In-Depth Tutorial
A formal, accessible 800–1000 word walkthrough of this topic, written for the serious aspirant. Switch to हिन्दी using the toggle on the right.
Why Division Deserves a Chapter of Its Own
Division is the slowest of the four basic arithmetic operations and consequently the most expensive in time when performed by the conventional long-division method. A three-digit number divided by a two-digit number using long division typically takes thirty to forty seconds; a four-digit dividend can take a full minute. In Banking Prelims and Staff Selection Commission Tier-I papers, between four and seven questions per paper require at least one division of this size — most commonly in percentage, ratio, time-and-work and average problems. Compressing this calculation through Vedic methods saves between two and four minutes per paper.
Vedic division rests on two principal techniques: Paravartya Yojayet for divisors that are slightly more than a base, and a running-sum technique for division by nine. Square roots are addressed through Vilokanam for perfect squares and the Dvanda Yoga method for general numbers. The same chapter also covers the inverse percentage and ratio shortcuts that grow naturally from division.
Division by Nine — The Running Sum Method
Division of any number by nine, regardless of length, can be performed in a single horizontal line using a running-sum technique. The candidate writes the dividend along the top line. The first quotient digit is the first digit of the dividend. Each subsequent quotient digit is the running sum of the dividend digits encountered so far, modulo nine. The remainder is the running sum of all dividend digits, modulo nine.
Consider 234 ÷ 9. The first quotient digit is 2. The next is 2 + 3 = 5. The remainder portion accumulates as 2 + 3 + 4 = 9, which becomes 0 with a carry of 1 to the previous quotient digit. The final answer is quotient 26, remainder 0. Consider 1234 ÷ 9: 1, 1+2=3, 1+2+3=6, remainder 1+2+3+4 = 10 → remainder 1, carry 1. Quotient 137, remainder 1. With ten practice problems this technique becomes automatic, and any division by nine — whatever the size of the dividend — is solved in under fifteen seconds.
Paravartya Yojayet — Division Near a Base
Paravartya Yojayet, meaning 'transpose and apply,' addresses divisions in which the divisor is slightly more than a base of ten, hundred or thousand. For a divisor such as 12, the procedure is to transpose the deficiency-related digits — that is, to invert the sign of the digits beyond the leading one — and to apply repeated addition to the dividend. The technique converts division into a sequence of small additions, which the candidate can perform mentally.
Consider 1234 ÷ 12. The divisor 12 is at base 10 with surplus 2; the transposed value is −2. The candidate writes the dividend in pairs of digits and applies the transposition iteratively, producing the quotient and remainder in a single horizontal pass. The full procedure requires five solved examples for full clarity, after which any division by a divisor between ten and twenty can be performed in twelve to fifteen seconds. The same logic extends to divisors near hundred, where the technique replaces a half-minute long division with a ten-second paper-and-pen calculation.
Vilokanam — Square Roots of Perfect Squares
Vilokanam, meaning 'by inspection,' produces the square root of a perfect square by simple observation. The candidate first determines the number of digits in the root, which equals the number of pairs in the original number when the digits are grouped from the right. The first digit of the root is the integer part of the square root of the leftmost group. The last digit of the root is determined by the unit digit of the original number, using the table of last-digit correspondences: 1 ↔ 1 or 9, 4 ↔ 2 or 8, 5 ↔ 5, 6 ↔ 4 or 6, 9 ↔ 3 or 7, 0 ↔ 0.
Consider √2025. There are two pairs (20, 25), so the root has two digits. The first digit is the integer square root of 20, which is 4 (since 4² = 16 ≤ 20 < 25 = 5²). The unit digit of 2025 is 5, so the last digit of the root is 5. The root is 45. To resolve the ambiguity when the unit digit corresponds to two possible last digits, multiply the leading group by the next integer; if the product is less than the leading group, take the larger candidate, otherwise the smaller. With twenty practice roots the candidate can identify any four-digit perfect square root in under ten seconds.
Dvanda Yoga — Square Roots of General Numbers
When the number is not a perfect square or when the question asks for an approximate root, the Dvanda Yoga method delivers the answer column by column. The candidate begins by grouping the digits from the right in pairs, identifying the first quotient digit by inspection of the leftmost group, and then computing each subsequent quotient digit using a duplex-style adjustment that incorporates the carry from the preceding step.
The full Dvanda Yoga procedure is most efficiently learned through five worked examples — one each for two-digit, three-digit, four-digit, five-digit and decimal results. Once the pattern is recognised, the candidate can extract any square root to two decimal places in under thirty seconds. For competitive examinations, however, the perfect-square case via Vilokanam is the more frequently used technique, and Dvanda Yoga serves chiefly for the data-interpretation block and for problems involving standard deviation.
A Five-Day Plan for Division and Roots
An efficient plan covers division and roots in five days. Day one introduces division by nine through the running-sum technique with twenty problems of increasing dividend length. Day two introduces Paravartya Yojayet for divisors near ten with twenty problems. Day three extends Paravartya to divisors near hundred. Day four covers Vilokanam square roots with twenty perfect squares from three to six digits. Day five is mixed practice with twenty division and ten square-root problems under timed conditions of fifteen seconds per problem.
By the end of the fifth day, the candidate can divide by nine, by any divisor near ten or hundred, and can extract perfect square roots up to six digits — all within strict examination time budgets. The combined time saving across one paper is between three and five minutes, sufficient for three to four additional confident attempts.