A price of a car is 200000 dollars. It goes on a sale twice. The price of it after the 2. sale is 149600 dollars. We know that both of the sales were between 10 and 20% discounts. What was the price of the car after the 1. discount?
EDIT: So, I left one detail out, the % discount is a whole number between 10% and 20%.
I tried guessing it, and I got one answer (maybe there are other answers), so the first discount was 12%, and the second discount was 15%. So the price after the first discount is 176000. But I still don't know how to get the answer without guessing.
Guessing and checking is actually a reasonable way to attach a problem like this. We can see right away (before the first guess) that the number of guesses required cannot be more than six guesses--one for each of the numbers $10$ through $15,$ representing the smaller discount. If you have a calculator it is easy to check each guess; even without a calculator it is not very difficult.
For a more methodical approach, we have $$ 200000\left(\frac{100-p}{100}\right)\left(\frac{100-q}{100}\right) = 149600, $$ where $p$ and $q$ are the two percentage discounts, $p$ and $q$ are both integers, $10\leq p\leq 20,$ and $10\leq q\leq 20.$
Simplifying the equation, we have $200000/(100\times100)=20,$ and dividing by $20$ on both sides we get $$ (100 - p)(100 - q) = 7480 = 2^3\times 5\times 11\times 17. $$ Since $100 - p$ and $100 - q$ are integers, and their product is divisible by the prime number $17,$ one of those two numbers must be a multiple of $17.$
Suppose it is $100 - p$ that is the multiple of $17.$ But since $10\leq p\leq 20,$ we know that $80\leq 100 - p \leq 90.$ The only multiple of $17$ that is in the range $[80,90]$ is $5 \times 17 = 85.$ (How do we know? We could just try a few multiples of $17$ --that is, we could guess-- or we could find the integer part of $80/17$ or $90/17$ first in order to find what our first "guess" should be. Since $90-80 < 17$ there can be at most one multiple of $17$ in range.) This implies that $p = 15.$
So now we have $$ 85(100 - q) = 2^3\times 5\times 11\times 17 = 2^3\times 11\times 85 $$ and therefore $$ 100 - q = 2^3\times 11 = 88, $$ so $q = 12.$
On the other hand if $100 - p$ is not divisible by $17,$ then $100 - q$ must be divisible by $17$ and we find that $q=15$ and $p=12$ using the same reasoning as above.
Tallying up the work we've done, we did a bunch of algebraic manipulation of the equation for the two discounts, we found the prime factorization of a four-digit number (which fortunately was guaranteed to have some small prime factors if there was to be any solution), we used the largest prime factor ($17$) to find all possible values of a number between $80$ and $90$ that is a factor of our four-digit number, and then we found the other factor and subtracted each factor from $100$ to get the discounts.
I'm not sure this is is a better approach then a simple guess-and-check trying different values of $p,$ but for a problem with different numbers there might be some advantage to applying the more complicated method.
Notice that $0.88 \times 0.85 = 0.85 \times 0.88,$ so there are two possible ways to get to the second sale price: first discount by $12\%$, then by $15\%,$ or first discount by $15\%$, then by $12\%.$ If we had some other way to know which discount was applied first (for example, if we knew the first discount was smaller), we could say for sure whether the first sale price was $200000\times 0.88 = 176000$ or $200000\times 0.85 = 170000.$