Theory
The whole and its parts
Multiply 18 by 22. Notice they sit symmetrically around 20: one is 2 below, the other 2 above. The sutra Vyastisamanstih, 'the parts and the whole', uses exactly this relationship, playing the whole (here, the middle value 20) against the parts (the gap of 2) to get the answer quickly.
This lesson teaches that accessible, verifiable application, multiplying via the average and the deviation, from Bharati Krishna Tirtha's 1965 book. The book uses the parts-and-whole idea more broadly (including factorisation); consult the prescribed text for those.
Theory
Average and deviation
When two numbers are symmetric about a middle value, write them as (m - d) and (m + d), where m is their average (the whole) and d is the deviation (the part).
Their product is the difference of squares: (m - d)(m + d) = m squared - d squared.
Example, 18 times 22: the average m is 20, the deviation d is 2. So the product is 20 squared - 2 squared = 400 - 4 = 396. And 18 times 22 is indeed 396. You replaced a two-digit multiplication with a square you know (400) minus a tiny square (4).
Theory
Another example
Try 17 times 23. They are symmetric about 20: 17 is 3 below, 23 is 3 above. So m is 20 and d is 3.
Product = m squared - d squared = 20 squared - 3 squared = 400 - 9 = 391. And 17 times 23 is 391.
The technique shines whenever two numbers straddle an easy middle value: pick the average as the whole, the gap as the part, and the product is just the easy square of the whole minus the small square of the part. Whole minus part, in squares.
Practical
Product via parts and whole (verified)
def product_symmetric(a, b):
m = (a + b) // 2 # the whole: their average
d = (b - a) // 2 # the part: the deviation from the average
return m*m - d*d # difference of squares
print(product_symmetric(18, 22)) # 396 (20^2 - 2^2)
print(product_symmetric(17, 23)) # 391 (20^2 - 3^2)This example runs in Gri-Learn on the web, where you can edit it and see the output.
Quiz
Using the parts-and-whole (average and deviation) method, what is 18 times 22?
- 400, because 20 times 20
- 396, because the average is 20 and the deviation is 2, so 20 squared - 2 squared = 400 - 4
- 404, because 400 + 4
- 360, because 18 times 20
Show the answer
396, because the average is 20 and the deviation is 2, so 20 squared - 2 squared = 400 - 4
18 and 22 are symmetric about their average 20, with deviation 2. The product is the difference of squares: 20 squared - 2 squared = 400 - 4 = 396, which equals 18 times 22. Option A gives only 20 times 20 (the square of the whole), forgetting to subtract the deviation's square. Option C adds instead of subtracts; the identity is (m - d)(m + d) = m squared MINUS d squared. Option D computes a different product entirely (18 times 20). The method: average is the whole (20), gap is the part (2), product is whole squared minus part squared.
Think first
Why does treating the numbers as whole-and-part help?
Why is 'average squared minus deviation squared' easier than just multiplying 18 by 22 directly? Then tap.
Show the answer
Because it replaces an awkward two-digit multiplication with a FAMILIAR square you already know plus a tiny subtraction, exploiting the symmetry the two numbers happen to have. Multiplying 18 by 22 directly means a full long multiplication with carries. But recognising that they sit an equal distance either side of 20 lets you use the difference-of-squares identity: their product is 20 squared minus 2 squared. Now 20 squared is 400, a fact you know instantly, and 2 squared is just 4, so the whole thing becomes 400 minus 4 = 396, arithmetic you can do in your head. The 'whole' (the average, 20) is chosen precisely because it is an easy, round anchor whose square you know, and the 'part' (the deviation, 2) is small, so its square is trivial. This is the same theme running through the unit: find the convenient structure in a problem, an easy anchor and a small adjustment, and let it carry the work, rather than grinding through the raw calculation. The method only helps when the numbers are symmetric about an easy middle (or can be made so), but that situation is common, and spotting it turns a hard product into an easy square minus a small square. Anchor at the whole, correct by the part.
Summary
Key takeaways
- Vyastisamanstih ('the parts and the whole') plays the whole against its parts.
- For two numbers symmetric about a middle value, use the whole (their average m) and the part (the deviation d).
- Their product is the difference of squares: (m - d)(m + d) = m squared - d squared.
- 18 times 22: m = 20, d = 2, so 400 - 4 = 396; 17 times 23: m = 20, d = 3, so 400 - 9 = 391.
- This replaces an awkward multiplication with an easy square minus a small square, using the numbers' symmetry.
- The book uses parts-and-whole more broadly (e.g. factorisation); consult the prescribed text.
- Memory hook: average is the whole, gap is the part, product is whole squared minus part squared.