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In a company of Clergymen, Doctors and Lawyers it is found that the sum of the ages of all present is 2160; their average age is 36; the average age of the Clergymen and the Doctors is 39; the average age of the Doctors the Lawyers is (32 + 8/11); the average age of the Clergymen and the Lawyers is (36 + 2/3).

If each Clergymen had been 2 years, each Lawyer 5 years, and each Doctor 7 years older, the average age of everyone present would have been greater by 5 years.

Determine the number of each profession present and their average ages.

Edited March 22, 2010 by K Sengupta

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Capital letters = number of people, lower case = average age of that group

Problem translates into the following equations:

(1): 2160 = c*C + d*D + l*L

(2): 36 = (c*C+d*D+l*L)/(C+D+L)

(3): 39 = (c*C + d*D)/(C + D)

(4): 32+8/11 = (d*D + l*L)/(D + L)

(5): 36 + 2/3 = (l*L + c*C)/(L+C)

(6):[(c+2)*C +(l+5)*L+(d+7)*D]/(C+D+L) = 36+5 = 41

From (1) and (2) we get:

(7): C + L + D = 2160/36 = 60

From (6),(2), and (7) we get:

(8): 2*C + 5*L + 7*D = 60*(41-36) = 300

From (3), (4), and (5) we get

(9): cC + dD = 39*(C+D)

(10):dD + lL = (32+8/11)*(D+L)

(11):lL + cC = (36 + 2/3)*(L+C)

Adding (9), (10), and (11) together we get

(12): 2*(dD + cC + lL) = C*(39+36+2/3) + D*(39 + 32 +8/11) + L*(32+8/11 +36+2/3) = C*(75+2/3) + D*(71+8/11) + L*(69+13/33)

Substituting (1) into (12), we have

(13): 4320 = C*(75+2/3) + L*(69+13/33) + D*(71+8/11)

Equations (7), (8), and (13) are sufficient to solve for C, D, and L by linear algebra.

Creating matrix:

[1,1,1,60;

2,5,7,300;

75+2/3,69+13/33,71+8/11,4320]

Reducing the matrix to row echelon form gives (C,L,D) = (16,20,24)

From (1), (9), and (10)

(14): 16*c + 20*l + 24*d = 2160

(15): 16*c + 24*d = 39*(40) = 1560

(16): 20*l +24*d= (32+8/11)*(44) = 1440

We can form another matrix:

[16,20,24,2160;

16,0,24,1560;

0,20,24,1440]

Reducing to row echelon form gives (c,l,d) = (45, 30, 35)

There are 16 clergymen, 20 lawyers, and 24 doctors.

The average clergyman is 45 years old, the average lawyer is 30 years old, and the average doctor is 35 years old.

Edited March 22, 2010 by mmiguel1