Logic, algebra, question to determine ages.

When Margaret reached her 17th birthday, Meredith was four times older than Marion.

When Margaret reaches her 25th birthday, eleven times the difference of Meredith and Marion’s ages will equal five times the sum of their ages.

a) Find the ages of Meredith and Marion on Margaret’s 17th birthday.

At Margaret age=17, let Meredith's age = x, Marion's age = y

When Margaret reached 17, $x = 4y$

When Margaret reaches 25, 8 years on,

$11((x + 8) - (y + 8)) = 5((x + 8) + (y + 8))\\ 11(x - y) = 5(x + y + 16)\\ 3x - 8y = 40$

Substitute $x = 4y => x=40,y=10$

b) Margaret’s current age divides Meredith’s age exactly. Given Margaret is younger than 25, determine her current age.

Let Margaret’s current age $= z$

$17 < z < 25$

Also Margaret is 23 years younger than Meredith (determined in (a) 40-17=23).

Given that Margaret's age divides Meredith's age exactly

=> Margaret is currently 23 years old, as 23 + 23 = 46 i.e. Meredith's age divisible exactly by 23.

Does that seem logical, or rather the most succinct solution?

(c) Claire is the daughter of Anne. The product of their ages is 1885. How old was Anne when Claire was born?

Let (current) Anne's age = x, Claire's age = y

$x > y\\ xy = 1885$

Finding factor pairs gives 29 × 65 = 1,885 (assume we shouldn't consider 13 × 145 = 1,885)

$x=65, y=29$

So when Claire was born, age=0, Anne's age was = 65-29 = 36.

Does that seem reasonable, logical and succinct?

I think (a) is complete. For (b) and (c), you show that your solutions are logical, but you don't demonstrate that there are no other logical/reasonable solutions.

For (b), I would suggest something along these lines:

Let $t \in \mathbb{Z}$ represent years since Margaret was 17 and Meredith was 40. We also know from the description that $t \ge 0$ (because Margaret being 17 was in the past), and $t \le 7$ because Margaret is not yet 25. Then, we have

$$\frac{40+t}{17+t} = n, $$

where $n \in \mathbb{Z}$.

Given $0 \le t \le 7$, the only integer solution to this equation is $n = 2$ and $t = 6$.

For (c), I would show what possible combinations of $x$ and $y$ are possible, then eliminate the ones that are unreasonable. The easiest way to do this is through the prime factorization of 1885:

$$ 1885 = 5\cdot 13 \cdot 29 $$

Then, since we know that Claire is Anne's daughter, and $x$ is Claire's age, $y$ is Anne's age, we also know $x>y$, so the set of possible current ages is:

$$ (x,y) \in \{ (1885, 1), (377, 5), (145, 13), (65, 29) \} $$

The only reasonable solution to this is $x = 65$ and $y = 29$, giving us $x - y = 36$.