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geniusboy [140]
3 years ago
13

Matt invests $1,669 in a saving account with a fixed annual interest rate of 2% compounded 12 times per year. How long will it t

ake for the account balance to reach 1,844.38?
Mathematics
1 answer:
vesna_86 [32]3 years ago
6 0

Answer:

It will take 4.84 years

Step-by-step explanation:

The initial amount that Matt invested was $1669. It means that principal is

P = 1669

It was compounded 12 times per year. So

n = 12

The rate at which the principal was compounded is 2%. So

r = 2/100 = 0.02

The formula for compound interest is

A = P(1+r/n)^nt

A = total amount in the account at the end of t years.

A = 1,844.38

Therefore

1,844.38 = 1669(1+0.02/12)^(12×t)

1,844.38/1669 = (1.0017)^(12t)

1.1051 = (1.0017)^(12t)

Taking log to base 10 of both sides, it becomes

Log 1.1051 = log 1.0017^(12t)

Log 1.1051 = 12tlog 1.0017

0.043 = 0.00074 × 12t

0.043 = 0.00888t

t = 0.043/0.00888

t = 4.84 years

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3 years ago
Brooklyn has a goal to save $8,000 to buy a new entertainment system. In order to meet that goal, she deposited $4,132.79 into a
densk [106]
To see how much interest she'll get after a quarter:

$4132.79 + ($4132.79 × 0.048) = $4331.16

After two quarters:
$4331.16 + ($4331.16 × 0.048) = $4359.06

You can keep going until eventually reaching $8000 then see how many quarters has passed. That's a lot of calculator work!

There's another way that uses less calculation, but more algebra. I call it the exponential formula method! There's this general formula for stuff that increases exponentially, like virus, population, and MONEY:

m= d {e}^{tc}

M is money, d is deposit, t is time taken, and c is just some unknown constant related to the interest rate. There's also the natural logarithm form of this equation, which will come in handy later:

ln( \frac{m}{d} ) = tc

Alright first we gotta find that constant c for this equation to be useful! Let's plug in stuff we know.

ln( \frac{4331.16}{4132.79} ) = (0.25)c

We know how much she'll have after one quarter (0.25 years), and we know how much she deposited initially.

After pressing some buttons on the calculator we'll find that c = 0.1875.

Great! Now we can use that formula to find how many years (t) it'll take to reach M=$8000. To save time I'm going to use the natural log form:

ln( \frac{8000}{4132.79} ) = t(0.1875)

That will give us t = 3.522 which means it'll take approximately 3.5 years for her deposit to reach $8000!
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3 years ago
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