Question

Which of the following numbers has terminal decimal representation?
A) \[\dfrac{1}{7}\]
B) \[\dfrac{1}{3}\]
C) \[\dfrac{3}{5}\]
D) \[\dfrac{{17}}{3}\]

Answer 1

Hint: First of all, we will do a short trick for such a question that a number has terminal representation if its denominator is of a specific type. Now, we will check the denominator of all the options and which one passes the condition for it will be the answer.

Complete step-by-step answer:
We know that a fraction has a terminal decimal representation if the denominator is of the form ${2^m} \times {5^n}$, where m and n are positive integers (Natural numbers).
Now, we have four options:
Option 1: \[\dfrac{1}{7}\]
Whatever we do, we can never eliminate 7 from the denominator. So, we can never write its denominator in the form ${2^m} \times {5^n}$. Therefore, it is not terminal.
Option 2: \[\dfrac{1}{3}\]
Whatever we do, we can never eliminate 3 from the denominator. So, we can never write its denominator in the form ${2^m} \times {5^n}$. Therefore, it is not terminal.
Option 3: \[\dfrac{3}{5}\]
We here have 5 in the denominator. We can write it as \[\dfrac{3}{5} = \dfrac{3}{5} \times \dfrac{2}{2} = \dfrac{6}{{2 \times 5}}\].
Now, its denominator is in the form ${2^m} \times {5^n}$ for m = 1 and n = 1.
Hence, it must be terminal.
And, we see that \[\dfrac{3}{5} = \dfrac{6}{{2 \times 5}} = \dfrac{6}{{10}} = 0.6\].
Option 4: \[\dfrac{{17}}{3}\]
Whatever we do, we can never eliminate 3 from the denominator. So, we can never write its denominator in the form ${2^m} \times {5^n}$. Therefore, it is not terminal.

Hence, we see that only one of the four options satisfy our requirements. Hence, the answer is (C) \[\dfrac{3}{5}\].

Note: The students must wonder that they have used the condition for a fraction to be terminal but how does this condition work? Let us ponder over it.
If we have a fraction with a denominator in the form of ${2^m} \times {5^n}$. We can always somehow multiply and divide that fraction with 2 or 5 depending upon whether n > m or m > n to make the denominator a power with base 10 and thus easily replicable into a decimal.
The students must not make the mistake of just multiplying or just dividing some number because that would change the number. If you multiply and divide by the same number, you are basically multiplying the number by 1, which would not result in any change in the number.