Why is my result not matching exactly with the solution of the problem?

Question

I am reading the following problem:

A farmer sells 10 tons of potatoes every year. He also grows enough seed potatoes for next year's planting. If his yield is exactly 20 times what he sows how many tons of potatoes does he need to plant to ensure a perpetual supply?

My approach:
Let $p$ be the seeds for regular potatoes.
Let $s$ be the seeds for seed potatoes.
Let $plant = p + s$ i.e. $plant$ is what the farmer sows.
We know that the yield is $20*plant = 20(p+s) = 20p + 20s $
We need to have $10$ tons of potatoes to sell and enough seed potatoes to be able to plant enough potatoes next year in order to have a perpetual supply.
Hence:
$20p = 10\space tons \equiv p = \frac{10\space tons}{20} \approx \frac{10000\space kg}{20} = 500\space kg$
Since we need enough seed potatoes to sow $500kg$ potatoes for next year we need:
$20s = 500\space kg \equiv s=\frac{500}{20} = 25 \space kg$ seed potatoes.
So total plant is $500 + 25 = 525kg$

So to review since the yield is $20x$ the sow we have:
$500\space kg *20 = 10000\space kg \approx 10 \space tons$ to sell
$25\space kg * 20 = 500\space kg$ to plant.
So it seems to check out.

The solution now is different and states:
Let $x$ be the weight of seed potatoes the farmer plants per year.
Then we know that:
$20x = 10 + x \equiv 19x = 10 \equiv x = \frac{10}{19} \space tons = 0.526 \space tons$

I see that my approach was needlessly convoluted as I spitted the types of potato seeds but it seems to me that the final result should match. I.e. my total is $500 + 25 = 525\space kg = 0.525 \space tons$ but it is not really the same as the solution's result ($0.526\space tons$).

So what are the errors in my reasoning since I should be getting $526\space kg$ and not $525kg$ as I have

Answer 1

Explanation of your method

I noticed why your answer doesn't work. Your answer involves a recursive procedure of growing seeds to replace the seeds that have been used, that is you reasoned as follows: I grew enough seeds to sell 10 tons, this is 1/2 ton. Now I need to grow enough for the 1/2 ton, this is 1/40 ton. But I need to grow enough for the 1/40 ton, this is 1/800 tons. You cannot stop because if you do, you will run out. So the actual calculation looks like this:

$$\frac 1 2+\frac 1{40}+\frac 1 {800}+...=\frac 1 2\left(1+\frac 1 {20}+\frac 1 {20^2}+...\right)\\ =\frac 1 2\sum_{i=0}^\infty(1/20)^i\text{ infinite geometric series}\\ =\frac 1 2\left(\frac 1{1-1/20}\right)\\ =\frac 1 2\cdot \frac{20}{19}\\ =\frac{10}{19}$$

Explanation of the given solution

The solution says $20p>10+p$ because you will output 20p tons of potatoes. You need this to be at least 10 tons plus p for next year (allowing you to plant p again next year).

> 10/19
[1] 0.5263158

Indeed,

$$\frac{10}{19}*20=10+\frac {10}{19}$$

so if you plant 10/19 tons, you will get 10 tons of potatoes as well as get 10/19 "back", in one fell swoop if I may add.

Answer 2

Your error is you didn't take into account the seed potatoes the farmer will have to plant next year. So $s$ is not $25$ kg, but a little larger.

That is, the farmer must plant enough to produce $10$ tons and plant again a quantity plant. So we have:

$$\mathrm{plant}=\frac{10\;\mathrm{tons}}{20}+\frac{\mathrm{plant}}{20}$$

Here, $\frac{\mathrm{plant}}{20}$ is what must be sowed to be able to plant again the quantity $plant$ next year.

$$\frac{19}{20}\mathrm{plant}=500\;\mathrm{kg}$$

$$\mathrm{plant}=\frac{20}{19}\times500\;\mathrm{kg}\simeq 526.3\;\mathrm{kg}$$