As a professional, how well can you organize your data?
Whenever I pose simplifications in my workshops such as:
{1-(1/4)}×{1-(1/9)}×{1-(1/16)}×...×{1-(1/225)},
the most common response is to find the individual fractions and then multiply
by reducing all the fractions together, or by, erm, using a calculator.
(3/4)×(8/9)×(15/16)×...×(224/225)
I dislike calculators, especially for puny calculations as the above.
But before plunging into it hurrily, if one could first take some time to
observe and organize this data, one might find a quicker way to achieve the
answer.
You all remember from elementary algebra that a²-b² = (a+b)(a-b), for two
numbers a and b.
The given problem could be broken down as:
(1+½)(1-½)×(1+⅓)(1-⅓)×(1+¼)(1-¼)×...×{1+(1/15)}{1-(1/15)}
= (3/2)(1/2)×(4/3)(2/3)×(5/4)(3/4)×...×(16/15)(14/15)
= (1/2)×(3/2)(2/3)×(4/3)(3/4)×(5/4)(4/5)×...×(15/14)(14/15)×(16/15)
If you lay these down on a piece of paper and remember simple cancellation,
you'll observe that everything in between becomes 1, and the final answer is
(1/2)(16/15) = 8/15.
The rearrangement is vital to visualize the pattern.
A well-organized set of data is powerful enough to often make useful patterns
visible in problems which otherwise seemed cumbersome.
Be quick, but don't hurry.
#riosophy
Posted by Supriyo Banerjee on LinkedIn
link: linkedin.com/in/supriyo-banerjee
Whenever I pose simplifications in my workshops such as:
{1-(1/4)}×{1-(1/9)}×{1-(1/16)}×...×{1-(1/225)},
the most common response is to find the individual fractions and then multiply
by reducing all the fractions together, or by, erm, using a calculator.
(3/4)×(8/9)×(15/16)×...×(224/225)
I dislike calculators, especially for puny calculations as the above.
But before plunging into it hurrily, if one could first take some time to
observe and organize this data, one might find a quicker way to achieve the
answer.
You all remember from elementary algebra that a²-b² = (a+b)(a-b), for two
numbers a and b.
The given problem could be broken down as:
(1+½)(1-½)×(1+⅓)(1-⅓)×(1+¼)(1-¼)×...×{1+(1/15)}{1-(1/15)}
= (3/2)(1/2)×(4/3)(2/3)×(5/4)(3/4)×...×(16/15)(14/15)
= (1/2)×(3/2)(2/3)×(4/3)(3/4)×(5/4)(4/5)×...×(15/14)(14/15)×(16/15)
If you lay these down on a piece of paper and remember simple cancellation,
you'll observe that everything in between becomes 1, and the final answer is
(1/2)(16/15) = 8/15.
The rearrangement is vital to visualize the pattern.
A well-organized set of data is powerful enough to often make useful patterns
visible in problems which otherwise seemed cumbersome.
Be quick, but don't hurry.
#riosophy
Posted by Supriyo Banerjee on LinkedIn
link: linkedin.com/in/supriyo-banerjee