Question

Write whether the rational number \[\dfrac{{51}}{{15000}}\] will have a terminating decimal expansion or a non-terminating repeating decimal expansion

Answer 1

Hint: Here we will use the concept of terminating and non-terminating decimal. We will first deduce the fraction into its lowest form and then convert the fraction into a decimal number by dividing the numerator with the denominator. We will analyze the decimal and find out whether it is a terminating decimal or non terminating decimal.

Complete step-by-step answer:
As we all know the meaning of terminating decimal and non-terminating repeating decimal, so when the fraction is at least from then if the denominator has no other prime factors except 2 and 5 then the fraction can be expressed as a terminating decimal.
So, firstly we have to deduce the given fraction \[\dfrac{{51}}{{15000}}\] in its lowest form, so we have to find LCM (Least Common Multiple) of 51 and 15000. 3 is the LCM of 51 and 15000.
Dividing both numerator by 3, we get
$\Rightarrow$ \[\dfrac{{\dfrac{{51}}{3}}}{{\dfrac{{15000}}{3}}} = \dfrac{{17}}{{5000}}\]
 is the least form of the given fraction.
Now we have to find the factors of denominator i.e. 5000.
$\Rightarrow$ \[5000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5\]
As we can clearly see that the denominator has only factors 2 and 5.
Therefore, a given fraction will have a terminating decimal expansion.
Now rewriting the numerator and denominator, we get
$\Rightarrow$ \[\dfrac{{51}}{{15000}} = \dfrac{{17}}{{5000}} = \dfrac{{17}}{{2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5}} = \dfrac{{17}}{{{2^3} \times {5^4}}}\]
Multiplying 2 to both numerator and denominator, we get
$\Rightarrow$ \[\dfrac{{51}}{{15000}} = \dfrac{{17 \times 2}}{{{2^4} \times {5^4}}} = \dfrac{{34}}{{{2^4} \times {5^4}}}\]
Now we will rewrite the denominator as a multiple of 10.
$\Rightarrow$ \[\dfrac{{51}}{{15000}} = \dfrac{{34}}{{10 \times 10 \times 10 \times 10}}\]
\[\therefore \dfrac{51}{15000}=0.0034\]
\[0.0034\] is the terminating decimal.

Note: We have to know the meaning of
Terminating decimals are those decimal numbers that contain a finite number of digits after the decimal point.
Non-terminating repeating decimals are those decimal numbers that contain an infinite number of digits repeating itself again and again after the decimal point.
Also we should know how to find the LCM (Least Common Multiple) of the numbers. LCM is the smallest positive integer that is divisible by both the numbers.