Question

Find the decimal expansion of \[\dfrac{{31}}{{16}}\].

Answer 1

Hint: Here we will first multiply and divide the given fraction with a square of some number. The number should be such that if it is multiplied to the denominator of the fraction then we get a multiple of 10 in the denominator. We will then simplify the fraction to get the required decimal expansion.

Complete step-by-step answer:
We have to find the decimal expansion of \[\dfrac{{31}}{{16}}\].
So, we will multiply and divide by 625 in the above value to get the denominator as power of base 10.
\[\dfrac{{31}}{{16}} = \dfrac{{31}}{{16}} \times \dfrac{{625}}{{625}}\]
As 16 is square of 4 and 625 is square of 25, we can write the above value as,
\[ \Rightarrow \dfrac{{31}}{{16}} = \dfrac{{31 \times 625}}{{{4^2} \times {{25}^2}}}\]
Now, we can see that product of 4 and 25 is 100 so we get,
\[ \Rightarrow \dfrac{{31}}{{16}} = \dfrac{{31 \times 625}}{{100 \times 100}}\]
Multiplying the terms, we get
\[ \Rightarrow \dfrac{{31}}{{16}} = \dfrac{{19375}}{{10000}}\]
Now dividing 19375 by 10000, we get
\[\dfrac{{31}}{{16}} = 1.9375\]
So \[1.9375\] is a decimal expansion of \[\dfrac{{31}}{{16}}\].

Note: Decimal expansion of a number is its representation in base-10. In decimal expansion, each decimal place contains a digit from 0-9 such that we multiply the digit by a power of 10 decreasing from left to right. We can get a non-terminating decimal expansion of irrational numbers or we can get a repeating decimal in such case we stop our calculation after some time because it is never-ending. Another method to find the decimal expansion is by directly dividing the numerator by the denominator but that requires more calculation and especially when the numbers are big it gets complicated so making the denominator in the power of 10 bases is more convenient.