"When you lose weight, where does it go?"
An interesting question, and one where most people haven't thought
about it, don't know, and really don't care. But when I was asked,
it set in motion a train of thought. Follow along, and see if it
makes sense.
Every breath you take ...
All the numbers here are approximate, and it would be an interesting
exercise to work out how you would verify them for yourself. Some
are easy, others less so. 
Assuming we are at rest, and we remain at rest, we take around ten
breaths a minute. There are 1440 minutes in a day, so about 14400
breaths per day, again, assuming we don't exert ourselves.
In each breath we take in about 800 ml or so, although it varies from
person to person. Again, it would be interesting to figure out how
to measure that with just stuff you can find around the house.
I repeat, this is for when you're at rest. It will all change
radically if you start exercising. 
So total volume of air breathed per day is about 11520 litres, or
11.5 $m^3.$
Of the air you breathe, about 20% is Oxygen, and when you breathe out
it's about 16%, or so I remember being told at one point. If that's
right, then roughly 4% of the air you breathe out is carbon dioxide.
That means you breathe out around 460 litres of CO${}_2$ per day,
and that's 0.46 cubic metres.
How much does that weigh? Good question.
Heavy ...
A lot of people are surprised by this ... how
would you check it? How would you measure it? 
Carbon dioxide weighs 0.001836 grams per cubic centimetre, which is
1.836 kilograms per cubic metre, so we have a total weight of around
845 grams.
(You should check my working so far.)
Now the atomic weight of carbon is 12, and of oxygen is 16, so the
atomic weight of one molecule of CO${}_2$ is 44. That means the
weight of carbon atoms we exhale in a day is 12/44 of that 845 grams,
or about 230 grams.
I should have done this calculation with glucose, not sucrose,
but you might like to try that for yourself and see what you get. 
The chemical formula for sucrose is $C_{12}H_{22}O_{11},$ so if we
assume that all the carbon we exhale comes from metabolising sucrose,
that means we need 230 grams worth of carbon. But the atomic weight
of sucrose is 12x12+22x1+11x16, which is 342, and the atomic weight
of the carbon in a sucrose molecule is 144. So the carbon is 144/342
of a sucrose molecule (which if you're interested simplifies to 8/19
(although it actually simplifies to that whether you're interested or
not!)), and that means we need 342/144 grams of sucrose to get each
gram of carbon.
So to get 230 grams of carbon we need (342/144)x230 grams of sucrose,
which is about 550 grams.
(You should check my working again.)
Energy ...
Depending on your source, you'll find that sucrose (which is just one
example of a sugar) is about 4 Calories (that's 4 kCals, the sort of
calories you count when dieting) per gram. So the number of (dietary)
Calories in 550 grams of sucrose is 2200.
Quoting from the NHS website[0]:
 An ideal daily intake of calories varies depending
on age, metabolism and levels of physical activity,
among other things.
 Generally, the recommended daily calorie intake is
2,000 calories a day for women and 2,500 for
men.
Interesting how our computed value for calories consumed based purely
on the carbon breathed out matches that so closely
So when you lose weight, where does it go?
The real situation is incredibly complex  don't for a moment
assume that this is everything. It's not, it's really not. but this
is a start on how we can think about these things. 
I haven't answered that directly, but the ideas are all here. The
nutrients in the food you eat are metabolised, and either get stored
as fat (or other energy stores), or broken down to provide energy,
and the byproducts are almost entirely water and carbon dioxide.
Broadly speaking, when you lose weight, you breathe it out.
But the amazing thing is that with some simply sums we can start to
analyse how this all works.
References
[0] https://www.nhs.uk/commonhealthquestions/foodanddiet/whatshouldmydailyintakeofcaloriesbe/
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