In an organization there are CE’s, ME’s and EE’s. The sum of
their ages is 2160; the average age is 36; the average age of
the CE’s and ME’s is 39; the average age of the ME’s and EE’s
is 32 8/11; the average age of the CE’s and EE’s is 36 2/3.
If each CE had been 1 year older, each ME 6 years and each EE
7 years older, their average age would have been greater by 5
years. Find the number of each group and their average ages.
Suppose there are C CE's,
Suppose there are M ME's,
Suppose there are E EE's,
Suppose the average age of the CE's is x years
Suppose the average age of the ME's is y years
Suppose the average age of the EE's is z years
>>...The sum of their ages is 2160...<<
Cx + My + Ez = 2160
>>...the average age is 36...<<
(Cx + My + Ez)/(C + M + E) = 36
>>...the average age of the CE’s and ME’s is 39...<<
(Cx + My)/(C + M) = 39
>>...the average age of the ME’s and EE’s is 32 8/11...<<
(My + Ez)/(M + E) = 32 8/11
>>...the average age of the CE’s and EE’s is 36 2/3...<<
(Cx + Ez)/(C + E) = 36 2/3
>>...If each CE had been 1 year older, each ME 6 years and
each EE 7 years older, their average age would have been
greater by 5 years...<<
[C(x+1)+M(y+6)+E(z+7)]/(C + M + E) = 36 + 5
So we have 6 equations in 6 unknowns:
(1): Cx + My + Ez = 2160
(2): (Cx + My + Ez)/(C + M + E) = 36
(3): (Cx + My)/(C + M) = 39
(4): (My + Ez)/(M + E) = 32 8/11
(5): (Cx + Ez)/(C + E) = 36 2/3
(6): [C(x+1)+M(y+6)+E(z+7)]/(C + M + E) = 36 + 5
Replacing the numerator of (2) by 2160 from (1)
2160/(C + M + E) = 36
2160 = (C + M + E)(36)
60 = C + M + E
(7) C + M + E = 60
Using (7), we can put 60 for the denominator in (6)
[C(x+1)+M(y+6)+E(z+7)]/60 = 36 + 5
[C(x+1)+M(y+6)+E(z+7)]/60 = 41
C(x+1)+M(y+6)+E(z+7) = 2460
Cx+C + My+6M + Ez+7E = 2460
Cx+My+Ez + C + 6M + 7E = 2460
Now from (1) we can replace Cx+My+Ez by 2160
2160 + C + 6M + 7E = 2460
(8) C + 6M + 7E = 300
Cross-multiplying (3), (4), and (5)
(9) Cx + My = 39(C + M)
(10) My + Ez = (32 8/11)(M + E)
(11) Cx + Ez = (36 2/3)(C + E)
Adding the left sides and the right sides, we get
2Cx + 2My + 2Ez = (227/3)C + (789/11)M + (2290/33)E
Multiplying thru by 33 to clear of fractions:
66Cx + 66My + 66Ez = 2497C + 2367M + 2290E
66(Cx + My + Ez) = 2497C + 2367M + 2290E
By (1) we can replace the parenthetical expression by 2160.
66(2160) = 2497C + 2367M + 2290E
or
(12) 2497C + 2367M + 2290E = 142560
So with (7), (8), and (12), we have three equations in three
unknowns:
C + M + E = 60
C + 6M + 7E = 300
2497C + 2367M + 2290E = 142560
Solving this system gives C=16, M=24, E=20
Now (1) becomes
16x + 24y + 20z = 2160
or dividing thru by 4
(13) 4x + 6y + 5z = 540
Now (9) becomes:
16x + 24y = 39(16 + 24)
16x + 24y = 1560
Dividing thru by 8
(14) 2x + 3y = 195
Now (11) becomes
16x + 20z = (36 2/3)(16 + 20)
or 16x + 20z = 1320
Dividing thru by 4
(15) 4x + 5z = 330
Now (13),(14), and (15) form a system of 3 equations in 3
unknowns
4x + 6y + 5z = 540
2x + 3y = 195
4x + 5z = 330
Solving that system gives x=45, y=35, z=30
Answer:
Average age of the 16 CE's is 45
Average age of the 24 ME's is 35
Average age of the 20 EE's is 30
Edwin