Advanced Arithmetic: Find Min Digits for Fraction Decimal

[Michele Nunes]

Homework Statement

What is the minimum number of digits to the right of the decimal point needed to express the fraction

as a decimal?
a) 4
b) 22
c) 26
d) 30
e) 104

Homework Equations

The Attempt at a Solution

One possible solution is: "We can rewrite the fraction as

. Since the last digit of the numerator is odd, a

is added to the right if the numerator is divided by [PLAIN]https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png, and this will continuously happen because [PLAIN]https://latex.artofproblemsolving.com/7/9/0/79069377f91364c2f87a64e5f9f562a091c8a6c1.png, itself, is odd. Indeed, this happens twenty-two times since we divide by https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png twenty-two times, so we will need

more digits. Hence, the answer is [PLAIN]https://latex.artofproblemsolving.com/8/9/7/897cab64b41a26c4bf59f579a975ec600cf2441b.png."

So I understand how they rewrote the fraction, however I'm totally lost by everything after that. I'm not exactly sure why a 5 is added to the right if the numerator is divided by 2 and how that has anything to do with the last digit being odd. Also, a calculator is not allowed on this. If someone could either clarify the explanation given or give a more simplified alternate solution, that would be appreciated.

 

[ehild]

I think it is a bit more clear if you do the division by 10⁴ at the end.

123456789=123456788 + 1. Divide it by 2 : it becomes an integer + 0.5 that means one decimal.
Divide that number by 2 gain, you get an integer, + 0.25, so you have two decimals. Do again: it is something + 0.125. Repeat: the number of decimals increases by one at every division by 2. The first term becomes odd after some divisions, but that will have less decimals than the division of 1.
So how many decimals are there if you divide by 2²² (that is, you do 22 divisions) ?

 

[Michele Nunes]

I think it is a bit more clear if you do the division by 10⁴ at the end.

123456789=123456788 + 1. Divide it by 2 : it becomes an integer + 0.5 that means one decimal.
Divide that number by 2 gain, you get an integer, + 0.25, so you have two decimals. Do again: it is something + 0.125. Repeat: the number of decimals increases by one at every division by 2. The first term becomes odd after some divisions, but that will have less decimals than the division of 1.
So how many decimals are there if you divide by 2²² (that is, you do 22 divisions) ?

Okay I'm kind of making sense out of it. If you divide 1 by 2²² then you will have 22 decimals, and then if you divide 123456788 by 10⁴ then you will have 4 decimals, and 22 + 4 = 26, I don't know though something about that process seems off, I feel like you can't just separate them like that, can you?

 

[ehild]

Is 123456789 = 123456788 +1?
Is (a+b)/2 = a/2 + b/2 true?
try with a smaller number : How many decimals are there in 9/2³?

 

[Michele Nunes]

Is 123456789 = 123456788 +1?
Is (a+b)/2 = a/2 + b/2 true?

Yes, but I would have 123456788/(2²² * 10⁴) + 1/(2²² * 10⁴)
and if I ONLY divided 123456788 by 10⁴, then where did the 2²² in the denominator go
and if I ONLY divided 1 by 2²², then where did the 10⁴ in the denominator go, do you see what I mean?

 

[ehild]

Yes, but I would have 123456788/(2²² * 10⁴) + 1/(2²² * 10⁴)
and if I ONLY divided 123456788 by 10⁴, then where did the 2²² in the denominator go
and if I ONLY divided 1 by 2²², then where did the 10⁴ in the denominator go, do you see what I mean?

$\frac {A}{b c}= \frac {A/b}{c}$. If you have a product of two numbers in the denominator, you can divide by one number, then you divide the result by the other.
Here, you divide by 2²² first, then you divide the result by 10⁴.
If you evaluate $\frac{36}{3*4}$ you can do it by dividing 36 with 12 or first dividing 36 by 3 (you get 12) then dividing 12 by 4 - the result is 3 with both methods.

 

[ehild]

Do you understand it now?

 

[Michele Nunes]

Do you understand it now?

Yes for the most part, thank you!