Theory
Round up, then adjust back
Adding 68 and 57 in your head is a little awkward because of the carry. But 70 plus 57 is easy, it is 127. Since 70 is just 2 more than 68, you subtract that 2 back: 127 minus 2 is 125. That is 68 plus 57.
That move, complete a number to a convenient round value, compute easily, then adjust for what you added, is the sutra Puranapuranabhyam, 'by the completion or non-completion'. This lesson teaches the arithmetic form and notes its algebraic cousin, completing the square, from Bharati Krishna Tirtha's 1965 book.
Theory
Completion in arithmetic
The method has three steps:
1. Complete an awkward number to a nearby round one. For 68, add 2 to make 70.
2. Do the easy calculation. 70 plus 57 is 127.
3. Adjust back by the amount you added. You added 2, so subtract 2: 127 minus 2 is 125.
Result: 68 plus 57 = 125. 'Completion' (purana) is filling a number up to a convenient value; 'non-completion' is the complementary idea of working with the shortfall. Either way, you trade an awkward calculation for an easy one plus a small correction.
Practical
Completion-based addition (verified)
def add_by_completion(a, b, round_to):
# complete 'a' up to round_to, add, then adjust back
added = round_to - a # 70 - 68 = 2
easy = round_to + b # 70 + 57 = 127
return easy - added # 127 - 2 = 125
print(add_by_completion(68, 57, 70)) # 125
# Same answer as 68 + 57, via an easier intermediate step.This example runs in Gri-Learn on the web, where you can edit it and see the output.
Formula
The same idea: completing the square
'Completion' is not only for arithmetic. In algebra, completing the square is exactly this idea: you take an expression like x squared + 6x and complete it to a perfect square, (x + 3) squared, then adjust by subtracting the extra: (x + 3) squared - 9. You added 9 to make the perfect square, so you subtract 9 to keep the value the same.
So Puranapuranabhyam names a single strategy with two faces: round up and correct in arithmetic, and complete-the-square in algebra. Both make a hard form easy by completing to a convenient shape and adjusting.
Quiz
Using completion, compute 68 + 57 by first rounding 68 up to 70.
- 129, because 70 + 57 = 127 and you add 2
- 125, because 70 + 57 = 127 and you subtract the 2 you added
- 127, because 70 + 57 = 127 and no adjustment is needed
- 123, because you subtract 4
Show the answer
125, because 70 + 57 = 127 and you subtract the 2 you added
Complete 68 to 70 by ADDING 2, giving the easy sum 70 + 57 = 127. Because you added 2 to 68, you must SUBTRACT 2 back to keep the total correct: 127 - 2 = 125, which is indeed 68 + 57. Option A adjusts the wrong way (adding the 2 again) and over-counts. Option C forgets to adjust at all, leaving the total 2 too high (that is 70 + 57, not 68 + 57). Option D subtracts the wrong amount. The rule: whatever you add to complete a number, subtract the same amount afterward, so the value is unchanged, just easier to compute.
Think first
Why is completing to a round number easier for our brains?
Why is 70 + 57 so much easier than 68 + 57, given they differ by only 2? Then tap.
Show the answer
Because round numbers (multiples of 10) avoid CARRYING, which is the main source of mental-arithmetic slips. When you add 68 + 57, the units 8 + 7 = 15 force a carry into the tens, and juggling that carry while also adding the tens is where mistakes creep in. But 70 ends in 0, so 70 + 57 has trivial units (0 + 7 = 7) and an easy tens sum (70 + 50 = 120), giving 127 with no carry to track, our brains handle 'add to a round number' almost instantly. Completion exploits this: it shifts the awkward number to the nearest round value, does the carry-free addition, and then makes a single small, clean adjustment (subtract the 2 you added) that is itself easy. In effect you replace one hard step (an addition with a carry) with two easy steps (a round addition plus a tiny subtraction), which is faster and more reliable for mental work. The same psychology underlies many mental-maths tricks and even how we make change or estimate bills, we round to convenient anchors and correct. And in algebra, completing the square uses the identical strategy for a different reason: a perfect square is a 'convenient shape' you can take roots of and reason about, so you complete to it and adjust. Round or regular shapes are simply easier to work with; completion gets you there and corrects the difference. Make it convenient, then fix the gap.
Summary
Key takeaways
- Puranapuranabhyam ('by completion or non-completion') completes a number to a convenient value, computes easily, then adjusts back.
- 68 + 57: complete 68 to 70 (add 2), compute 70 + 57 = 127, subtract the 2 back -> 125.
- Whatever you add to complete a number, subtract the same amount afterward to keep the value unchanged.
- The same idea is completing the square in algebra: x squared + 6x becomes (x + 3) squared - 9.
- Completion works because round numbers avoid carrying, making the arithmetic easier and more reliable.
- It is one strategy with two faces (arithmetic rounding and algebraic completion); from the 1965 book.
- Memory hook: round up, compute easily, adjust back by what you added.