The avg of 3 children in a family is 20% of the avg age of the father and eldest child.The total age of the mother and the youngest child is 39 years. If the father age is 26 years. What is the age of second child?

• SIMPLER SOLUTION WITH DETAILED EXPLANATION
  F = age of father = 26
  M = age of mother
  A = age of eldest child
  B = age of second child
  C = age of youngest child
  the average of 3 children in a family is 20% of the average age of the father and the eldest child.
  (A+B+C)/3 = 20% OF ((F+A)/2)
  Since 20% = 1/5
  (A+B+C)/3 = 1/5 ((F+A)/2)
  Multiply fractions on the right:
  (A+B+C)/3 = (F+A)/10
  Cross-multiply:
  10(A+B+C) = 3(F+A)
  Distribute:
  10A+10B+10C = 3F+3A
  Subtract 3A from both sides:
  7A+10B+10C = 3F
  Since the father is 26, substitute 26 for F
  7A+10B+10C = 3(26)
  Factor 10 out of 2nd and 3rd terms
  on the left. Multiply the right
  7A+10(B+C) = 78
  7A = 78-10(B+C)
  A = (78-10(B+C))/7
  That must be an integer. So the numerator
  must be divisible by 7.
  To get that numerator, we are subtracting a
  multiple of 10 from 78. Therefore the resulting
  numerator must end in 8. And it must be divisible
  by 7.
  The only two-digit multiple of 7 that ends in 8 is 28,
  so the numerator 78-10(B+C) must = 28, which makes
  
  A = = (78-10(B+C))/7 = 28/7 = 4
  So the eldest child is 4
  78-10(B+C) = 28
  Subtract 28 from both sides:
  50-10(B+C) = 0
  Add 10(B+C) to both sides:
  50 = 10(B+C)
  Divide both sides by 10
  5 = B+C
  So the sum of the two younger children's
  ages is 5
  The sum of the two younger children = 5, and since
  they are younger than the eldest, who is 4, they
  are 3 and 2. So we have the answer, but let's
  get all the ages.
  Since the total age of the mother and the youngest
  child is 39 years, and the youngest child is 2, that
  makes the mother have to be 37.
  The father is 26, the mother is 37, the eldest child is 4,
  the second child is 3, and the youngest child is 2.
  Answer: The second child is 3
  
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