Question

It is given that if:
$1.\overline{9}=2$
Then find the value of 2, 00, 00, 000?

Answer 1

Hint: In the above problem, the information we are given is that $1.\overline{9}=2$. And we have to find the value of 2, 00, 00, 000 in terms of $1.\overline{9}$. The bar on 9 means 9 is repeating endlessly. Now, we know that $1.\overline{9}=2$ so we can write 2 in 2, 00, 00, 000 as $1.\overline{9}$ and then multiply the remaining number of zeros in 2, 00, 00, 000.


Complete step by step answer:
It is given in the above problem that:
$1.\overline{9}=2$
The bar on 9 shows that 9 is repeating endlessly as follows:
$1.99999999......=2$
And the above problem has rounded off the above number in which 9 is repeating as 2.
And also we are asked to find the value of 2, 00, 00, 000 using the relation $1.\overline{9}=2$.
As it is given that $1.\overline{9}=2$ so we can write 2 in 2, 00, 00, 000 as $1.\overline{9}$.
Writing 2 in 2, 00, 00, 000 as $1.\overline{9}$ we get,
$1.\overline{9}\times 10000000$ ……….. Eq. (1)
As you can see that there are 7 zeros in the above number so we can compact the 7 zeros by writing 10 to the power of 7.
$1.\overline{9}\times {{10}^{7}}$
Hence, we have written 2, 00, 00, 000 using the relation $1.\overline{9}=2$ as $1.\overline{9}\times {{10}^{7}}$.

Note:
You might have thought that in the above solution in the step which we have shown above by eq. (1) as:
$1.\overline{9}\times 10000000$
We can multiply the 7 zeros and then the decimal will shift 7 places forward from its current position.
$1.9999999.....\times {{\left( 10 \right)}^{7}}$
Now, if we shift the decimal point in the forward direction by 7 places we get,
$19999999.99999.....$
Rewriting the above number by putting a bar on the 9 which is written after the decimal point we get,
$19999999.\overline{9}$
But the problem in writing this solution is that the number does not contain $1.\overline{9}$ in it. That’s why we did not write the number 2, 00, 00, 000 in this form $19999999.\overline{9}$.