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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

Re: Which is greater 105^19 or 100^20? [#permalink]
18 Jan 2013, 07:50

2

This post received KUDOS

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

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. _________________

Thanks and Regards!

P.S. +Kudos Please! in case you like my post.

Last edited by goutamread on 20 Jan 2013, 02:23, edited 1 time in total.

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. _________________

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 _________________

--------------------------------------------------------------- Consider to give me kudos if my post helped 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. _________________

--------------------------------------------------------------- Consider to give me kudos if my post helped you.

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! _________________

--------------------------------------------------------------- Consider to give me kudos if my post helped you.

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

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? _________________

--------------------------------------------------------------- Consider to give me kudos if my post helped you.

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