[quote="FightToSurvive"]How much more interest will maria receive if she invests 1000$ for one year at x % annual interest, compounded semianually, than if she invest 1000$ for one year at x percent annual interest, compounded annually?

a> 5x
b> 10x
c> x^2/20
d> x^2/40
e>10x+ (x^2/40)

I dont from what source these questions are from.
See from simplification sake, take x =10%
(i) investing 1000$ for one year at x % annual interest, compounded semiannually
1000*1.05*1.05=1102.5
Interest amount = 1102.5-1000=102.5
(ii)investing 1000$ for one year at x percent annual interest, compounded annually
1000*1.1=1100
Interest amount = 1100-1000=100
Difference in two cases = 102.5-100 = 2.5

Putting x=10 in the above option , D gives you the answer

Just take x = 200%
You can solve it in a minute!

Got this wrong on an official practice test. Resolving here for retention:

Method 1 - Plug-in values for x (quickest method)

x= 20

Compounded
New balance after compounding is equal to 1000(1+20/200)^2
= 1000(11/10)^2
= 1000(121/100)
= 1210

Simple interest
New balance with interest 1000(1+20/100)^1
= 1000(6/5)
= 1200

'How much more interest' = 1210 - 1200 = 10 more

Plug-in 20 to each answer choice to get 10
A - 5(20) = 200 --> Wrong
B - 10(20) = 200 --> wrong
C - (20)^2/ 20 = 20 --> wrong
D - (20)^2/40 = 10 --> correct
E - (10(2) + (20)^2/40) = 30 --> wrong


Method 2 - Algebraic, but since we have variables, breakdown the compounding periods instead of calculating it all in one go)


Compounding:
1000(1+x/100/2)^1
= 1000 (1 + x/200)^1
= 1000 + 5x
New balance after first compounding period = 1000 + 5x

Second compounding period
Principle *(Interest rate/#compound periods + 1)^compounding period
(1000 + 5x)(1 + x/200)^1
= 1000 + 5x + 5x + (5x^2)/200
= 1000 + 10x + x^2/40
Now we need to eliminate the principle to get the interest portion
= 1000 + 10x + x^2/40 - 1000
= 10x + x^2/40

Simple interest amount
1000(1+ x/100)^1
= 1000 + 10x
Interest portion = 1000+10x - 1000
= 10x

How much more interest?
= 10x + x^2/40 - 10x
= x^2/40
_________________