We shall learn the idea of terminating decimals below.
Terminating Decimals –
In order to express a fraction x/y in decimal form, we divide ‘x’ by ‘y’. If after a finite number of steps, no reminder is left, then we call it a terminating decimal.
For example –
1 3 34
a) ------ = 0.50, b) ------- = 0.375 , c) -------- = 6.8
2 8 8
are all terminating decimals.
An important result, a fraction x/y would be considered as a terminating decimal only when prime factors of ‘y’ are out of 2 & 5 only.
Example.1) Without actual division, find out, which of the following fractions are terminating decimals.
a) 11/20
Ans.) The given number is 11/20 and HCF of (11, 20) = 1
so, 11/20 is in simplest form.
now, 20 = 2 X 2 X 5
so, the prime factors of 20 are 2 & 5. Hence 11/20 is a terminating decimal
b) 43/60
The given number is 43/60 and HCF (43, 60) = 1
so, 43/60 is in the simplest form.
Now, 60 = 2 X 2 X 3 X 5
Thus the prime factors of 60 are 2, 3, and 5
Hence, 43/60 is a non-terminating, repeating decimal.
c) 66/180
the given number is 66/180 and HCF (66, 180) = 6
66 66 ÷ 6 11
so, -------- = ------------ = --------
180 180 ÷ 6 30
Thus, 11/30 is simplest form. Now, 30 = 2 X 3 X 5
So, the prime factors of 30 are 2, 3, and 5
Hence, 11/30 and hence 66/180 is non-terminating, repeating decimals.
d) 121/252
the given number is 121/252 and HCF (121, 252) = 1
so, 121/252 is in simplest form.
Now, 252 = 2 X 2 X 3 X 3 X 7
Thus, the prime factors of 252 are 2, 3, and 7.
Hence, 121/252 is a non-terminating, repeating decimals.