Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties Class 9

Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties Class 9

Question 1.

Perform the long division for $\frac{1}{13}$. Identify the repeating block of digits. Does it show cyclic properties if you evaluate $\frac{2}{13}$? Now compute $\frac{3}{13}$ , $\frac{4}{13}$ , etc. What do you notice? Class 9

Solution:

∴ $\frac{1}{13}$ = 0.076923076923...

Repeating block = 076923

Similarly,

$\frac{2}{13}$ = 2 × $\frac{1}{13}$

= 2 × 0.076923076923...

= 0.153846153846...

Repeating block = 153846

∴ It does not show cyclic properties similar to the repeating block of $\frac{1}{13}$.

$\frac{3}{13}$ = 3 × $\frac{1}{13}$

= 3 × 0.076923076923...

= 0.230769230769...

$\frac{4}{13}$= 4 × $\frac{1}{13}$

= 4 × 0.076923076923...

= 0.307692307692...

All decimal expansions are repeating decimals.

∴ The repeating blocks of $\frac{1}{13}, \frac{3}{13}, \frac{4}{13}$ show cyclic patterns: 076923, 230769, 307692.