TL;DR

A puzzle with insufficient information.

<< When Holly was twice the age Ivy was 15 years ago, Ivy was half the age Holly will be in 12 years time.

How many years ago was that? >>

OK, but why isn't the age of Holly and Ivy given?

TL;DR Summary: A puzzle with insufficient information.

This is from the front cover of Mathematical Conundrums, CRC Press, 2023.

<< When Holly was twice the age Ivy was 15 years ago, Ivy was half the age Holly will be in 12 years time.

How many years ago was that? >>

OK, but why isn't the age of Holly and Ivy given?

Can you set up the relevant Equations? Start with ##H## and ##I## as their current ages.

##
\begin{bmatrix}
1 & -2 \\
-\frac{1}{2} & 1
\end{bmatrix}

\begin{bmatrix}
H \\
I
\end{bmatrix}

=

\begin{bmatrix}
-30 \\
6
\end{bmatrix}
##

Nope, I can't. It's not insufficient information, it's contradictory information.

$$
\begin{bmatrix}
1 & -2 \\
-\frac{1}{2} & 1
\end{bmatrix}

\begin{bmatrix}
H \\
I
\end{bmatrix}

=

\begin{bmatrix}
-30 \\
6
\end{bmatrix}
$$

PS: Sorry, don't mean to derail the tread. I'll figure out LaTex later...

I have clearly made a hash of the problem...

In addition to @PeroK's suggestion to write ##H## and ##I## as their current ages I suggest defining ##Y## as "the number of years ago that was" and then re-reading the question carefully.

2(i+12) = h+12
holly's age (h) plus 12 years
should be equal to
double (ivy's age (i) plus 12 years)

"How many years ago was that?"

That doesn't make sense. It was no years ago.

The answer is zero.

h-15 = 2*(i-15)
holly's age (h) minus 15 years
should be equal to
double (ivy's age (i) minus 15 years)

2(i+12) = h+12
holly's age (h) plus 12 years
should be equal to
double (ivy's age (i) plus 12 years)

We probably shouldn't go too far down discussing the solution in this thread before the OP returns, so I'll just repeat my hint that you need to define ##Y##. PM me if you want to discuss further.

The question's wording is confusing but seems to mean this:

In (unknown) year -x, Holly was twice the age Ivy was in the year -15.
Also in year -x, Ivy was half the age Holly will be in year 12.

There are three unknowns (x, H and I) but only 2 equations. However, it turns out (with only simple algebra) that there is a unique value for x (but not so for H and I).

Why is x anything other than zero?
One equation deals with events 15 years ago and the other deals with events 12 years hence.
What is forcing x to not be "now"?
Some detail I'm missing here.

The initial word "When" can be taken to mean"15 years ago".
Or it can be taken to mean "At some unknown time, x years in the past".

With the latter interpretation (and using the conventions noted in Post #15) the phase:
"When Holly was twice the age Ivy was 15 years ago..."
generates the equation:
H-x = 2(I-15)

That's because x years ago Holly's age was H-x. And 15 years ago, Ivy's age was I - 15.

The other part of the question:
"...Ivy was half the age Holly will be in 12 years time."
is referring to Ivy's age x years ago, generating the equation:
I-x = (H+12)/2

The wording is (deliberately?) ambiguous. Consider the phrase:
"When Holly was twice the age Ivy was 15 years ago..."

The initial word "When" can be taken to mean"15 years ago".
Or it can be taken to mean "At some unknown time, x years in the past".

With the latter interpretation (and using the conventions noted in Post #15) the phase:
"When Holly was twice the age Ivy was 15 years ago..."
generates the equation:
H-x = 2(I-15)

That's because x years ago Holly's age was H-x. And 15 years ago, Ivy's age was I - 15.

The other part of the question:
"...Ivy was half the age Holly will be in 12 years time."
is referring to Ivy's age x years ago, generating the equation:
I-x = (H+12)/2

The question is, essentially: "what is the reference year?" AKA Ibix's Y variable in post 9.

It seems a trick question, or goose chase - the reference year is arbitrary. We might as well pick 15 years after the first scenario and 12 years before the second scenario, since that's how the scenario is described. i.e. now.

So what am I missing? Clearly, what I am missing is the very crux of the riddle, so to forestall further clutterment, I will step back and lurk.

I don't see how that addresses my confusion.

The question is, essentially: "what is the reference year?" AKA Ibix's Y variable in post 9.

It seems a trick question, or goose chase - the reference year is arbitrary. We might as well pick 15 years after the first scenario and 12 years before the second scenario, since that's how the scenario is described. i.e. now. So what am I missing?

When Holly was 20 she was twice the age that Ivy was 15 years ago. That seems clear enough.

It reminds me of the opening line of One Hundred Years of Solitude:

"Many years later, as he stood before the firing squad, Colonel Aureliano Buendia was to remember that distant time when his father took him to discover ice. "

For convenience, take the current year to be 0 and Holly and Ivy’s current ages to be H and I respectively.

The question's wording is confusing but seems to mean this:

In (unknown) year -x, Holly was twice the age Ivy was in the year -15.
Also in year -x, Ivy was half the age Holly will be in year 12.

There are three unknowns (x, H and I) but only 2 equations. However, it turns out (with only simple algebra) that there is a unique value for x (but not so for H and I).

It reminds me of the opening line of One Hundred Years of Solitude:

"Many years later, as he stood before the firing squad, Colonel Aureliano Buendia was to remember that distant time when his father took him to discover ice. "

"When Holly was twice the age Ivy was 15 years ago, Ivy was half the age Holly will be in 12 years time"

Taking the current year as 2025, the question can be re-expressed as:

x years ago, Holly's age was twice (what Ivy's age was in 2010)
and
also x years ago, Ivy's age was half (what Holly's age will be in 2037).
Find x.

"When Holly was twice the age Ivy was 15 years ago, Ivy was half the age Holly will be in 12 years time"

Taking the current year as 2025, the question can be re-expressed as:

x years ago, Holly's age was twice (what Ivy's age was in 2010)
and
also x years ago, Ivy's age was half (what Holly's age will be in 2037).
Find x.