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