In a family of husband, wife and a daughter, the sum of the husband's age, twice the wife's age and thrice the daughters' age is $85$; while the sum of twice the husband's age, $4$ times the wife's age and $6$ times the daughter's age is $170$. It is also given that the sum of $5$ times the husband's age, ten times the wife's age and $15$ times the daughter's age equals $450$. The number of possible solutions, in terms of the age of husband, wife and daughter, to this problem is

Let the ages of the husband, wife, and daughter be H, W, and D. The first condition yields the equation H + 2W + 3D = 85, and the second condition yields 2H + 4W + 6D = 170. Notice that multiplying the first equation by 2 results in exactly the second equation, meaning they are the same line. However, the third condition gives 5H + 10W + 15D = 450, and multiplying the first equation by 5 results in 5H + 10W + 15D = 425, which contradicts the third equation. Because this system contains a contradiction, there are 0 possible solutions.

• Thanks
• Wrong explanation