Can the decimal point values of fractions be predicted?

Question

Is there anyway to know what the decimal point value of a given fraction or a string of fractions will be? For example if you know 21/4 = 5.25 and 21/5 = 4.2 can you predict that 21/6 = 3.5? With some sort of equation or a function? Or do you just have to actually divide it and see the result? It's not the whole value I'm interested in just the decimal points but anything would be helpful

Answer 1

The fractional part $.25$ is the same for all of the following: $$ 1/4, \quad 5/4, \quad 9/4, \quad 13/4, \quad 17/4, \quad 21/4, \quad 25/4 \ldots $$ In this sense, it can indeed be predicted:

$\quad$The fractional part of $x$ is the same as the fractional part of $x-1$.

$\quad$The fractional part of $x$ is the same as the fractional part of $x-[x]$.

(Here $[x]$ denotes the integer part of $x$, i.e. the greatest integer not exceeding $x$.)

In some cases we can predict that the fractional part is precisely zero using divisibility rules; for example, the fractional part of 90108/9 is zero because 90108 is divisible by 9.

Also, we can predict with 100% certainty that for any ordinary fraction, the decimal will either be repeating or terminate.

In addition, we can always predict the result when the denominator of your fraction is 10, 100, 1000, etc. For example: $$ 12345/10 = 1234.5, \qquad 12345/100 = 123.45, \qquad 12345/1000 = 12.345 \ldots $$

Otherwise, you just have to actually divide it and see the result.

Answer 2

Consider p/q, q > 1, 0 < p < q.

If the denominator, q, is relatively prime to both 2 and 5, then for some natural n and m, mq = 10^n-1. The decimal form of the fraction will then be mp repeated every n digits.

If the denominator, q, only has factors of 2 and 5, then it terminates.

Else, the fraction can be split into 2 fractions(p/q=a/b+c/d), where b is relatively prime to both 2 and 5, and d has only factors of 2 and 5.