The extra spaces on either side of the decimal point are there to make it easy to distinguish the
digit string to its left from the digit string to its right.
First, take the part of the decimal expression to the right of the point. Put those digits
into the numerator of a fraction. Then count the number of digits in the string. Suppose
that number is n. In this case, you have 7601811. That’s a string of seven digits, so n= 7.
In the denominator of the fraction, write a 1 and then n ciphers. The fractional part is
therefore
7,601,811/10,000,000
Second, take the part of the decimal expression to the left of the point. Put those digits into
the numerator of a new fraction. In the denominator, put 1. In this case, the result is
3,588/1
The third part of the process is a little tricky! Add n ciphers after the 1 in the denominator you
just put down, so that denominator is identical to the denominator you “built” for the deci-
mal part of the expression. Then also add n ciphers in the numerator for the whole-number
part. When you do this, you multiply the whole-number part of the expression by a certain
number and then divide that number by itself. That’s just a fancy (or maybe you’d rather say
messy) way of multiplying by 1. In this case you get
35,880,000,000/10,000,000
Fourth, add the fraction you “built” for the whole-number part of the decimal expression to
the fraction you “built” for the decimal part. This should be easy, because you have engineered
things to get a common denominator! In this case, it’s 10,000,000. Adding the numerators
produces
35,880,000,000+ 7,601,811 = 35,887,601,811
That’s the numerator of the ratio you want. The denominator is 10,000,000, so the complete
ratio is
35,887,601,811/10,000,000
If you’d like to check this, divide the ratio out on a calculator that can display at least 11 digits.
You should get the original decimal expression.
Endless repeating decimal to ratio
The solution to the “challenge” problem in the last section should give you an idea of how to
convert any endlessly repeating decimal to a ratio of two integers. You can generalize on the
number of digits in the repeating pattern, from one up to as many as you want.
When you encounter a decimal expression that has a sequence of digits that repeats with-
out end, first split the whole-number part from the decimal part. Call the whole-number part a.
Conversions 105