Question

Autumn is depositing $575 at the beginning of every six months into an account with an interest rate of 3.67%, compounded semiannually. She wants to renovate her kitchen in seven years. How much will Autumn have in the account at the time of the renovation?

Answer 1

@mathmate @just_one_last_goodbye @Legends @leonardo0430 @DarkMoonZ

Answer 2

Are you familiar with the compound interest formula?

Answer 3

yes

Answer 4

Can you post it?

Answer 5

We will work together to get the answer.

Answer 6

Cool

Answer 7

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

Answer 8

Excellent! Do you know what each symbol represents?

Answer 9

P = 575
r = 3.67% or 0.0367
n = 2
t = 7

Answer 10

Exactly!

Answer 11

575(1+0.0367/2)^14 = $741.70

This, however, is not an answer choice.

Answer 12

Oh, sorry!
I have asked you for the compound interest formula, but since she puts in money every 6 months, the total will be much bigger.
We need to use the amortization formula. Would you have that handy?

Answer 13

No... :(

Answer 14

Let me figure it out and post it.
Give me a couple of minutes.

Answer 15

ok, here it is.
Let R=1+r/n=1+0.0367/2
n=14 as before

Answer 16

Let P=payment for every 6 months, and
A=amount at the end of 7 years.

Since she puts money in the bank at the beginning of each 6-month period, so the amount she gets a the total amount is:
\(A = PR + PR^2 + PR^3.... + PR^n\)
ok so far?

Answer 17

@MafHater ?

Answer 18

Ok, so...

Answer 19

P = 575

Answer 20

What is R?

Answer 21

interest?

Answer 22

Remember we know everything on the right hand side, so in principle, we can calculate the answer.

Answer 23

R=1+r/n = 1+0.0367/2=1.01835 for each period of six months.

Answer 24

PR means the amount she gets from the LAST payment
PR^14 means the amount she gets from the FIRST payment.

Answer 25

Because the first payment has stayed in the account for 14 periods.

Answer 26

Wow, I hate this. LOL

Answer 27

In math, try to understand, and not memorize. Once you understand, everything clicks.

Answer 28

Okay, would you mind giving me the whole equation? I will then solve it.

Answer 29

BTW, the difference between evaluate and solve:
If and when I give you the whole formula, you will evaluate it, because you know everything on the right hand side.
When you have an equation, you solve for the unknown that is somewhere in the equation, not necessary only on the left-hand side.

Answer 30

Going back to:
\(A = PR + PR^2 + PR^3.... + PR^n\)
we factorize out the common factor:
\(A = PR(1+R + R^2 + R^3.... + PR^{n-1})\)

Answer 31

Yes, correct. I have about four of these types of problems. So, if you give me the whole equation for this question (with the variables plugged in), I will be able to evaluate and solve the remaining three.

Answer 32

596.10 + 596.10^2 + 596.10^3 .... etc?

Answer 33

and use an identity that simplifies to:
\(\huge A = PR\frac{R^n-1}{R-1}\)
Note that this formula is different from the ones you have used/learned because she puts in he money at the BEGINNING of the period.

Answer 34

Yes, you could do it the other way to check the answer as well! Good thought!

Answer 35

SO...

A = 596.10 * 21.10^14 - 1/21.10 - 1

Answer 36

Your other way is not quite right, you need to do
585.551+596.294+607.238+....741.697
Make sure you calculate using at least 3 digits after the decimal to avoid rounding errors.

Answer 37

Be careful with parentheses:
\(\huge A = 575*1.01835 \frac{(1.01835^{14} - 1)}{(1.01835 - 1)}\)

Answer 38

How did you get 1.01835?

Answer 39

R=1+r/n=1+0.0365/2=1.01835
That is the equivalent cumulative rate of interest for every compounding period of six-months, including the principal.

Answer 40

It's the same as the (1+r/n) in the compound interest formula, but makes the formula simpler.

Answer 41

Ahhh.

Answer 42

What about this one?

The economy today grows at a rate of 2.2% semiannually. Ayden has an annuity that pays $1,655 at the beginning of every six month period. What is the value of Ayden’s annuity if he receives it in a lump sum now instead of over the next 18 years?

Answer 43

Did you get the previous one, what did you get?

Answer 44

9251.03

Answer 45

Yes, that's what I got too! excellent.

Answer 46

For this annuity problem, do you have the annuity formula? See if you can find it.

Answer 47

PV of the annuity = 1655*(1 - 1.022^-36) / .022

Answer 48

what would be the answer?

Answer 49

40860.15833, but this is not an answer choice.

Answer 50

P = PMT [(1 - (1 / (1 + r)n)) / r]

Where:

P = The present value of the annuity stream to be paid in the future

PMT = The amount of each annuity payment

r = The interest rate

n = The number of periods over which payments are to be made

Answer 51

That looks more like it. It's not the same as your previous.

Answer 52

Ahh

Answer 53

Hold on.

Answer 54

wait, you used 1.022 instead of 1.011!

Answer 55

I take it back, .022 is semi-annual interest.

Answer 56

I'll do it myself and get back to you.
Gimme a minute.

Answer 57

It a bit similar to the previous problem because it's paid in advance (at the beginning of the period), so the future value is
FV=1655*1.022(1.022^36-1)/0.022=91407.79
and convert it to PV
PV=FV/(1.022^36)=41759.08
@mathater