Solve Maths Competition Problems: Sum a+b & Logarithm X

[Kushal]

these are numbers i could not solve. please help me.

Homework Statement

1) If the repeating decimal 0.8451$$\bar{51}$$ is represented by the fraction a/b, where a and b are positive integers with no common factors greater than 1, find the sum a + b.

2) if log(log(log(log x))) = 0 and log represents the base ten logarith, what is the value of x.

The Attempt at a Solution

1) i don't understand how to proceed. some hints would be appreciated.

2)i kno that if, log_ba = x

then, a = b^x

i got 10¹⁰⁰ as answer but the paper says the answer is 10^{10,000,000,000} is the answer.



thanks

 

[nicksauce]

For b why don't you show us how you can your answer, writing out every step in full.

For a, it is the same as finding a number 0.51515151... (and then adding 84 and dividing by 100). Do you know how you would find such a number?

 

[Kushal]

2) log(log(log(log x))) = 0

log(log(log x)) = 10⁰ = 1

log(log x) = 10¹ = 10

log x = 10¹⁰

x = 10¹⁰¹⁰oooh ok... now i see my mistake! i tried doing the calculation in my head... so foolish of my parterrm, for #1
d'you mean that i should break the 0.8451515151 into 0.84 + 0.005151515151 ?!

thnks

 

[nicksauce]

errm, for #1
d'you mean that i should break the 0.8451515151 into 0.84 + 0.005151515151 ?!

thnks

Yes that's how I would approach the problem. Now can you find a fraction to represent 0.005151515... ?
If you don't know how, you should read the section entitled "Fraction from a repeating decimal" here
http://en.wikipedia.org/wiki/Recurring_decimal