Math: Shopping Problem

Today I went to the mall. I was going to buy some body lotions and hand sanitizer at my favorite store, Bath and Body Works. Then, I was faced with a dilemma. I have two coupons but I cannot use them at the same time. I had to choose to use one of them.

Coupon A: 20% off your purchase.
Coupon B: $10 off when your purchase is $30 or higher.

Which one to choose?

This seems simple, but it's deeper than I thought. To be able to answer this, I simply revert younger self in school. Referring to the basic math. And the answer is.... depends on your purchase.

Purchase#1:
If the purchase is lower than $30, then definitely use Coupon A. You can't use Coupon B for anything less than $30 anyway.

Purchase#2:
If the purchase is equal or more than $30 and less than X amount, it will be more profitable to use Coupon B. Why? Because:
Coupon B: $10/$30 = 33% discount!
Coupon A: 20% x $30 = $6 discount.--> If you use Coupon A on a $30 purchase, you'll only get 20% discount.
Where I come from, 33% discount ($10) is better than 20% discount($6)! I think everywhere in the world, that principle is still valid.

So now here comes the serious question. What is the break even point? Until what point it'll be more profitable to use Coupon B? I mean... if your purchase is $100, it'll be better to use Coupon A, because:
Coupon A: 20% x $100 = $20 discount.
Coupon B: $10 discount of $100 purchase = 10%.
$20 discount (20% off) is better than $10 discount (10% off).

To be able to answer this, let's just calculate the X. The break even point will be the $10.
Coupon B: $10 = $30 x 33%
Coupon A: $10 = X x 20%. ---> X = $10/0.2. X = $50.

So, if your purchase is $50, then here's what happen:
Coupon A = 20% x $50 = $10.
Coupon B = $10 off $50 = $10/$50 = 20%.
No matter which coupon you use, it doesn't matter, it'll be the same discount, both dollar wise and percentage wise.

Purchase#3:
Using the basic principles above, for any purchase above $50, it'll be more profitable to use Coupon A. Why? Simple.
Coupon A = Let's say your purchase is $51. Then: 20% x $51 = $10.2 discount. 2 pennies still higher than 0 pennies.
Coupon B = $10/$51 = 19.6%. Which is 0.04% less than 20% that's offered from Coupon A.

Purchase#4 (my actual purchase):
My final shopping resulted in a $40 shopping. Which coupon you think I choose? By using simple logic and math, I did my calculation quickly before surrendered one of the coupon to the clerk:
Coupon A: 20% x $40 = $8 discount.
Coupon B: $10/$40 = 25% discount.
In this actual shopping experience, Coupon B wins because it was 25% discount (higher than 20%) and I literally saved $10 (instead of $8).

I really have to thank my math teachers.

Thank you to all math teachers everywhere you are!!!