Question 174113
Let the ages, from youngest to oldest be
{{{a}}}, {{{b}}}, {{{c}}}, and {{{d}}}
Given:
(1) {{{a + b + c + d = 384}}}
(2) {{{c - a = 14}}}
(3) {{{b = a + 3}}}
(4) {{{d = (a + b)/2 + 20}}}
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(3) {{{b = a + 3}}}
 {{{a = b - 3}}}
 {{{b - a = 3}}}
(2) {{{c - a = 14}}}
Subtract (3) from (2)
(5) {{{c - b = 11}}}
 {{{c = b + 11}}}
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Substitute (4) and (5) in (1)
(1) {{{a + b + c + d = 384}}}
(1) {{{b - 3 + b + b + 11 + (b - 3 + b)/2 + 20 = 384}}}
{{{3b + 8 + (2b - 3)/2 + 20 = 384}}}
Multiply both sides by {{{2}}}
{{{6b + 16 + 2b - 3 + 40 = 768}}}
{{{8b = 715}}}
{{{b = 89.375}}}
{{{a = b - 3}}}
{{{a = 86.375}}}
{{{c = b + 11}}}
{{{c = 89.375 + 11}}}
{{{c = 100.375}}}
(4) {{{d = (a + b)/2 + 20}}}
{{{d = 107.875}}}
The ages from youngest to oldest are
86.375, 89.375, 100.375, and 100.875
check:
(1) {{{a + b + c + d = 384}}}
{{{86.375 + 89.375 + 100.375 + 107.875 = 384}}}
{{{384 = 384}}}
OK
(2) {{{c - a = 14}}}
{{{100.375 - 86.375 = 14}}}
{{{14 = 14}}}
OK
(3) {{{b = a + 3}}}
{{{89.375 = 86.375 + 3}}}
{{{89.375 = 89.375}}}
OK
(4) {{{d = (a + b)/2 + 20}}}
{{{107.875 = (86.375 + 89.375)/2 + 20}}}
{{{107.875 = 107.875}}}
OK