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October 9th, 2018, 12:56 AM   #1
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stick ratio problem

A stick is broken up into 2 parts. The ratio of the lengths of the bigger part and the smaller part is equal to the ratio of the length of the full stick and bigger part. What is this ratio?

How to solve this stick ratio problem?

Last edited by skipjack; October 9th, 2018 at 03:08 AM.
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October 9th, 2018, 01:13 AM   #2
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The answer is known as the golden ratio.

You need to solve the equation x = (1 + x)/x, where x the ratio to be found is x:1.

This equation is equivalent to x² - x - 1 = 0. Can you solve that?
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October 9th, 2018, 02:11 AM   #3
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Quote:
Originally Posted by skipjack View Post
The answer is known as the golden ratio.

You need to solve the equation x = (1 + x)/x, where x the ratio to be found is x:1.
How you got x = (1 + x)/x? How can you tell answer is golden ratio?

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This equation is equivalent to x² - x - 1 = 0. Can you solve that?
I can solve it.
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October 9th, 2018, 03:30 AM   #4
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By the definition of x, (bigger part):(smaller part) is x:1. Taking the smaller part's length to be 1 unit of length, the larger part has length x units and so the original full stick had length 1+x units.

Hence the question implies x:1 = (1 + x):x, which is equivalent to x = (1 + x)/x.

By its definition, x is positive, so x is the positive root of the quadratic equation.

The golden ratio (or some other term with equivalent meaning) has been known about and studied for thousands of years.
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October 9th, 2018, 03:40 AM   #5
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Quote:
Originally Posted by skipjack View Post
By the definition of x, ......
What is that definition of x?
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October 9th, 2018, 04:08 AM   #6
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The question asks for the ratio of the lengths of the bigger part and the smaller part, which can be written as (bigger part's length):(smaller part's length) or as (bigger part's length)/(smaller part's length):1.

I defined x by stating "where x the ratio to be found is x:1", which was intended to be "where the ratio to be found is x:1". I originally typed "where x is the ratio to be found", then slightly mis-edited it.
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