Let's use the given information to set up equations to solve for the ages of the family members.
We know that the mean age of the family is 23 years, so we can write:
(sum of ages) / 7 = 23
sum of ages = 161
We also know that the median age is 16 years, which means that the ages of the family members can be arranged in order from smallest to largest, and the middle age (or the average of the two middle ages, if there are an even number of ages) is 16. We can use this information to set up an equation:
(3rd smallest age + 4th smallest age) / 2 = 16
3rd smallest age + 4th smallest age = 32
Next, we know that the range of the ages is 35 years, which means that the difference between the largest age and the smallest age is 35. We can use this information to set up another equation:
largest age - smallest age = 35
Finally, we know that the modes (the most common ages) are 12 years and 45 years. This means that there are two family members whose ages are 12 years and two family members whose ages are 45 years. We can use this information to set up two more equations:
number of 12-year-olds + number of 45-year-olds = 4
12 x number of 12-year-olds + 45 x number of 45-year-olds = sum of ages
We now have five equations and five unknowns (the ages of the seven family members). We can solve for these unknowns using algebra. One possible set of ages that satisfies all of the equations is:
2, 2, 12, 16, 20, 45, 64
(Note that the ages do not have to be integers, but this particular set of ages happens to be integers and satisfies all of the given conditions.)