12 Days of Christmas GMAT Competition - Day 5: A group of students : Problem Solving (PS)

IssacChan

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20 Dec 2024, 21:58

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Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A group of students, with an average (arithmetic mean) weight of p pounds, is boarding a plane. The plane has space for only one more person: either Sally or Harry. If Sally, who weighs 105 pounds, joins the group, the average weight decreases by 6 pounds. If Harry, who weighs 195 pounds, joins instead, the average weight increases by 12 pounds. What is the value of p?

A. 129
B. 135
C. 141
D. 147
E. 153

This question was provided by GMAT Club
for the 12 Days of Christmas Competition

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Let x be the number of person in the group

Add Sally
(xp +105)/(x+1) = p-6 (1)

Add Harry
(xp + 195)/(x+1) = p+12 (2)

(2) - (1)
(xp+195-xp-190)/(x+1) = p+12-p+6
90/(x+1) = 18
x+1 = 90/18
x+1 = 5
x = 4

sub x = 4 into (1)
(4p + 105)/(4+1) = p-6
4p + 105 = (p-6)*5
4p + 105 = 5p - 30
p = 135

Therefore the answer is B

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20 Dec 2024, 22:40

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Hi All,

according to me,

given that if a person with weight 105 is added then the average reduces 6 and the if the weight added is 195 then the avg increases by 12,

following this ,

--> Total weight difference/ Difference in average , from this we will be able to find per how much weight the average increases/decreases

--> (195-105)/(6+12)

-->90/18

-->5

Therefore by this we can say that if the weight is increased/decreased by 5 then the average shall increase/decrease by 1.

Applying this to the given statements,

For weight addition of 105, the average weight is getting reduced by 6. So in terms of weight, it will be reduced by 6 x 5 =30.

So by this we know that if the weight addition was of 105 + 30, there wouldnt have been any average weight increase/descrease.

If we calculate for 195 also it will give us that increase 12 avg leads to increase of 12 x 5 =60. and 195 -60 --> 135.

Hence 135 is our answer

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20 Dec 2024, 23:21

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The difference between 195-105 =90 pounds .. equals 18 points difference. So each point is 90/18=5 pounds.
Therefore p = 105+5*6 = 195-5*12 =135 pounds.
So B.

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Solve this step by step:

Let's say there are n students in the original group
- Original total weight = n × p
If Sally joins:
- New average = (original total + 105)/(n + 1) = p - 6
- (n × p + 105)/(n + 1) = p - 6
If Harry joins:
- New average = (original total + 195)/(n + 1) = p + 12
- (n × p + 195)/(n + 1) = p + 12
From Sally's equation:
- n × p + 105 = (n + 1)(p - 6)
- n × p + 105 = n × p - 6n + p - 6
- 105 = -6n + p - 6
- 111 = -6n + p
From Harry's equation:
- n × p + 195 = (n + 1)(p + 12)
- n × p + 195 = n × p + 12n + p + 12
- 195 = 12n + p + 12
- 183 = 12n + p
Now we have two equations:
- 111 = -6n + p
- 183 = 12n + p
Subtracting them:
- -72 = -18n
- n = 4
Plug n = 4 back:
- 111 = -6(4) + p
- 111 = -24 + p
- p = 135

Therefore, the average weight is 135 pounds.

The answer is B.

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Let say,
Total students = t
Total weight = w = p*t = pt

1) If Sally boards:
Then new avg = p - 6
=> pt + 105 = (p-6)*(t+1)
=> pt + 105 = pt + p - 6t - 6
=> 111 = p - 6t ...(i)

2) If Harry boards:
Then new avg = p + 12
=> pt + 195 = (p+12)*(t+1)
=> pt + 195 = pt + 12t + p + 12
=> 183 = p + 12t ...(ii)

Now, multiply (i) by 2 and add it to (ii)

222 + 183 = 3p
=> 3p = 405
=> p = 135

Answer: B. 135

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21 Dec 2024, 00:01

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A group of students, with an average (arithmetic mean) weight of p pounds, is boarding a plane. The plane has space for only one more person: either Sally or Harry. If Sally, who weighs 105 pounds, joins the group, the average weight decreases by 6 pounds. If Harry, who weighs 195 pounds, joins instead, the average weight increases by 12 pounds. What is the value of p?

from the question we can write these equations by assuming the number of students in the group = n, average weight = p pound, so total weight = n*p

When Sally is added to the group, the total weight can be written as
(np+105) = (p-6)* (n+1)

When Harry is added to the group, the total weight can be written as
(np+195)= (p+12)* (n+1)

Solving these 2 equations for p will give us

P= 135

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21 Dec 2024, 00:28

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When the student weighs 105 pounds the average weight is p - 6 pounds;
When the student weighs 195 pounds the average weight is p + 12 pounds.

So, the increase in total weight of 195 - 105 = 90 pounds corresponds to the increase in average weight of (p + 12) - (p - 6) =18 pounds, which means that there are 90/18 = 5 students (including the new one). So, initially there were 4 student.

Total weight = 4p + 105 = 5(p-6) --> p = 135 pounds.

Answer: B.

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21 Dec 2024, 00:52

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Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A group of students, with an average (arithmetic mean) weight of p pounds, is boarding a plane. The plane has space for only one more person: either Sally or Harry. If Sally, who weighs 105 pounds, joins the group, the average weight decreases by 6 pounds. If Harry, who weighs 195 pounds, joins instead, the average weight increases by 12 pounds. What is the value of p?

A. 129
B. 135
C. 141
D. 147
E. 153

This question was provided by GMAT Club
for the 12 Days of Christmas Competition

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Did a very algebraic method

try equating it and solving with linear equations

you will get answer as 135

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Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A group of students, with an average (arithmetic mean) weight of p pounds, is boarding a plane. The plane has space for only one more person: either Sally or Harry. If Sally, who weighs 105 pounds, joins the group, the average weight decreases by 6 pounds. If Harry, who weighs 195 pounds, joins instead, the average weight increases by 12 pounds. What is the value of p?

A. 129
B. 135
C. 141
D. 147
E. 153

This question was provided by GMAT Club
for the 12 Days of Christmas Competition

Win $40,000 in prizes: Courses, Tests & more

Let no of students = x

Total Weight/x=p(given)
Total Weight = px
Now, if Sally joins the group,
px+105/x+1 = p-6
px+105=(x+1)(p-6)
p=6x+111 (1)

If Harry joins the group, then
px+195/x+1 = p+12
px+195=(p+12)(x+1)
p = 183-12x (2)

From eqn 1 & 2 we get

6x+111 = 183-12x
x = 4

Substitute the value of x in either eq1 or 2, we get
p = 135

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21 Dec 2024, 01:03

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Hi everyone

This is a statistics question with weighted averages.
P=Average , Q = number of students
First equation: [Q*P+195][/Q+1]
Second equation: [Q*P+105][/Q+1]
The difference between them: [Q*P+195][/Q+1] - [Q*P+105][/Q+1] = 6+12
Q*P+195-Q*P+105=18(Q+1)
90=18Q+18 72=18Q
Q=4

If Q=4 then:
[Q*P+195][/Q+1] - [Q*P][/Q] = 12
[4P+195][/5] - [4P][/4] = 12
[4P+195][/5] - P = 12
4P+195-5P=60
P=135
Our Answer B

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21 Dec 2024, 01:11

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Answer is B. 135

Let's assume the number of students = x
The sum of all their weights is px

We can express the effect of Sally or Harry on the group as such:

Sally: (px + 105)/(x+1) = p-6
Harry: (px + 195)/(x+1) = p+12

We can solve these equations, which gives us x = 4 and p = 135

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Let the sum of students' weight be S
Total number of students be N
Average (P) = S/N
-> S = PN

If Sally joins -

(S+105)/(n+1) = p-6
(S+105) = (p-6)(n+1)
pn + 105 = pn -6n +p -6

6n - p +111 = 0

If Harry joins -

(S+195)/(n+1) = p+12
(S+195) = (p+12)(n+1)
pn + 195 = pn + 12n +p +12

12n - p +183 = 0

Now, we have 2 equations-

6n - p +111 = 0
12n - p + 183 = 0

We can easily solve and get the answer for P as 135.

FINAL ANSWER - Option B

Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A group of students, with an average (arithmetic mean) weight of p pounds, is boarding a plane. The plane has space for only one more person: either Sally or Harry. If Sally, who weighs 105 pounds, joins the group, the average weight decreases by 6 pounds. If Harry, who weighs 195 pounds, joins instead, the average weight increases by 12 pounds. What is the value of p?

A. 129
B. 135
C. 141
D. 147
E. 153

This question was provided by GMAT Club
for the 12 Days of Christmas Competition

Win $40,000 in prizes: Courses, Tests & more

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let the total weight of the group of students= W and no. of the students in the group= n.

So, Initially- W/n= p --- [arithmetic mean of weights of n students], Therefore- W=pn

When Sally joins, mean weight- (W+105)/(n+1)= p-6 => (pn+105)/(n+1)= p-6 => p=6n+111..(i)
When Harry joins, mean weigh- (W+195)/(n+1)= P+12 => (pn+195)/(n+1)= p+12 => p= 183-12n..(ii)

Solving (i)and(ii) we get, p=135

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we have n students
average weight of the group is p
Total weight = np

(1) We had one person to the group weighing 105 pounds -> n+1
total weight = (np + 105) / n+1
new average weight = (np + 105) / n+1 = p-6
p=6n+111

(2) We had one person to the group weighing 195 pounds -> n+1
total weight = (np + 195) / n+1
new average weight = (np + 105) / n+1 = p=12
p=183-12n

from (1) and (2) we deduce n=4 and therefore p=135

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Initially, there are n students with avg weight 3 pounds.

If Sally joins, avg weight decreases by 6 pounds.

If Harry joins, avg weight increases by 12 pounds.

Using this info, I set up equations and solved for n.

Turns out, n = 12.

So, if Sally or Harry joins, there are 12 + 1 = 13 people.

The answer is not 13, but 13 × 10 - 1 = 129.

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21 Dec 2024, 02:58

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From the answers provided, only two have a minus 6 and a plus 12 as part of the answers that is 135 and 141. So test both options
Starting with 135. Assume total number of students is x so (135x+105)/x+1=129 and (135x + 195)/x+1=147
Solving the above two equations yields a common answer 4 so 135 must be the answer B. (141 does not give an interger)

Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A group of students, with an average (arithmetic mean) weight of p pounds, is boarding a plane. The plane has space for only one more person: either Sally or Harry. If Sally, who weighs 105 pounds, joins the group, the average weight decreases by 6 pounds. If Harry, who weighs 195 pounds, joins instead, the average weight increases by 12 pounds. What is the value of p?

A. 129
B. 135
C. 141
D. 147
E. 153

This question was provided by GMAT Club
for the 12 Days of Christmas Competition

Win $40,000 in prizes: Courses, Tests & more

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21 Dec 2024, 03:03

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x=students

p-105=6(x+1)=6x+6 -> p=6x+111
195-p=12(x+1)=12x+12 -> p=-12x+183

18x=72 -> x=4 -> p=135

Answer B

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