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How to check if two multiplications are equal to each other or greater or lesser without actually multiplying them?
For example, compare (254)(847) and (383)(536)

EDIT:
While trying to find a rule i got one
(5)(11) < (6)(10)
or
(x)(y) < (x+1)(y-1) when y > x > 0
and another rule is that if adding and subtracting 1 equates them the difference is one
(3)(5) + 1 = (4)(4)
(x)(y) + 1 = (x+1)(y-1) when y + 2 = x , y > x >= 0

  • 4
    I'd say add up their logarithms and compare, but that's "mosquito-nuking" territory...2010-12-14
  • 2
    Approximate each factor with the nearest round number that you can easily perform multiplications with: $(254)(847)\approx(250)(850)=212.500$ and $(383)(536)\approx(400)(550)=220.000$. If the approximation is good, the corrections will be many orders smaller than the estimate and can safely be neglected. Now, since the results are pretty close in this case, I think you'll have to compute the corrections anyway. Another trick would be to write them as $AB=((A+B)^2-A^2-B^2)/2$, maybe this can help.2010-12-14
  • 0
    Found another trick, instead of comparing the products, compare $847/383$ and $536/254$.2010-12-14
  • 1
    I am preparing for GRE test and it includes many questions like that. In GRE there is no time to multiply and check. Easier way should be a rule.2010-12-15
  • 0
    @raskolnikov and which multiplication will be bigger after using second method?2010-12-15
  • 0
    Divisibility Rules can also be really handy for simplifying both sides of the inequality, especially if both sides have a factor of 2 as in the example. It is interesting that my suggestion for comparing a product is to factor it further...2010-12-15
  • 3
    @LifeH2O: I know you didn't want to multiply, but if you multiply from the left you get the most significant digits first. You can then stop as soon as you can see the difference. Memorizing special rules is the road to madness.2010-12-16
  • 0
    @Ross:That's a beautiful advice :-)2011-08-21

6 Answers 6