When will amount A be double then B?

[kokky]

Hi, I know how to solve this in my own way but it takes too long. Need the way to solve this quickly, formula or something. Sorry for bad English.

Person A deposited 500$ and has interest rate yearly 18% that is compounding monthly, and person B deposited 1000$ and has interest rate yearly 9% that is compounding quarterly. When will amount of person A be double in size then amount of person B?

 

[stapel]

HPerson A deposited 500$ and has interest rate yearly 18% that is compounding monthly, and person B deposited 1000$ and has interest rate yearly 9% that is compounding quarterly. When will amount of person A be double in size then amount of person B?

Need the way to solve this quickly, formula or something.

How about plugging into the compound-interest formula?

 

[kokky]

But that is too slow, I would have to calculate it for both persons and many different years so it would take at least 10-15 mins.

On the exam I will have 3 minutes per question so I have to solve this quick. There must be formula to do that, perhaps with logarithms or something.

 

[kokky]

Yes, when will A amount become twice as big as B amount. Calculator allowed.

So perhaps it would be something like this:

500 * 1,18(Nth potention) = 2 * 1000 * 1,09(Nth potention) and then calculate N (how would I do that)?

And why is there monthly/quarterly compounding difference? How to put that into formula?

 

[HallsofIvy]

The formula for compound interest is this: starting from amount A, if the interest each compounding period is r= 100r%, then the amount after n compounding periods is \(\displaystyle A(1+ r)^n\). In the first case, the interest is 18% per year and the compounding is monthly. There are 12 months in a year so the interest "each compounding period", that is, each month, is \(\displaystyle \frac{18}{12}= 1.5%\) so r= 0.015. After n months that $500 will have grown to \(\displaystyle $500(1.015)^n\). In the second case the interest is 9% per year and the compounding period is a quarter year of 3 months. The interest per quarter is \(\displaystyle \frac{9}{4}= 2.25%\) so r= 0.0225. After n months here will have been \(\displaystyle \frac{n}{3}\) quarters so $1000 will have grown to \(\displaystyle $1000(1.0225)^{n/3}\).

The amount A has will be twice the amount B has when \(\displaystyle 500(1.015)^n= 2000(1.0225)^{n/3}\).

Solve that equation for n.n You will need to use logarithms to do that.