More Interests word problems

[Guest]

Jessica's grandmother has given her the sum of $12 000 to pay for her college education. Jessica expercts to start college four years from now when, she estimates the cost will total $17 000. At what interest rate, compounded quarterly, does she invest her money?

I did this: A=P(1+i)^n
17000=12000(1+i)^16 but I don't tihk it's right..and if it is, how do i isolate i?


Tim is planning his post secondary education. He estimates he will need $21 000, 3 years from now. He has saved $8000 that he is investing in a fund that pays 8.5%, compounded quarterly. His aunt has promised to add $7000 to his fund when some of her investments come due in 18 months. Will Tim have enough money?

I got this: A= P(1+i)^n
= 8000 (1+ .085/4)^12
= 10 296.15 + 7000
= 17296.15
but the answer is suppose to be $18 237.43, so I don't know what I did wrong. Help!


Lastly for now,
Spongebob and Patrick Star are going to open their own restaurant in three years time after they have completed chef training. They need
$12 500 for the opening. SPongebob has found an investment fund that pays 8.1% interest, compouned anually. Patrick Star has found one that only pays 7.95%, but is compounded weekly. If they wish to put aside the necessary money today, which is better investment and by how much?

I don't get how to do this question at all. -_-'


Thanks for all the help,
Anna

 

[soroban]

Hello, anna!

Jessica's grandmother has given her the sum of $12 000 to pay for her college education.
Jessica expercts to start college four years from now when she estimates the cost will total $17 000.
At what interest rate, compounded quarterly, does she invest her money?

I did this: A = P(1 + i)ⁿ
17000 = 12000(1 + i)¹⁶

but I don't think it's right . . . . It's correct!
and if it is, how do i isolate i?

You need to be familiar with logarithms and exponential function.

We have: . . 12000(1 + i)¹⁶ . = . 17000

. . . . . . . . . . . . . . . (1 + i)¹⁶ . = . 17/12

Take logs: . . . . log(1 + i)¹⁶ . = . log(17/12)

and we have: . 16log(1 + i) . = . log(17/12) . = . 0.151267675

Divide by 16: . . . .log(1 + i) . = . 0.00945423

Exponentiate: . . . . . . .1 + i . = . 10^0.00945423 . = . 1.02207846

And we have: . . . . . . . . . . i . = . 0.02207846

Hence, the quarterly interest rate must be about 2.2%

. . . Therefore, the <u>annual</u> rate will be 8.8%

 

[Guest]

What does Log do on the calculator? I havn't learned Log yet

 

[soroban]

Hello, anna"!

Tim is planning his post secondary education.
He estimates he will need $21 000, 3 years from now.
He has saved $8000 that he is investing in a fund that pays 8.5%, compounded quarterly.
His aunt has promised to add $7000 to his fund in 18 months.
Will Tim have enough money?

I got this: A = P(1 + i)^n

A = 8000(1 + 0.085/4)^12 = 10 296.15 + 7000 ? = 17296.15

but the answer is suppose to be $18 237.43
I don't know what I did wrong.

You forgot that the aunt's money will also be earning interest.

Her $7000 will become: . A .= .7000(1 + 0.985/4)^6 .= .$7941.28
. . . (Her money will earn interest during the last 18 months, or 6 quarters.)

Together with Tim's $10,296.15, you'll get their total.

 

[soroban]

Hello, anna!

Spongebob and Patrick Star are going to open their own restaurant in three years time.
They need $12 500 for the opening.
Spongebob has found an investment fund that pays 8.1% interest, compouned annually.
Patrick Star has found one that only pays 7.95%, but is compounded weekly.
If they wish to put aside the necessary money today, which is better investment and by how much?

They will put aside P dollars now.

Spongebob's plan: . P(1 + 0.081)³ . = . 12500
. . . Solve for P: . P .= .$9895.39

Patrick Star's plan: . P(1 + 0.0795/52)¹⁵⁶ . = . 12500
. . . Solve for P: . P .= .$9849.40

Patrick Star's plan is "better" . . . it requires $45.99 less right now.

[But maybe Spongebob's bank gives away free calendars . . . ]

 

[Guest]

How do you use Log?? I got to do it on my calculator, except It said Error..how do you put in "i" in the calculator?? I used the varible A instead but it ended up saying syntax error on the calculator

 

[Denis]

What does Log do on the calculator? I havn't learned Log yet

(1 + i)^16 = 17/12

Soroban used logs to solve that; you can solve without logs:

remember: if a^b = c, then a = c^(1/b)

so: 1 + i = (17/12)^(1/16)
1 + i = 1.022~
i = 1.022~ - 1 = .022~ : capish?

AND the annual rate is not .022 * 4;
it is 1.022^4 - 1 = 1.09098~ - 1 = .09~