To slightly challenge the framing of your question, I'd like to suggest that it will be much easier if you start by deciding what percentage of the value of the hoard is in coins of each type.
For example, let's say that 75% of the value is in gold, 20% in silver and 5% in copper, and the total value of the hoard is 800 gp.
- 75% of 800 gp is 600 gp, so your hoard contains 600 gold coins.
- 20% of 800 gp is 160 gp, so your hoard contains 160 × 10 = 1600 silver coins.
- 5% of 800 gp is 40 gp, so your hoard contains 40 × 200 = 8000 copper coins.
With reasonable round numbers like this, you can even do the math in your head. (I did, above.) And your players will probably appreciate the round numbers too.
If that still feels too unrealistic to you, just arbitrarily fudge the numbers a bit after you've done the math and say that the hoard contains, say, 589 gold coins, 1712 silver coins and 8475 copper coins.
(As long as you only fudge the numbers by less than 10%, and make sure not to fudge them all in the same direction, the total value of the hoard won't change too much. The randomly fudged numbers above, for example, sum to 589 + 171.2 + (84.75 / 2) = 802.575 gp, which is actually surprisingly close to 800 gp.)
By the way, note how the copper coins in the example above ended up being almost 80% of the total number of coins, despite having only 5% of the total value of the hoard.
That's a thing that naturally happens when you have a mixture of coins of vastly different values: a single gold coin is worth 200 coppers, so if more than about 0.5% of the total value is in copper, there will necessarily be more copper coins than gold coins in the hoard.
It's up to you to decide if that's what you want, and that decision is likely to depend on who set up the hoard and why.
In general, a useful rule of thumb is that high-value coins like gold are the best for storing lots of wealth in a small space, while low-value coins like copper are probably the most commonly circulated and easiest to come by.
So something like a pirate's buried treasure chest, or a rich merchant's hidden emergency stash, is likely to be mostly gold and maybe some silver to fill up the space. Meanwhile, a store of money in active use, like a lord's treasury or a trader's coin purse, will likely have quite a few copper coins in it, because they'll keep getting more in trade as fast as they can trade them away.
And as for a dragon's hoard or a bandit camp's loot, who knows? Did they indiscriminately gather all the coins and other loot they could get (implying lots of silver and copper), or did they just pick the most valuable and easy-to-carry stuff and leave the rest behind (implying little if any copper at all)? And, come to think of it, how exactly does a dragon transport coins to their lair anyway?
Anyway, if you've already decided on specific percentages of coins by number, as in your question, you can also convert between value percentages and coin percentages.
The easiest way to do that is a two-step process. For instance, to convert from value percentages to coin percentages:
First multiply each percentage by the number of the corresponding coins that are worth one gold piece (i.e. 1 for gold, 10 for silver, 200 for copper).
This first step will give you the correct proportions of the value in each type of coin, but these proportions will generally not sum up to 100%.
Thus, the second step is to add the scaled percentages from the previous step together, divide each scaled percentage by this sum and multiply it by 100%.
So for example, in my example hoard above, gold, silver and copper coins appear in proportions of 75 gold coins to 20 × 10 = 200 silver coins to 5 × 200 = 1000 copper coins. These proportions sum up to 75 + 200 + 1000 = 1275, so we need to divide them by 12.75 to obtain the normalized percentages of 75 / 12.75 ≈ 5.88% gold coins, 200 / 12.75 ≈ 15.69% silver coins and 1000 / 12.75 ≈ 78.43% copper coins.
(Of course, we can round those to, say, 5% gold coins, 15% silver coins and 80% copper coins for convenience.)
Conversely, to convert from coin percentages to value percentages you can use the same two steps, but divide instead of multiplying in the first step.
(Or, equivalently, you can instead multiply each percentage by the number of copper pieces the corresponding coin is worth, i.e. 200 for gold, 20 for silver and 1 for copper. This will give numbers that are 200 times larger than what you'd get using the division method, but their relative proportions will be the same, and the second step will normalize them to the same percentages anyway.)
For instance, for your original example hoard of 13% gold coins, 60% silver coins, and 27% copper coins, we find that the value is distributed in proportions of 13 gp in gold to 60 / 10 = 6 gp in silver to 27 / 200 = 0.135 gp in copper.
Dividing those proportions by their sum of 19.135 gp and multiplying by 100%, we find that the value of the hoard is made up of approximately 67.94% gold, 31.36% silver and 0.7055% copper.
You can then multiply these percentages with any total value of the hoard in gp and convert the resulting share of the value to the appropriate type of coins.
For example, with a total value of 873 gp, this means that your hoard contains:
- 873 × 67.94% / 100% ≈ 593 gp in gold coins,
- 873 × 31.36% / 100% ≈ 273.8 gp silver, for a total of 2738 silver coins, and
- 873 × 0.7055% / 100% ≈ 6.16 gp in copper, for a total of 1232 copper coins.
Also note that, as seen in the examples above, the number of copper coins in a hoard mixed with silver and gold typically makes very little difference to the total value.
So, as long as you don't insist on getting the total value exactly right down to the last cp, one way to simplify the math is to first figure out the approximate number of gold and silver coins in the hoard and then just add whatever number of copper coins feels appropriate to you. As long as it's not more than, say, 20 times the number of gold coins, it won't change the value of the hoard significantly.
(And if the number of copper coins in the hoard is more than 20 times the number of gold coins, that means there's either very little gold or a huge amount of copper in the hoard. In the latter case, transporting all that heavy copper out of whatever godforsaken dungeon it was found it in might become a valid concern — unless of course your players just decide to leave it behind and only take the more valuable silver and gold!)