Question

Mr. Jimenez deposited money into an account in which interest is compounded quarterly at a rate of 2.6%.

How much did he deposit if the total amount in his account after 4 years was $7160.06, and he made no other deposits or withdrawals?

Answer 1

In 4 years there are 4*4 = 16 quarters.
Since each quarter, his principle increases by 1.026 % after 16 quarters, he has
$$
\large{
P_0\times1.026^{16}=7,160.06
}
$$
Where his initial deposit was \(P_0\).
So,

$$
\large{
P_o=\cfrac{7,160.06}{1.026^{16}}
}
$$
Does this make sense?

Answer 2

A little confusing :P so what process am i suppose to go through to get the answer ?

Answer 3

We start with \(P_0\) dollars.

At the end of the 1st quarter we have \(1.026\times P_0\) dollars.

Are you with me?

Answer 4

okay yes

Answer 5

In the 2nd quarter we have this and more
$$
(1.026 \times P_0)\times 1.026
$$

Got that?

Answer 6

See where we are going?

Answer 7

yes

Answer 8

Ok, now. For ever quarter, we are multiplying by 1.026, because this represents the increase in the value of our previous amount. So after n quarters we have:
$$
\large
P_0\times1.026^n
$$

Right?

Answer 9

For example, after 2 quarters, the value from his initial investment is
$$
\large
P_0\times1.026^2
$$

Answer 10

Because this equals
$$
(1.026 \times P_0)\times 1.026
$$

As we discussed above.

Answer 11

would i do this for 12 quarters?

Answer 12

In 4 years there are 16 quarters?

Do you agree?

Answer 13

right! so i would do that 16 times

Answer 14

You would do this 16 times, yes.

Do it 16 times means
$$

\large{
P_0\times 1.026^{16}
}
$$
Because, n, the number of quarters is 16 in 4 years. We are compounding every quarter.

Answer 15

We know that value of our investment after 16 quarters, right? So this number we just developed must equal 7160.06

Answer 16

The only thing left for us to do is to find our initial investment \(P_0\).

Do you have this?

Answer 17

Are we in sync?

Answer 18

yes

Answer 19

k, so let's find \(P_0\). Can you help me?

Answer 20

would it be 4748.53

Answer 21

I don't know, what did you do? Tell me.

Answer 22

Did you use our work here?

Answer 23

Whatever this turns out to be is the answer...
$$
\large{
P_o=\cfrac{7,160.06}{1.026^{16}}
}
$$

Answer 24

yes thats what i did

Answer 25

lemme double check for you

Answer 26

Yes maam! You are correcto!

Answer 27

okay

Answer 28

great!