Theory
Scale what you already know
If you happen to know that 16 times 16 is 256, then 16 times 32 should be easy, because 32 is just double 16, so the answer is double 256. That reasoning by ratio is the sutra Anurupyena, 'proportionately'.
It captures a simple, powerful idea: reuse a known result and scale it in proportion to reach a related one. It also lets the Nikhilam method use a convenient 'working base'. This lesson teaches the accessible proportion form, with verified examples, from Bharati Krishna Tirtha's 1965 book.
Theory
Proportion in action
Start from a known product: 16 times 16 = 256.
- 16 times 32: since 32 is 2 times 16, the product is 2 times 256 = 512. (And 16 times 32 is indeed 512.)
- 16 times 48: since 48 is 3 times 16, the product is 3 times 256 = 768. (And 16 times 48 is 768.)
Each new answer comes from the known one by a single multiplication, the ratio, instead of a fresh full calculation. That is proportion: change one factor by a ratio, and the product changes by the same ratio.
Practical
Proportional scaling in Python (verified)
def by_proportion(known_product, ratio):
# If a known product changes one factor by 'ratio', scale the product too.
return known_product * ratio
base = 16 * 16 # 256
print(by_proportion(base, 2)) # 16*32 -> 512
print(by_proportion(base, 3)) # 16*48 -> 768
# Because product(a, k*b) = k * product(a, b): change a factor by k, scale by k.This example runs in Gri-Learn on the web, where you can edit it and see the output.
Formula
Also: a working base for Nikhilam
Anurupyena has a second, related use. Nikhilam is easiest near a base like 100 or 1000, but sometimes your numbers sit near an awkward value like 50 or 200. You can use a convenient working base (a multiple or sub-multiple of the main base) and then adjust proportionately for the difference in scale.
We keep the focus here on the clear proportion idea, because it is the most transferable. For the detailed working-base multiplication method, consult the prescribed text. The unifying thread is the same either way: use a ratio to move between a convenient case and the one you actually want.
Quiz
You know that 16 times 16 = 256. Using Anurupyena (proportion), what is 16 times 32?
- 272, by adding 16 to 256
- 512, because 32 is 2 times 16, so the product is 2 times 256
- 256, because one factor is still 16
- 1024, because you square 256
Show the answer
512, because 32 is 2 times 16, so the product is 2 times 256
Since 32 is 2 times 16, changing that factor by a ratio of 2 scales the product by 2: 2 times 256 = 512, which is exactly 16 times 32. Option A adds 16 instead of applying the ratio; proportion multiplies, it does not add. Option C ignores that one factor doubled (16 times 32 is not the same as 16 times 16). Option D squares the known product for no reason. The proportion rule: if you multiply one factor by k, the product is multiplied by k, so doubling a factor doubles the answer.
Think first
Why is reasoning by proportion so useful in mental maths?
Why lean on a known result and a ratio, rather than just computing directly each time? Then tap.
Show the answer
Because it lets you TRADE a hard calculation for an easy one by anchoring to something you already know, which is often far faster and less error-prone. Direct computation treats every problem as brand new, but many problems are close relatives of ones you can do instantly. If you know a convenient anchor, like 16 times 16 = 256, or 25 times 4 = 100, then a whole family of related products (16 times 32, 16 times 48, 16 times 64) becomes a single easy multiplication by a ratio, rather than a full long multiplication each time. This is exactly how skilled mental calculators work: they keep a stock of known results and reach new answers by scaling, halving, doubling, and combining them proportionately. It also generalises well beyond arithmetic, proportion is the backbone of scaling recipes, converting units, adjusting quantities, and reading maps, so the habit of thinking 'this is that, times a ratio' pays off everywhere. And it connects to the Nikhilam working-base idea: when your numbers are near an awkward value, you compute near a convenient base and adjust proportionately, again trading a hard case for an easy one plus a ratio. So proportion is not just a trick for these examples; it is a general strategy, anchor to the known, scale to the wanted, that makes computation lighter. Know one, scale to many.
Summary
Key takeaways
- Anurupyena ('proportionately') scales a known result by a ratio to reach a related one.
- From 16 times 16 = 256: 16 times 32 = 2 times 256 = 512 (32 is double 16); 16 times 48 = 3 times 256 = 768.
- The rule: multiply one factor by k and the product is multiplied by k.
- In code, a related product is just known_product times the ratio.
- Anurupyena also lets Nikhilam use a convenient working base and adjust proportionately (consult the prescribed text).
- Proportion trades a hard calculation for a known result plus an easy ratio; it is a technique from the 1965 book.
- Memory hook: anchor to a known product, scale by the ratio.