Question

Write the following rational number as a decimal form and find out the block of repeating digits in the quotient.
$\dfrac{10}{13}$

Answer 1

Hint: For solving these types or problems, we need to have a clear understanding of rational numbers and how can they be converted to decimals. Converting the given rational number into decimal form, we can easily identify the block of repeating digits which gives the answer.

Complete step-by-step answer:
In mathematics, a rational number is a number such as \[-\dfrac{3}{7}\] that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, a numerator p and a non-zero denominator q. They can be positive or negative. There are two types of rational numbers:
A) finite or terminating decimal B) Non terminating decimals.
Non terminating decimals are the ones where a single number or a block of numbers get repeated in the decimal form infinitely. According to the given problem we need to find that repeating block when $\dfrac{10}{13}$ is converted into the decimal form.
Now, converting $\dfrac{10}{13}$ in the decimal form gives \[0.7692307692307692307\] .
By closely analysing the decimal form of the number, we can say that the block of numbers or the number \[769230\] repeats itself infinitely.
Hence, the block of repeating digits in the quotient is \[769230\] .

Note: These questions are pretty easy to solve, but we need to convert the rational number into its decimal form very carefully. A slight miscalculation may not result in getting the repeating block of numbers. We also need to write the repeating block as it is instead of using the decimal point preceding it.