Question

When 7/11 is written as a decimal, what is the sum of the first 20 digits after the decimal point? Pls Help

Answer 1

There's a neat trick to finding the rational form of a repeating decimal number. Take for instance $x=0.123123123\ldots$. Then

$x = 0.123123123\ldots \\\\ \implies 1000x = 123.123123\ldots \\\\ \implies 1000x - x = 123.123123\ldots - 0.123123\ldots \\\\ \implies 999x = 123 \\\\ \implies x = \dfrac{123}{999}$

It's easy to reverse this method to find the repeating decimal form of $\frac7{11}$. Let

$x = \dfrac7{11}$

Multiply the numerator and denominator by 9,

$x = \dfrac{63}{99}$

It follows that

$x = \dfrac{63}{99} \\\\ \implies 99x = 63 \\\\ 100x - x = 63.6363\ldots - 0.6363\ldots \\\\ \implies x = 0.636363\ldots$

The first 20 digits after the decimal are made up of 10 each of 3 and 6, so the sum is 10 × (3 + 6) = 90.

Answer 2

Answer:

90

Step-by-step explanation:

Using long division, we find that  Every group of two digits after the decimal point has a sum of 9 so the sum of the first 20 digits after the decimal point is 10*9=90