how long will take for $1200 to amount to $1800 at an annual rate of 6% compounded quarterly?

Answer 1

At the end of 7th year $1200.00 would be $1820.66 @ 6% compounded quarterly.

Answer 2

When they compound interest in non-annual periods, instead of giving you 6% at the end of every year, they give you a fraction of 6% (in this 6%/4) 4 times a year. Thus, 4 times a year, you add 1.5% of what's in your account to your account total. In other words, you take 1 (which stands for your account total) and add 1.5% of 1 (or .015). Since your account total isn't actually $1 (hopefully), you then multiply by your account total. So, each quarter, you multiply your account total by 1.015 to get your new total.
If you keep your money in the account for 1 quarter, your total is $1200*1.015 = $1218
Another quarter will give you
$1230*1.025 = $1200 * (1.015) * (1.015) = $1236.27.
Hopefully, you noticed that instead of multiplying by 1.015 over and over again, you can multiply by a power of 1.015. For instance, in the second equation (after two quarters of compound interest), your balance is really just $1200*(1.015), or (your principle)*(1+your rate)^(the number of compounding periods), which mathematicians write as p*(1+r)^t=B (your balance after time t).
Here, you've been given p = $1200, 1+r = 1.015, and B = 1800.
Now, you just have to solve for t.
1200 * (1.015)^t = 1600.
(1.015)^t = 1800 / 1200 = 1.5.
This is an exponential equation. In order to solve it, you need to know how to manipulate logarithms. If a^x = b, we write x = log_a(b) (the a is normally a subscript). In the equation above, t = log_1.015 (1.5).
Your calculator probably has a log_10 button and and log_e button (written as ln for "natural logarithm"). Unfortunately, it doesn't have a log_1.015 button. However, there's a neat rule we can use that says log_a(b) = ln(b) / ln(a). (This also works if you replace ln with log_10, or any other logarithm.)
From t = log_1.015 (1.5), we get t = ln (1.5) / ln (1.015).
You can use your calculator to figure out that ln (1.5) / ln (1.015) = 27.233236695384091569073584523324.
Now, your interest is compounded quarterly, and these units are in quarters. However, the new amount doesn't go into your account until the end (or beginning) of each quarter. So your time can't be a fraction of a quarter. Here, we always round up (even if its closer to the lower one), because you won't have 1800 dollars or more until after the 28th quarter (in this case).
Obviously 28 quarters = 7 years, and that's your answer.

Answer 3

so ,each quarter of a year the increase is 1.5 °, this means that each quarter the total amount is 1.015 that of the precedent

you want to know then the total relative to the initial amont is

1800/1200 =1.5

there are different manners ; I show one

write 1.5 = (1.015)^x

you use the ln and ln 1.5 = x ln1.05

0.405= x*0.01489

and x = 27.2 quarters or 6 years and 3 quarters