SOLUTION: A man is four years older than his wife and 5 times than his son. When the son was born, the age of wife is six-seventh that of the husband’s age. Find the age of the wife.

Question 1145748: A man is four years older than his wife and 5 times than his son. When the son was born, the age of wife is six-seventh that of the husband’s age. Find the age of the wife.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i get the following solution.

the age of the man is 33.6
the age of the wife is 29.6
the age of the son is 5.6

the man is 4 years older than his wife (33.6 - 29.6 = 4)
the man is 5 times older than his son ((33.6-5.6)/5.6 = 5)

when the son was born, the wife was 6/7 * the age of the man.
29.6 - 5.6 = 24
33.6 - 5.6 = 28
24/28 = 6/7

you said "and 5 times than his son"
i took that to mean "5 times older than his son"
if that's not what you meant, than you need to clarify what you meant.

5 times older than his son means that that the father is 6 times as old as his son.

if you meant "5 times as old as his son", then the solution winds up being the same, but with different numbers as shown below.

the man is 35.
the woman is 31.
the son is 7 (5 * 7 = 35)

7 years ago, the son was born.
the women was 31 - 7 = 24
the man was 35 - 7 = 28
24 / 28 = 6/7

the method of solution, assuming the man was 5 times as old as his son, was as follows:

let m = the age of the man
let w = the age of the woman
let s = the age of the son.

you are given that:

m = w + 4
m = 5 * s
since m = w + 4 and m = 5 * s, then you get w + 4 = 5 * s
solve for s to get s = (w + 4) / 5

the son was born s years ago.
when the son was born, the woman was 6/7 * the man's age.
you get (w - s) = 6/7 * (m - s)
multiply both sides of this equation by 7/6 and you get:
7/6 * (w - s) = (m - s) which can also be shown as:
(m - s) = 7/6 * (w - s)
multiply both sides of this equation by 6 to get:
6 * (m - s) = 7 * (w - s)
simplify to get:
6m - 6s = 7w - 7s

this last equation is used to solve the problem.
start with 6m - 6s = 7w - 7s
replace m with w + 4 and replace s with (w + 4) / 5 to get:
6 * (w + 4) - 6 * (w + 4) / 5) = 7 * w - 7 * (w + 4) / 5
multiply both sides of this equation by 5 to get:
30 * (w + 4) - 6 * (w + 4) = 35 * w - 7 * (w + 4)
simplify to get:
30 * w + 120 - 6 * w - 24 = 35 * w - 7 * w - 28
combine like terms to get:
24 * w + 96 = 28 * w - 28
subtract 24 * w from both sides and add 28 to both sides to get:
124 = 4 * w
solve for w to get:
w = 31

since m = w + 4, the m = 35
since m = 5 * s, then s = 7

you get:
m = 35
w = 31
s = 7

the son was born 7 years ago.
the man was 28
the woman was 24
24/28 = 6/7

your solution, assuming that the man is 5 times as old as his son, is that the wife is 31 years old.

similar logic was applied if i assumed that the man was 6 times as old as his son.
in that case, w + 4 = 6s which makes s = (w + 4) / 6.

assuming the man is 5 times as old as his son, rather than assuming the man is 5 times older than his son, gave me a cleaner answer.

that's why i'm thinking you meant 5 times as old as his son, rather than 5 times older than his son.

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