Fractions, decimals & series
Split a fraction into partial fractions by inspection
The textbook method sets up simultaneous equations for the unknown numerators — the cover-up shortcut skips all of it.
Spot it when
A proper fraction with a factored denominator, e.g. (3x+13)/((x+1)(x+2)) — common before integration.
Worked examples
The method builds the answer directly — worked results:
When you’ll actually hit this
Real moments — exams, homework, competitions — where this shortcut earns its keep.
A calculus question needs a fraction broken up before you can integrate it.
Paraavartya: cover (x+1) and put x = −1 → 10; cover (x+2) and put x = −2 → −7. Answer: 10/(x+1) − 7/(x+2).
An algebra exam wants a decomposition and there is no time for the full equate-coefficients method.
Substitute each root into the covered fraction → 4/(x+3) − 3/(x+4).
A board demonstration of partial fractions where the teacher wants the numerators read off instantly.
Each numerator is the rest of the fraction evaluated at that factor’s root — one substitution apiece.
The method behind it
परावर्त्य योजयेत्
Parāvartya Yojayet
“Transpose and apply”
Division by a near-10 divisor: transpose the sub-digits of the divisor (negate them) and use them as a flag that helps you divide step by step.
Conventional long division of : 5 written steps. Paraavartya flag division: 4 mental steps with no written remainders.
Use when
- • Divisor has leading digit 1, or is close to 10^k
Avoid when
- • Divisor is arbitrary (use standard long division)