Q: $1068.68 = $1000[1 + (r/12)]^12; A: r = 0.06: How?

[tscott036]

the question is

$1068.68 = $1000[1+(r/12)]^12

The answer is r = .06 but how do I do it!!!!????????

 

[Trebor]

Firstly, divide both sides by $1000:

1.06868=[1+(r/12)]^12

Then cancel out the power (^12) by taking the 12th root of both sides (^(1/12)):

1.06868^(1/12)=1.0055507...=1+(r/12)

Take away 1 from both sides:

0.0055507...=r/12

Multiply both sides by 12:

r=0.66608...=0.667(3.s.f.)

 

[arthur ohlsten]

1068.68 = 1000[1+r/12]^12

take logs of each side and solve for r.
divide both sides by 1000
1.06868=[1+r/12]^12
take log of each side

log[1.0688]= 12 log[1+r/12]
.2279=12 log[1+r/12]
.001899=log[1+r/12]
raise both sides to power of 10
1.0044= 1+r/12
12[.0044]=r
r=.0528
r=5.3%

please check math and calculate with more accuracy than I did

Arthur

1068.68=1000[1+r/12]^1/2
divide both sides by 1000
1.06868=[1+r/12]^1/2
square both sides
1.1449=1+r/12
.1449=r/12
multiply both sides by 12
1.74

Arthur
r=

 

[soroban]

Re: I feel like an idiot!!!!

Hello, tscott036!

I'll take baby-steps.
. . And I got a different answer . . .

Solve for \(\displaystyle r:\;\$1068.68 \:= \:\$1000\left(1\,+\,\frac{r}{12}\right)^{12}\)

The answer is: \(\displaystyle r\,=\,0.06\) but how do I do it?

We have: \(\displaystyle \L\:1000\left(1\,+\,\frac{r}{12}\right)^{12}\:=\:1068.68\)

Divide by 1000: \(\displaystyle \L\;\left(1\,+\,\frac{r}{12}\right)^{12}\:=\:1.06868\)

Raise both sides to the \(\displaystyle \frac{1}{12}\) power: \(\displaystyle \L\;1\,+\,\frac{r}{12}\:=\1.06868)^{\frac{1}{12}} \:=\:1.005550701\)

Subtract \(\displaystyle 1\) from both sides: \(\displaystyle \L\:\frac{r}{12}\:=\:0.005550702\)

Multiply both sides by \(\displaystyle 12:\;\;\L r\;=\;0.06608426\)


Therefore: \(\displaystyle \L \:r\:\approx\:0.066\) . . . which is closer to \(\displaystyle 0.07\)


Too fast for me, trebor!

 

[tscott036]

thanks guys, ya i get .666 too

 

[Denis]

REMEMBER:
if a^x = b, then a = b^(1/x)

 

[arthur ohlsten]

I am curious. How did you take the 12th root of the number?
Arthur

 

[Trebor]

You raise it to the power of 1/12. This goes for all nth roots; just raise it to the power of 1/n.

 

[tkhunny]

Of course, any modern calculator ith an exponent key or a logarithm key is more than sufficient. Consider this an historical reference. Something fun to do on a Saturday morning.

I used to utilize the binary expansion of the exponent and the square root button on my super cheap calculator from 20 years ago. I was taking exams that did not allow higher function calculators at the time, so I devised something with what they would allow.

1/12 = 1/16 + 1/64 + 1/256 + 1/1024 + 1/4096 + 1/16384 + 1/65536 + ...

So

\(\displaystyle a^{1/12}\;=\;a^{1/16}a^{1/64}a^{1/256}a^{1/1024}a^{1/4096}a^{1/16384}a^{1/65536}...\)

You work the square root key and the internal memory quite a bit, but it can be done in a pinch. Also, we're only at four decimal places after 65536 on this one.

 

[Mrspi]

Arthur, I've got a cheap scientific calculator (one a high school student discarded at the end of the semester....probably cost $15 ten years ago). It's a TI-35....and it will do any kind of powers.....

To find n^(1/12) power, you would use the "y^x" key:

n (y^x) (1/12) =

Of course, all these things can be done using logs, but for probably $10 now, you don't have to go through all the grief.

 

[Denis]

Arthur, I've got a cheap scientific calculator (one a high school student discarded at the end of the semester.....

Oh Mrspi....REEEEEEEEALLY :shock:

 

[arthur ohlsten]

I thought the student had gotten the 12th root without a calculator.
He would have taken logs, or taken the cube root and the square root twice.
I asked because I didn't think the schools taught how to take cube roots algebraicly anymore.
I never had a student who knew how to get any root above 2 without referring to logs or calculators.
Arthur

 

[tkhunny]

My slide rule still works, too, but it doesn't do symbolic calculus.

 

[arthur ohlsten]

finding the square root is a algebraic process, as is finding the cube root.
You do not need a calculator, or slide rule, to calculate the roots to any accuracy you need. It can be done solely with paper and pencil.
Arthur

 

[Denis]

You do not need a calculator, or slide rule, to calculate the roots to any accuracy you need. It can be done solely with paper and pencil.

Ok, but why, Arthur?
Like, you don't need a car to travel from Toronto to Detroit; you can walk :shock:

 

[arthur ohlsten]

Of course there could be times when you musy know what you are doing. An example: you must program a computer to find the third root of a number, either a base 10 or say base 3 number.
Or suppose the square root of a binary number
Students shouldn't rely on calculators exclusively.
Arthur