Terminating decimals

Only 106 out of the 343 or less than 31% of the competitors in the primary group in the 3^rd MATH-Inic Vedic Mathematics National Challenge was able to get the correct answer to this question which can be easily solved in just a few seconds.

Fractions expressed in their lowest terms, when converted to decimals may be classified into three groups: terminating, partially recurring and perfectly recurring decimals.

By inspecting the factors of the denominator of a fraction, we can determine to which group its decimal equivalent will belong.

A fraction has a terminating decimal form if its denominator is 2, 5 or a product of 2s and 5s.  1/2 , 1/5 , 1/10 and 1/500 are all convertible into terminating decimals.

1/2  = 0.5

1/5 = 0.2

1/10 = 1/(2×5) = 0.1

1/500 = 1/(5x10x10)  

It has a partially recurring decimal string if its denominator has prime factors other than 2 and 5, in addition to 2s or 5s or products of 2s and 5s. 1/6  and 1/15 have partially recurring decimal strings.

1/6 = 1/(2×3) = 0.1666…

1/15 = 1(3×5) = 0.0666…

It is said to have a perfectly recurring decimal equivalent if its denominator has no factor which is 2, 5 or products and powers of 2s and 5s. Examples are 1/7, 1/11 and 1/13

1/7 = 0.142857142857…

1/11 = 0.090909…

1/13 = 0.076923076923…

In the IVMO 2022 question, all choices are expressed in their lowest terms. Let us consider them one by one:

1. 9/10; 10 = 2 x 5
2. 19/20; 20 = 2 x 2 x 5
3. 29/30; 30 = 3 x 2 x 5
4. 39/40; 40 = 2 x 2 x 2 x 5
5. 49/50; 50 = 5 x 5 x 2

Clearly only choice c’s denominator 30 has a prime number factor, 3, which is not 2 or 5.