I don't really follow your example. Here's how I would think of it:
It sounds like the company (before the new investment) is valued at $20k. Let's say that the founders put in $20k of personal money into the company. And let's say the company has 20,000 shares issued, with each share valued at $1.
Generally the way the new investment would be described is that the new investor sets valuation on the company. From you example, it sounds like the new investor thinks the company is worth $500k, and wants to own 10% of it for $50k. In this example, before the investment the company is worth $450k, and after, it is worth $500k (including the new cash). The company presumably has increased in value from the original $20k to $450k thanks to your hard work. Congratulations.
So the original 20k shares are worth $22.50 each before the new investment, reflecting the company value of $450k. The company sells 2,222 shares at $22.50 each for a total investment of $50k. Now there are 22,222 shares outstanding, and the share price was $22.50.
There a lots of variations on this theme, depending on whether the value of the company is set before the investment or after; and whether the new shares have any different properties (voting rights, etc) than the old shares.
In your example, I'm not clear on what "decide to increase the company's equity by 10%" means, nor what the concept of a "reserve" is here.
Hope this example helps.