SOLUTION: the four oldest people in Golden City lived a total of 384 years put together. The difference in ages for the youngest and the second oldest is 14. The second youngest is 3 years o

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Question 174113: the four oldest people in Golden City lived a total of 384 years put together. The difference in ages for the youngest and the second oldest is 14. The second youngest is 3 years older than the youngest. The oldest is 20 years older than the average of the second and youngest. What are their ages?
Found 3 solutions by ptaylor, gonzo, josmiceli:
Answer by ptaylor(2198)   (Show Source):
You can put this solution on YOUR website!
Let x=oldest
y=2nd oldest
z=2nd youngest
w=youngest
x+y+z+w=384------------------eq1
y-w=14----------------------------eq2
z=w+3--------------------------------eq3
average of second and youngest=(y+w)/2
x=((y+w)/2)+20 multiply each term by 2
2x=y+w+40---------------------------------eq4
substitute z=w+3 from eq3 and y=w+14 from eq2 into eq1:
x+w+14+w+3+w=384;
x+3w=384-17;
x+3w=367-----------------------eq1a
substitute y=w+14 into eq4
2x=w+14+w+40;
2x=2w+54
x=w+27----------------------------eq4a
substitute x=w+27 from eq4a into eq1a;
w+27+3w=367
4w=367-27
4w=340
w=85----------------------------youngest
substitute w=85 into eq3
z=85+3=88---------------------------second youngest
substitute w=85 into eq2
y-85=14
y=85+14=99-------------------------------second oldest
substitute w=85 into eq4a
x=85+27=112------------------------oldest
CK
112+99+88+85=384
384=384
and
99-85=14
14=14
and
88=85+3
88=88
and
((99+85)/2)+20=112
92+20=112
112=112
Does this help???-----ptaylor

Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website!
youngest = a
second youngest = b
second oldest = c
oldest = d
-----
all four lived a total of 384 years.:
a + b + c + d = 384
-----
difference in ages for the youngest and the second oldest is 14:
c - a = 14
-----
the second youngest is 3 years older than the youngest.
b = a + 3
-----
The oldest is 20 years older than the average of the second and youngest.
*****
question here is second what?
second youngest or second oldest.
i'll have to solve for both to see.
if second oldest, then:
d = (c+a)/2 + 20
if second youngest, then:
d = (b+a)/2 + 20
-----
equations you have to work with are:
a + b + c + d = 384
c = a + 14 (derived from c-a = 14)
b = a + 3
----
before we go any further, we will substitute:
a + 14 for c, and:
a + 3 for b
-----
a + b + c + d becomes:
a + a + 3 + a + 14 + d = 384
if we combine like terms, this equation becomes:
3*a + 17 + d = 384
-----
we have also:
d = (c+a)/2 + 20
or:
d = (b+a)/2 + 20
-----
we'll look at d = (b+a)/2 + 20 first.
we start with:
d = (b+a)/2 + 20
we substitute a+3 for b:
d = (a + 3 + a)/2 + 20
we combine like terms:
d = (2*a + 3)/2 + 20
we substitute for d in the total equation:
3*a + 17 + d = 384
becomes:
3*a + 17 + (2*a+3)/2 + 20 = 384
we combine like terms:
3*a + 37 + (2*a+3)/2 = 384
we multiply by 2 to remove fractions:
6*a + 74 + 2*a + 3 = 768
we combine like terms:
8*a + 77 = 768
we subtract 77 from both sides:
8*a = 768 - 77
we combine like terms:
8*a = 691
a = 86.375
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i was looking for an integer.
before going further, i'll try the second option.
that was:
d = (c+a)/2 + 20
-----
we start with:
d = (c+a)/2 + 20
we substitute for c:
d = (a + 14 + a)/2 + 20
we substitute for d in the total equation:
3*a + 17 + d = 384
becomes:
3*a + 17 + (2*a + 14)/2 + 20 = 384
we remove parentheses:
3*a + 17 + 2*a/2 + 14/2 + 20 = 384
we simplify:
3*a + 17 + a + 7 + 20 = 384
we combine like terms:
4*a + 44 = 384
we subtract 44 from both sides:
4*a = 384 - 44
we simplify:
4*a = 340
we divide both sides by 4
a = 340/4 = 85
-----
this option looks more promising so we'll pursue further.
we used the equation for:
The oldest is 20 years older than the average of the second OLDEST and youngest.
to solve for d:
d = (c+a)/2 + 20
-----
we have a = 85
c = a + 14 = 85 + 14 = 99
c = 99
-----
b = a + 3 = 85 + 3 = 88
b = 88
-----
d = (c+a)/2 + 20 = (99 + 85)/2 + 20 = 184/2 + 20 = 92 + 20 = 112
d = 112
-----
a = 85
-----
a + b + c + d = 85 + 88 + 99 + 112 = 384
-----
the equation appears to be satisfied.
youngest is 85
second youngest is 88
second oldest is 99
oldest is 112
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i also solved for:
d = (a+b)/2 + 20
which was:
The oldest is 20 years older than the average of the second YOUNGEST and youngest.
the answer came out different but i was still able to solve it and got an answer that also makes sense except that the ages are not in whole integers.
here's how that worked:
d = (a+b)/2 + 20
we substitute for b:
d = (a + a + 3)/2 + 20
we substitute for d in the total equation:
3*a + 17 + d = 384
becomes:
3*a + 17 + (2*a + 3)/2 + 20 = 384
we combine like terms:
3*a + 37 + (2*a + 3)/2 = 384
we subtract 37 from both sides:
3*a + (2*a + 3)/2 = 347
we multiply both sides by 2 to get:
6*a + 2*a + 3 = 2*347
we combine like terms and simplify:
8*a + 3 = 694
we subtract 3 from both sides:
8*a = 691
we divide both sides by 8:
a = 691/8 = 86.375
b = a + 3 = 89.375
c = a + 14 = 100.375
d = (a+b)/2 + 20 = (86.375 + 89.375)/2 + 20 = 175.75/2 + 20 =87.875 + 20 = 107.875
-----
we have:
a = 86.375
b = 89.375
c = 100.375
d = 107.875
add them all up together and we get:
384
-----
whether we used second oldest or second youngest in solving for d, we had a solution either way.
-----
choose the one that applies to your problem.
-----

Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website!
Let the ages, from youngest to oldest be
, , , and
Given:
(1)
(2)
(3)
(4)
-------------------
(3)

(2)
Subtract (3) from (2)
(5)

-------------------
Substitute (4) and (5) in (1)
(1)
(1)

Multiply both sides by

(4)

The ages from youngest to oldest are
86.375, 89.375, 100.375, and 100.875
check:
(1)

OK
(2)

OK
(3)

OK
(4)

OK