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# The ratio of the average age of the class without the teacher to the

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Manager
Joined: 29 Nov 2018
Posts: 200
Concentration: Marketing, Strategy
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10 Jan 2019, 08:00
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Difficulty:

55% (hard)

Question Stats:

59% (02:26) correct 41% (02:36) wrong based on 30 sessions

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The ratio of the average age of the class without the teacher to the average age including the teacher is 8 : 9. If the ratio of the teacher’s age and the average age of the students is 3 : 1, find the number of students.

A. 8
B. 9
C. 10
D. 12
E. 15
Manager
Joined: 01 May 2017
Posts: 82
Location: India
The ratio of the average age of the class without the teacher to the  [#permalink]

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Updated on: 10 Jan 2019, 23:10
The ratio of the average age of the class without the teacher to the average age including the teacher is 8 : 9. If the ratio of the teacher’s age and the average age of the students is 3 : 1, find the number of students.

Let number of students in the class = x
Let the average age of the students = y
Let the age of the teacher = z

Given,
1) ratio of the average age of the class without the teacher to the average age including the teacher = 8 : 9
Average age of students and teacher = $$(No.Of Students*Avg Age + Age of teacher)/(No.of students + 1)$$
= $$(x*y + z)/(x+1)$$
Given this ratio = 8:9
$$y/((x*y + z)/(x+1)) = 8/9$$
$$y(x+1)/(x*y + z) = 8/9$$
$$xy + 9y = 8z$$ -----------------------equation 1

2) Ratio of the teacher’s age and the average age of the students is 3 : 1
$$z/y = 3/1$$
$$z = 3y$$ --------------------------------- equation 2
equation 2 in equation 1 implies,
$$xy + 9y = 8(3z)$$
$$xy = 15y$$
$$y(x-15) =0$$
Since y avg age can't be zero
$$x = 15$$
Number of students = 15

Option E is correct

Originally posted by Dare Devil on 10 Jan 2019, 18:55.
Last edited by Dare Devil on 10 Jan 2019, 23:10, edited 1 time in total.
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Re: The ratio of the average age of the class without the teacher to the  [#permalink]

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10 Jan 2019, 21:13
3
cfc198 wrote:
The ratio of the average age of the class without the teacher to the average age including the teacher is 8 : 9. If the ratio of the teacher’s age and the average age of the students is 3 : 1, find the number of students.

A. 8
B. 9
C. 10
D. 12
E. 15

Solve it using the logical approach.

Avg Age of students : Avg age of students incl teacher = 8:9

Age of teacher : Avg age of students = 3:1 = 24 : 8 (to make it comparable to ratio above.)

Say if teacher's age is 24, the students avg age is 8 which becomes 9 when the teacher is added to the group. The increase of 1 is because of the excess 16 (24 - 8 = 16) that the teacher brought in to divide evenly among all students and herself. Since everyone got 1 (the avg increased by 1), there must have been total 16 people (15 students + 1 teacher).

Hence number of students must be 15.

Answer (E)

To read more on this approach, check: https://www.veritasprep.com/blog/2012/0 ... etic-mean/
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Posts: 7763
Re: The ratio of the average age of the class without the teacher to the  [#permalink]

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10 Jan 2019, 21:47
cfc198 wrote:
The ratio of the average age of the class without the teacher to the average age including the teacher is 8 : 9. If the ratio of the teacher’s age and the average age of the students is 3 : 1, find the number of students.

A. 8
B. 9
C. 10
D. 12
E. 15

(A)A logical way..

(I) If s students are there, the strength with the addition of a teacher becomes s+1..
(II) However, teacher's age is 3 times the average of students average age, we can take instead of teacher we are adding the 3 students to total students, say s... so s+3

Therefore, final ratio 8:9 will be similar to ratio s=1 : s+3..
$$\frac{8}{9}=\frac{s+1}{s+3}.....9(s+1)=8(s+3)......9s+9=8s+24......s=24-9=15$$

(B)If you do not understand the reasoning behind the above solution..
# of students = s and average age = a... so total age = as
# after addition of teacher = s+1, and age of teacher = 3a...so total age = as+3a... average age now = $$\frac{as+3a}{s+1}=\frac{a(s+3)}{s+1}$$

Thus initial average age : average age after addition = a : $$\frac{a(s+3)}{s+1}$$ = 8:9.....1 : $$\frac{(s+3)}{s+1}$$ = 8:9..
$$9*1=\frac{(s+3)}{s+1}*8....9(s+1)=8(s+3)......9s+9=8s+24......s=24-9=15.$$

E
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Joined: 07 Jan 2019
Posts: 1
Re: The ratio of the average age of the class without the teacher to the  [#permalink]

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10 Jan 2019, 22:43
1
Dare Devil wrote:
The ratio of the average age of the class without the teacher to the average age including the teacher is 8 : 9. If the ratio of the teacher’s age and the average age of the students is 3 : 1, find the number of students.

Let number of students in the class = x
Let the average age of the students = y
Let the age of the teacher = z

Given,
1) ratio of the average age of the class without the teacher to the average age including the teacher = 8 : 9
Average age of students and teacher = $$(No.Of Students*Avg Age + Age of teacher)/(No.of students + 1)$$
= $$(x*y + z)/(x+1)$$
Given this ratio = 8:9
$$(x*y + z)/(x+1) = 8/9$$
$$xy + 9y = 8z$$ -----------------------equation 1

2) Ratio of the teacher’s age and the average age of the students is 3 : 1
$$z/y = 3/1$$
$$z = 3y$$ --------------------------------- equation 2
equation 2 in equation 1 implies,
$$xy + 9y = 8(3z)$$
$$xy = 15y$$
$$y(x-15) =0$$
Since y avg age can't be zero
$$x = 15$$
Number of students = 15

Option E is correct

Thank you for the solution. I have a small question to ask. I see no element of average age of students without teacher being used while deriving the 1st equation. I see that the ratio has been directly equated to the equation of new average age with teacher. However the solution is correct. Where am I thinking wrong? Or what am I missing here. Can you kindly help me out? Thanks in advance

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Manager
Joined: 01 May 2017
Posts: 82
Location: India
Re: The ratio of the average age of the class without the teacher to the  [#permalink]

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10 Jan 2019, 23:39
Bharat1993 wrote:

Thank you for the solution. I have a small question to ask. I see no element of average age of students without teacher being used while deriving the 1st equation. I see that the ratio has been directly equated to the equation of new average age with teacher. However the solution is correct. Where am I thinking wrong? Or what am I missing here. Can you kindly help me out? Thanks in advance

Posted from my mobile device

Hey Bharat, I did the problem basically on the paper first So I wrote the final step after resolving the equation as equation 1 ...you are right I missed avg of students.. I arrived at the right answer..including the avg of students. I have edited it, inline in the first post with the correct steps and thanks for pointing out the mistake.
Re: The ratio of the average age of the class without the teacher to the   [#permalink] 10 Jan 2019, 23:39
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# The ratio of the average age of the class without the teacher to the

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