Any repeating decimal can be converted to an exact fraction by aligning the repeating block through multiplication, then subtracting to cancel the infinite tail. **Method** (non-repeat length *k*, repeat length *m*): - Multiply *x* by 10^k and by 10^(k+m) - Subtract to eliminate the repeating part - Solve: *x = integer / (10^(k+m) − 10^k)* **Worked example** — 1.05222… (k=2, m=1): ``` 1000x = 1052.222... 100x = 105.222... ───────────────── 900x = 947 → x = 947/900 ``` **Pure repeat** (k=0): 0.̄3 → 10x − x = 3 → x = 1/3. The method works because any rational number either terminates or repeats — the algebraic subtraction extracts the exact fraction from the infinite decimal. ## Cross-Domain Applications **Finance**: Exact interest rates (e.g. 6.̄6% = 20/3 %) matter for compound interest calculations. **Computer science**: Floating-point representation loses precision on repeating decimals; knowing the exact fraction avoids accumulated rounding error. ## References - Math Academy, Mathematical Foundations I (2026-04-22)