Class 9 Math Exercise 1.3 | Question 6

Question 6: Look at several examples of rational numbers in the form \\frac{p}{q}) (q ≠ 0), where p and q are integers with no common factors other than 1 and have terminating decimal expansions. What property must q satisfy?

Explanatory Answer | Exercise 1.3 Question 6

Let us look at a few examples
Example 1: \\frac{3}{25}) = 0.12 which is the same as \\frac{12}{100})
Example 2: \\frac{11}{4}) = 2.75 which is the same as \\frac{275}{100})
Example 3: \\frac{17}{100}) =0.17
Example 4: \\frac{29}{125}) = 0.232 which is the same as \\frac{232}{1000})
Example 5: \\frac{13}{16}) = 0.8125 which is the same as \\frac{8125}{10000}) and
Example 6: \\frac{91}{10}) = 9.1.

What is common to all the denominators?

From the six examples, we can deduce that the denominator of all of these fractions are powers of 10.
So, the denominator will have equal powers of 2s and 5s in it.
However, in that form p and q may have common factors other than 1.
Look at example 1 or example 2. In \\frac{12}{100}), 4 is a common factor between p and q.
So, when reduced to the form where p and q have no common factors other than 1 (a few 2s and or 5s will get cancelled), q need not be a power of 10. But it will be a number whose prime factors must be only 2 and/or 5.
In other words when p and q have only 1 as their common factor and if 'q' has any prime factor other than 2 or 5, the fraction will not be a terminating decimal.