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Which is greater 105^19 or 100^20?

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Which is greater 105^19 or 100^20? [#permalink] New post 11 Jan 2013, 02:13
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Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.
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Last edited by Orko on 16 Jan 2013, 06:46, edited 1 time in total.
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Re: Question [#permalink] New post 11 Jan 2013, 03:01
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100^{20} = 100^{19} * 100 i.e 100 times 100^{19}

= 100^{19} + 100^{19} + 100^{19} + ...... and so on till hundred times --> (1)

now,

105^{19} = (100 + 5) ^{19}

Now, as we know (a + b)^n = a^{19} + ^nC1*a^{(n-1)}*b^1 + ^nC2*a^{(n-2)}*b^2 + ........ + ^nCn-1*a^1*b^{(n-1)} + b^n

(100 + 5) ^{19} = 100^{19} + 19*100^{18}*5^1 + 171*100^{17}*5^2 + .... + 5^{19}

= 100^{19} + 85*100^{18} + ... preceding all the terms will be < 100^{19} --> (2)

so compare (1) and (2) now, we get

100^{20} > 105^{19}
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Last edited by goutamread on 12 Jan 2013, 00:04, edited 3 times in total.
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Which is greater 105^19 or 100^20? [#permalink] New post 11 Jan 2013, 03:56
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Let me try :


1st expression : 105^{19} = (100 *1.05)^{19} = 100^{19}*1.05^{19} --- [ as (a*b)^n = (a^n)*(b^n) ]

2nd expression : 100^{20} = 100^{19} * 100^1 = 100^{19}*100 --- [ as a^n = (a^1)*(a^{n-1}) ]

now as 100^{19} is common,

Compare 1.05^{19} with 100.

as 1.05 is slightly greater than 1

surely, 1.05^{19} < 100


thus

105^{19} < 100^{20}

Let me know if this worked for you. :!:
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Last edited by goutamread on 12 Jan 2013, 00:20, edited 2 times in total.
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Re: Question [#permalink] New post 11 Jan 2013, 21:18
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ObsessedWithGMAT wrote:
Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.


goutamread has given you a great approach. I would use a similar one though I just try to simplify first. It helps you think faster in some cases.

105^{19} and 100^{20} have 5^{19} common. Cancel out the common 5^{19} and you are left with

105^{19} = 5^{19}*21^{19}
and 100^{20} = 5^{19}*20^{19}*100

Now compare 21^{19} and 20^{19}*100

It is easy to make some sense out of 20^{19}*100. It is 100 times 20^{19} i.e. you can get this expression if you add 100 terms of 20^{19} together

20^{19}*100 = 20^{19} + 20^{19} + 20^{19} ... (100 terms) ..............(I)

To compare, we can use binomial for 21^{19}. If you do not know how to use binomial, check out this link:
http://www.veritasprep.com/blog/2011/05 ... ek-in-you/

21^{19} = (20 + 1)^{19}

Let's try to expand (20 + 1)^{19} using binomial. We will get 19+1 = 20 terms when we expand.

(20 + 1)^{19} = 20^{19} + 19*20^{18} + 19* 18/2 * 20^{17} + ... + 1^{19} ............... (II)

Notice that this expression has 20 terms and none of the terms will be greater than 20^{19}.

Compare (I) with (II).
(I) has 100 terms, all of them 20^{19}
(II) has 20 terms, all of them equal to or less than 20^{19}.

Hence, (I) will be greater than (II). Or we can say that 100^{20} will be greater than 105^{19}.
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Re: Which is greater 105^19 or 100^20? [#permalink] New post 18 Jan 2013, 07:50
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ObsessedWithGMAT wrote:
Thanks a lot goutamread! This one is really better. But, I want to know that what made you to split 105 into 100*1.05. This is some thing that is really very important to know because I believe that these are the tactics that one has to think of while attacking a question.



I'm not an expert, rather I'm just another aspirant on the forum. I could try to explain:

See, whenever we deal with a big number/expression, we try to simplify the expression to make it realistic :P

There could be numerous ways to solve a question, what matters is which one you feel at your ease... now as you mentioned inequality,

assume 105^{19} > 100^{20} lets try to prove whther our assumption is correct or not. If yes, then 105^{19} > 100^{20} and if not then, 105^{19} < 100^{20}

105^{19} > 100^{20}
Now, don't these expressions scare us. yes they are scary.. so lets try and simplify.
lets divide both side by 100^{20}, (as 100^{20} is +ve --> inequality sign wont change)
\frac{105^{19}}{100^{20}} > 1
\frac{105^{19}}{{100^{19}*100}} > 1
(\frac{105}{100})^{19}*\frac{1}{100} > 1
\frac{(1.05) ^{19}}{100} > 1 OR (\frac{26}{25}) ^{19}* \frac{1}{100} > 1 ==> 26 ^{19} > 100 * 25^{19}

From here its already discussed earlier how to solve the problem. Take away could be : There are various ways to solve a problem, what matters is what suits you, but you should be aware of more than one trick.
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Last edited by goutamread on 20 Jan 2013, 02:23, edited 1 time in total.
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Re: Question [#permalink] New post 11 Jan 2013, 21:05
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ObsessedWithGMAT wrote:
Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.


Could you change the main subject as 'Which is greater 105^19 or 100^20?' :) So, It may be helpful for others too, in case they need to dive into the problem.
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P.S. +Kudos Please! in case you like my post. :)

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Re: Which is greater 105^19 or 100^20? [#permalink] New post 17 Jan 2013, 10:52
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Here is the smart solution. Lets assume both are equal. If after some manipulation we can show LHS > RHS or LHS < RHS we would have got the answer.

105^19 = 100^20

Take 100^19 to the LHS (Divide both sides by 100^19)

We have 1.05^19 = 100

From Binomial theorem we know than (1+x)^n = 1 + nx when x<<1

Hence 1.05^19 = (1 + 0.05*19) = 1.95 < 100

Hence 105^19 < 100^20
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Re: Which is greater 105^19 or 100^20? [#permalink] New post 18 Jan 2013, 08:39
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For more practice : which-is-greater-54-200-or-5145.html
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Re: Question [#permalink] New post 11 Jan 2013, 03:38
goutamread wrote:
100^20 = 100^19 * 100 i.e 100 times 100^19
= 100^19 + 100^19 + 100^19 + ...... and so on till hundred times --> (1)

now,

105^19 = (100 + 5) ^19
Now, as we know (a + b)^n = a^19 + (nC1)(a^(n-1))(b^1) + (nC2)(a^(n-2))(b^2) + .... + (nCn-1)(a^1)(b^(n-1)) + b^n
(100 + 5) ^19 = 100^19 + (19)(100^18)(5^1) + (171)(100^17)(5^2) + .... + 5^19
= 100^19 + (85)(100^18) + ... preceding all the terms will < 100^19 --> (2)

so compare (1) and (2) now, we get

100^20 > 105^19


Hi Goutamread,

Thanks for the response and effort! Although it seems that you have tried to explain in the best possible way that you are aware of, the explanation has just gone over my head :roll:
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Re: Which is greater 105^19 or 100^20? [#permalink] New post 16 Jan 2013, 06:46
goutamread wrote:
Let me try :


1st expression : 105^{19} = (100 *1.05)^{19} = 100^{19}*1.05^{19} --- [ as (a*b)^n = (a^n)*(b^n) ]

2nd expression : 100^{20} = 100^{19} * 100^1 = 100^{19}*100 --- [ as a^n = (a^1)*(a^{n-1}) ]

now as 100^{19} is common,

Compare 1.05^{19} with 100.

as 1.05 is slightly greater than 1

surely, 1.05^{19} < 100


thus

105^{19} < 100^{20}

Let me know if this worked for you. :!:


Thanks a lot goutamread! This one is really better. But, I want to know that what made you to split 105 into 100*1.05. This is some thing that is really very important to know because I believe that these are the tactics that one has to think of while attacking a question.
_________________

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Consider to give me kudos if my post helped you.

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Re: Question [#permalink] New post 16 Jan 2013, 06:47
goutamread wrote:
ObsessedWithGMAT wrote:
Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.


Could you change the main subject as 'Which is greater 105^19 or 100^20?' :) So, It may be helpful for others too, in case they need to dive into the problem.


Done! :-)
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Re: Question [#permalink] New post 16 Jan 2013, 06:50
VeritasPrepKarishma wrote:
ObsessedWithGMAT wrote:
Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.


goutamread has given you a great approach. I would use a similar one though I just try to simplify first. It helps you think faster in some cases.

105^{19} and 100^{20} have 5^{19} common. Cancel out the common 5^{19} and you are left with

105^{19} = 5^{19}*21^{19}
and 100^{20} = 5^{19}*20^{19}*100

Now compare 21^{19} and 20^{19}*100

It is easy to make some sense out of 20^{19}*100. It is 100 times 20^{19} i.e. you can get this expression if you add 100 terms of 20^{19} together

20^{19}*100 = 20^{19} + 20^{19} + 20^{19} ... (100 terms) ..............(I)

To compare, we can use binomial for 21^{19}. If you do not know how to use binomial, check out this link:
http://www.veritasprep.com/blog/2011/05 ... ek-in-you/

21^{19} = (20 + 1)^{19}

Let's try to expand (20 + 1)^{19} using binomial. We will get 19+1 = 20 terms when we expand.

(20 + 1)^{19} = 20^{19} + 19*20^{18} + 19* 18/2 * 20^{17} + ... + 1^{19} ............... (II)

Notice that this expression has 20 terms and none of the terms will be greater than 20^{19}.

Compare (I) with (II).
(I) has 100 terms, all of them 20^{19}
(II) has 20 terms, all of them equal to or less than 20^{19}.

Hence, (I) will be greater than (II). Or we can say that 100^{20} will be greater than 105^{19}.


Dear Karishma,

Thanks a lot for taking out time to reply to my question! I am still struggling with the fact that this question is from Inequalities and neither you nor goutamread applied any concept of Inequalities or is it that I am not able to identify?
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Re: Question   [#permalink] 16 Jan 2013, 06:50
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