Understanding is key [HELP]

Vraces:

I need to understand why for example
[A radioactive kalashnikov gets less radioactive every year. After 60 years only 25% of the radio-activeness remains. What is the yearly reducing of radio-activeness in percent?]

I'm wondering. Since(Based on my assumption) 0.75^(1/60) should be in english: 75% of of the radioactiveness in one year. Since 75% was removed during 60 years shouldn't 0.75^(1/60) be the answer?[1 - 0.25^(1/60) = 0.75^(1/60)? ] And also. Why should it be ^(1/60) and not the percentage divided by 60?What does ^(1/60) actually mean? What happens when it calculates? I'm aware it multiples itself the times stated. For example: 3^3 = 27 ; 3^3(1/3) = 9 I must understand these things. It's good to know how to solve something, but if you don't know the meaning of it you can't solve another thing. I'm going to guess only Melody could explain this, but others are also welcome. IF you actually know the significance. Do not guess while trying to explain as it would only confuse me. And by the way, I may have confused you a little bit with what I'm trying to say. Good luck and thanks.

Hi Vraces,
Firstly I am really glad that you tried to explain. I wish everybody did that.
Your initial problem is very similar to the one you had last time.
Your formula is
S=P(1-r) ⁿ
P is present value (or initial value in this case) = P
S is the letter that is usually used for future value = 0.25P
It is -r because the value is getting smaller all the time =unknown
n is number of years = 60

oh I am sorry Vraces I read your response a little wrong. please forgive me, it is after midnight here and I am very tired. i just thought you would appreciate a quick answer.
The 75% that you found is true but it is not relevant to the answer. All that is important is that after 60 years you have 0.25 of your original value.

just use the standard depreciation formula
0.25P=P(1-r) ⁶⁰ now the p's cancel out
0.25=(1-r) ⁶⁰

0.25 ^1/60=1-r [add r to both sides and subtract 0.25 ^1/60 from both sides.]
r = 1 - 0.25 ^1/60

this time, the 1 is not in a bracket.
on you calc it is
1- 0.25^(1/60) = 0.022840031 => 2.28% (correct to 3 significant figures)
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Now let me see if i can help you with some of your other issues.

3^2=9 you know that
9^(1/2) means square root of 9 so the answer is 3
so are power of 2 and a power of 1/2 cancel each other out.
sqrt(3 ²)=sqrt(9)=3
written in a different way, this is
(3 ²) ^1/2 = 9 ^1/2 = 3

2 ⁵ = 2x2x2x2x2 = 32
therefore
32 ^1/5 = 2
because
32 ^1/5 means, what times be itself 5 times = 32.

Think about it. If you have more questions post again. You are right, it is important that you understand this.
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I should tell you that we are now fortunate enough to have at least 2 other regular people on the forum who could have helped you with this.
They are reinout-g and I like Serena. I think that some of the others could have helped you too.
But I am glad that I did. I want to keep answering questions for the people I know, especially when I know that they are trying hard to learn.

Ok, I got most of that.
But, I'm still wondering why is 0.75 irrelevant? And how do you [^] a number on this page?

Vraces:

I need to understand why for example
[A radioactive kalashnikov gets less radioactive every year. After 60 years only 25% of the radio-activeness remains. What is the yearly reducing of radio-activeness in percent?]

I'm wondering. Since(Based on my assumption) 0.75^(1/60) should be in english: 75% of of the radioactiveness in one year. Since 75% was removed during 60 years shouldn't 0.75^(1/60) be the answer?[1 - 0.25^(1/60) = 0.75^(1/60)? ] And also. Why should it be ^(1/60) and not the percentage divided by 60?What does ^(1/60) actually mean? What happens when it calculates? I'm aware it multiples itself the times stated. For example: 3^3 = 27 ; 3^3(1/3) = 9 I must understand these things. It's good to know how to solve something, but if you don't know the meaning of it you can't solve another thing. I'm going to guess only Melody could explain this, but others are also welcome. IF you actually know the significance. Do not guess while trying to explain as it would only confuse me. And by the way, I may have confused you a little bit with what I'm trying to say. Good luck and thanks.

Dear Vraces,

I understand your confusion, and since Melody has just called it a day, I will do my best to give you the most clear and understandable explanation I can give you.

First let me calculate the following things for you;

a) 0.75/60 = 0.0125

b) 0.25^(1/60) = 0.9771599684342459

Suppose we have 100 kilograms of radioactive material, then we know that after 60 years only 25 kilograms will be left.

the first percentage is what we use if something reduces with 0.0125 of the original amount each year.

So in the case of the radioactive material this would mean that after 1 year the radioactive material would be

100-100*0.0125 = 98.75

after year 2 it would be

98.75 - 100*0.0125 = 97.5

which is the same as

100 - 100*0.0125 - 100*0.0125 = 97.5

which is the same as

100 - 2 * 100*0.0125 = 97.5

so after the 60 years it would be

100 - 60*100*0.0125 = 25 kg

As you can see, the radioactive material would then reduce with an amount of 1.25 kg each year


However, this is not what actually happens, because what actually happens is that the radioactive material reduces with the same percentage each year

If we calculate the percentage that is reduced each year in the above case we get;
1.25/100 * 100% = 1.25%
1.25/98.75 * 100% = 1.27%
1.25/97.5 * 100% = 1.28%
etc.

We want to reduce the material with a percentage per year,
Now let me give an example which will make you understand,

Suppose the radioactive material reduces with 2% each year
Then after 1 year we know we have 100kg * (1-0.02) = 100kg * 0.98 = 98kg
After 2 years we know we have 98kg*(1-0.02) = 98kg*0.98 = 96.04kg
or we can calculate it as 100kg * (1-0.02) ² = 96.04kg
After 5 years we know we have 100kg * (1-0.02) ⁵ = 90.39kg

(remember 2% reduction means 98% remains, which is why it is (1-0.02)=0.98)

Now lets turn the question around,
Suppose we do not know by how many percent the radioactive material reduces per year but we do know that after 5 years there is only 90.39 kg of 100kg left.

Then we know 100kg * (the percentage that is left per year) ⁵ = 90.39
We can rewrite this to (the percentage that is left per year) ⁵ = 90.39/100
and finally (the percentage per year) = (90.39/100) ^(1/5) = 0.98 = 1-0.02
So 1-(90.39/100) ^(1/5) gives us the percentage the radioactive material reduces by each year!

Now to get back to your question,

we know that in 60 years only 25% will be left, hence to calculate what percentage will be left every year we have (0.25) ^(1/60) = 0.977

Suppose again we have 100kg, in year 1 the amount that will be left can be calculated by
100*0.977 = 97.7 (so it reduced by 2.3 kg)

in year 2 the amount that will be left can be calculated by

100*(0.997) ² = 95.5 (so this year it reduced by 2.2 kg)

As you can see, the amount that it reduces by is now different, yet the percentage it reduces by is the same, which is 1-0.977 = 0.023%

Now for why i use 0.25^(1/60) instead of 0.75(1/60), 0.25 stands for, 25 percentage is left after 60 years, while 0.75 stands for 75 percent remains after 60 years,


This is exactly the difference between 0.25^(1/60) and 1-(0.75/60)

I hope this helps you

Vraces:

Ok, I got most of that.
But, I'm still wondering why is 0.75 irrelevant? And how do you [^] a number on this page?

I see melody beat me too it, yet I can help you with this question,

First look at the following equation;

1-0.25 ^(1/60) =NOT= 0.75 ^(1/60)

why are they not equal?

look at this equation (which makes it a little easier to explain)

1-0.25 ^(1/2) =NOT= 0.75 ^(1/2)
(1-0.25 ^(1/2)) ² = NOT= 0.75 ^(1/2)*2
1-2*0.25 ^(1/2)+0.25 ^(1/2)*2 = NOT = 0.75 ^(1/2)*2
1-2*0.25 ^(1/2)+0.25= NOT = 0.75
1-2*sqrt(0.25)+0.25 = NOT = 0.75

The same things happens when I take both sides of 1-0.25 ^(1/60) =NOT= 0.75 ^(1/60) to the power of 60

To give you an explanation 0.25 ^(1/60) gives what percentage is left each year if there is 25% left after 60 years.
1-0.25 ^(1/60) = what percentage the radioactivity reduces by each year.
0.75 ^(1/60) gives what percentage is left each year if there is 75% left after 60 years.

edit: you can click the button sup to make 'superscript' which is similar to ^, the other button sub gives subscript which you can use if you for example want to do this X _{the amount of chips} + Y _{the quality of TV} = Z _{damage to couch}

Thanks! That was exactly what I needed to know.

*
Thanks reinout-g

Vraces has been struggling with this for a while.
That was a great explanation.
I especially like the very first line.

1-0.25(1/60) =NOT= 0.75(1/60)

I will etch that into my mind and use it myself next time.

And Vraces,
The sub (subscript) and sup (superscript) buttons are located just above the Smilies.

If I want to enter 3 squared I can press
3 sup (the curser will then be put exactly where it nees to be and you type 2. Then you just have to move the curser to the end and keep going.
or alternatively you can type
32 then highlight the 2 and press sup
either of those will give you 3 ²



I often give hints like this in my nightly wrap ups - maybe you could take a look sometimes

I haven't struggled with since I got helped out. But I wanted to know why it's 1 - 0.25^(1/60) instead of 0.75^(1/60), so that in the future I wouldn't have problems.

Hio Vraces,

Do you understand it all now?

By the way, I love your signature, it's cool!