Question

find the rational number whose decimal representation is 0.777...​

Answer 1

Answer:

7/9

Step-by-step explanation:

Step I: x = 0.7777

Step II: After examining we find that repeating digit is 7.

Step III: Place the repeating digit (7) to the left of decimal point. To do so, we need to move the decimal point 1 place to the right. This can also be done by multiplying the given no. by 10.

So, 10x = 7.777

Step IV: After step 3 place the repeating digits to the right of decimal point. In this case if we place the repeating digits to the right of decimal point it becomes the original number.

x = 0.7777

Step V: The two equations are-

x = 0.7777,

⟹ 10x = 7.777

Now we have to subtract the right and left hand sides-

10x - x = 7.777- 0.7777

⟹ 9x = 7.0

⟹ x = 7/9

Hence, x= 7/9 is the required rational number.

Answer 2

Let A = 0.777...

10A = 7.777...

Now,

10A - A = 7.777... - 0.777..

9A = 7

A = $\frac{7}{9}$