650 – 2a < 3a + 300

Now split it into two inequalities: 3a < 650 – 2a and 650 – 2a < 3a + 300

Inequality 1: 3a < 650 – 2a

5a < 650

a < 130

Inequality 2: 650 – 2a < 3a + 300

5a > 350

a > 70

So we get that 70 < a < 130. Since a is an integer, can we say that a can take all values from 71 to 129? No. What we are forgetting is that b is also an integer. We know that

b = (650 – 2a)/3

For which values will be get b as an integer? Note that 650 is not divisible by 3. You need to add 1 to it or subtract 2 out of it to make it divisible by 3. So a should be of the form 3x+1.

b = (650 – 2*(3x+1))/3 = (648 – 6x)/3 = 216 – 2x

Here, for any positive integer x, b will be an integer.

From 71 to 129, we have the following numbers which are of the form 3x+1:

73, 76, 79, 82, 85, … 127

This is an Arithmetic Progression. How many terms are there here?

Last term = First term + (n – 1)*Common Difference

127 = 73 + (n – 1)*3

n = 19

a will take 19 distinct values so the last term i.e. (a+100) will also take 19 distinct values.

Method 2: Using Transition Points

Note that a < b < a+100

Since a < b, let’s find the point where a = b, i.e. the transition point

2a + 3a = 650

a = 130 = b

But b must be greater than a. If we increase b by 1, we need to decrease a by 3 to keep the average same. But decreasing a by 3 decreases the largest number i.e. a+100 by 3 too; so we need to increase b by another 1.

We get a = 127 and b = 132. This give us the numbers as 127, 132, 132, 132, 227. Here the average is 150

Since b < a+100, let’s find the point where b = a+100

2a + 3(a+100) = 650

a = 70, b = 170

But b must be less than a+100. If we decrease b by 1, we need to increase a by 3 to keep the average same. But increasing a by 3 increases the largest number, i.e. a+100 by 3 too, so we need to decrease b by another 1.

We get a = 73 and b = 168. This gives us the numbers as 73, 168, 168, 168, 173. Here the average is 150

Values of a will be: 73, 76, 79, ….127 (Difference of 3 to make b an integer)

This is an Arithmetic Progression.

Last term = First term + (n – 1)*Common difference

127 = 73 + (n – 1)*3

n = 19

a will take 19 distinct values so the last term i.e. (a+100) will also take 19 distinct values.

Answer (B)
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numbers are a,b,b,b,100+a
2a+3b+100=150*5=750
\(a= \frac{650 - 3b}{2}\) , or \(a= 325 - \frac{3b}{2}\) , note that 'b' has to be an even integer in order to get a an integer value .

we know b>a
so b> \(\frac{650 - 3b}{2}\)
so b>130

we know that b<100+a
so b < \(100 + \frac{650 - 3b}{2}\)
so b<170

130<b<170
and since b is even , we will get only 19 such values.

Answer 19.

Hi,

I was not able to solve but tried till some point and no idea how to proceed further.

Assume numbers : X, Y, Y, Y, X+100

average (x+y+y+y+x+100) = 5*150
2x+3y = 650

y = (650 – 2x)/3

we know that x < y < x+100

x < (650 – 2x)/3 < x+100
3x < 650 – 2x < 3x + 300

split into two inequalities: 3x < 650 – 2x and 650 – 2x < 3x + 300
1: 3x < 650 – 2x
5x < 650
x < 130

2: 650 – 2x < 3x + 300
5x > 350
x > 70

So we get that 70 < X < 130.
Number x should fall between 71 to 129.

I'm lost here, how to proceed further.
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I think it would be 19.
Once we have got the range for X, we need to find out the numbers between 70 and 130 that will be divisible by 3.
From 70 to 130, there are 19 multiples of 3. So, every third value would give us an integer value for Y.
Eg: X-73 Y-168
X-76 Y-166
Correct me if I am wrong.

Lets say numbers are in this order: x < y = y = y < x+100. Question is, how many different values can x take.
x+3y+x+100 = 750
2x+3y = 650
We have condition x < y < x + 100 so lets use it
2x + 3y < 2x + 3*(x+100) = 5x + 300
2x + 3y > 2x + 3x = 5x
2x + 3y = 650

So we got double inequality:
5x < 650 < 5x + 300 which turns into
350 < 5x < 650
70 < x < 130
x can take 130-70-1 = 59 values which corresponds to answer E
edit: forgot to take into account that 2x + 3y = 650 and both x and y = integers.
y = (650 - 2x)/3 = 2/3*(325-x) - integer, so 325 - x has to bi divisable by 3 to be correct
Now if I input all my available X (from 71 to 129) I'll notice that with a period of 3 my x values will yield me the result I need, I just need to carefully count them.
x = 71: 325 - 71 = 254, not divisable by 3 - fail
x = 72: 325 - 72 = 253, not divisable by 3 - fail
x = 73: 325 - 73 = 252, divisable by 3 - good
same goes for 249, 246, 243 and etc.
That means with x go with period of 3 starting from 73 and finishing with 73 + 3*18 = 127 making it total 19 numbers and corresponding to option B afterall.
edit: zzz, it seems like one of the main problems on the GMAT is actually remembering what type of numbers you are working with.

Hi All,

This question has some rather specific 'restrictions' that you have to pay attention to.

1) All 5 numbers are positive INTEGERS
2) The average of the 5 is 150 (so the sum of those 5 integers is 750)
3) The greatest of the 5 is 100 greater than the smallest of the 5
4) The remaining 3 integers are all the SAME and are somewhere between the biggest and smallest.

The question asks for the number of different POSSIBLE values for the LARGEST integer.

To start, I'm going to name the simplest example that fits the above information:
100, 150, 150, 150 and 200

To figure out the other options, we have to "slide" the numbers. There are some numbers that are IMPOSSIBLE though. For example.....

101 _ _ _ 201

101+201 = 302
So the sum of the other 3 integers would have to be 448. Since the 3 integers have to be the SAME number, there is no way for them to sum to 448. Thus, the largest number CANNOT be 201

This same "problem" happens with....
102 _ _ _ 202

102+202 = 304
So the sum of the other 3 integers would have to be 446. Since the 3 integers have to be the SAME number, there is no way for them to sum to 446. Thus, the largest number CANNOT be 202

We DO come across another solution though with....
103 _ _ _ 203

Here, the 3 other integers would be 148
103, 148, 148, 148, 203

This provides the basis for the pattern that we need to answer this question: We can increase/decrease the smallest and largest integers by 3 and decrease/increase the other integers by 2.

The options that "fit" would start with....
103
106
109
112
115
118
121
124
127, 130, 130, 130, 227

At this point, there are no other options, since the "middle 3" integers have to be BIGGER than the smallest integer and that would not happen beyond that last example.

This proves that there are 9 options beyond the first one (the 100, 150,150,150, 200 option). We can use this same pattern to SUBTRACT 3 from the smallest and largest values and ADD 2 to the other values....

Those options would be
97 152 152 152 197
94
91
88
85
82
79
76
73 168 168 168 173

Just as in the prior list, there are no other options past the last one. This is another 9 options.

In total, we have 1+9+9 = 19 options.

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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need not to mention that for each unique value of 'b' there will be a unique value of 'a' and '100+a' .

note 2x+3y = 650 --> \(y= \frac{650 - 2x}{3} ----> \frac{(650+x) - 3x}{3}\) implies 650+x should be multiple of 3 .
X is of the form (2k-1) . have a look at it.

Here is my solution:

n=5; average=150 => sum=750

we have a n n n a+100 => 2a+3n=750-100=650

also we know that a<n<a+100. than means that n=a+T, where T is some number from 1 to 99

then we have 2a+3a+3*T and this equals to 650
5a+3*T=650; 5a=650-3*T.

a=13-3*T/5. As "a" must be integer then T must be a multiple of 5. Thus, T can be 5,10,15,...,95.

Lets count this possibilities:

(95-5)/5+1=19

Thus T, and therefore a and a+100 have 19 different values.

Answer: (B)

Hope this helps

I used Diophantine equations for solving this question. May be somebody will find this approach easier.
One integer greater by 100, rest are the same. So we have:
a+b+b+b+a+100=750
2a+3b=650 Now let’s solve this Diophantine equation taking into consideration following restriction:
a<b<a+100
3b=650-2a => 3b=2(325-a)
b=2n, a=325-3n
a should be a positive integer, so 325-3n>0, solving this we have first inequality n<108
Now we apply second restriction – a<b
325-3n<2n => 325<5n => n>65
And final restriction b<a+100
2n<325-3n+100 => 5n<425 => n<85
Combining above inequalities we have 65<n<85 => 66≤n≤84 from where number of possible values including the boundaries is 84-66+1=19
Answer B
Hope this will help. Regards

let the 5 integers be a,b,b,b,a+100
to find lowest values that work, let a+100=b
thus, 5b=850➡b=170
70+170+170+170+170=750
this won't work, but if we add 3 to a and subtract 2 from b we get
73+168+168+168+173=750, the lowest set, and
76+166+166+166+176=750, the next lowest set
let x=the number of different values the largest number may take
73+3x=168-2x
5x=95
x=19
B

5 numbers will be a, a+100, a+x, a+x, a+x (x can be between 1 and 99)

5a + 100 + 3x = 750
5a = 650-3x

650-3x needs to be a multiple of 5. This happens when 3x is a multiple of 5 => x is a list of multiples of 5 (between 1,99)

5,10,15,---- 95
19 values

(B)

I used this approach and felt its the easiest:

x+y+y+y+(x+100) = 5*150 = 750 ----- (1)

2x + 3y = 650 ---------------------------- (2)

now we want to higher and lower limit, so we take, x=y :

5x = 650
x = 130

this implies, the set will be --- 130,130,130,130,230

lets try, 129 ---- 129+3y+229 = 750
3y = 750-358 ----- y will not be an integer as 352 is not divisible by 3

so, if you start thinking, even without doing the substitution and all... straight away analyse whether x + (x+100) is divisible by 3 or not

you will get 127 + 227 =354 ------ divisible by 3

so our one limit is 127,132,132,132,227

similarly by taking y = x + 100

we will get the other limit ---> 73,168,168,168,173

the highest value has decreased from 227 to 173, with a jump of 3 units ----- 227-173 = 57 ----- 57/3=19

hence A=19 is the right answer..

Hope this helped
Kudos if it did..

I feel a becomes negative after certain values of T hence may be a problem

Since the set of 5 integers has an average of 150, the total is 750
Let us say the integers are x,y,y,y,x+100 as the question says two integers differ by 100 and the three others are same. Moreover it is also mentioned that y is in between x and x+100. So the numbers can be x+1, x+2....x+99. Say x+k where k can be anything between 1 and 99
So we have x,x+k,x+k,x+k,x+100=750
5x+3k=650
5x=650-3k which means 650-3k is a multiple of 5.
We know 650 is a multiple of 5. So 3k has to be a multiple of 5 where k={1,2....99}
Since there are 19 multiples of 5 in this range 5,10,...95 the answer is 19
Hence B

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I did like this:

x+3y+x+100 -> 5x = 650 -> x = 130

We can also pick the largest number as x ->

x+3y+x-100 -> 5x = 850 -> x = 170

This is the interval in which the middle numbers can slide. And if the middle numbers slide 20 less or more from 150 then the two extreme values must slide 30 less or more from the "intuitive starting point" (100-150-150-150-200) to adjust the mean.

However, the two cases when (1) the smallest number is 130 or when (2) the largest number is 170 is forbidden according to the inequalities.

Now because two numbers slide in one direction and three equal numbers slide in another direction, the distances from the respective end points must be 3-2-2-2-3 or any multiples of these within our range, for example:

127-132-132-132-227

From 132-168, finally, we can count the possible instances for the middle numbers and the answer will be 19. Of course, this is the same number of instances as for the extreme values.

132, 134, 136, 138, 140 = 5
to 150 -> 10
to 160 -> 15
to 168 -> 19

I'm unable to understand how we get 127 as the end term of the sequence? (a's value). What is the logic? please someone help me to understand it...