MHB How to Simplify Complex Problems: Expert Tips and Tricks

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To simplify complex problems, start by clearing complex fractions, as demonstrated with the example of \(\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}}\). Multiply by a common factor to simplify the expression further, transforming it into \(\frac{y - x}{y + x}\). Then, apply a strategic multiplication to achieve a denominator of the form \(y^2 - x^2\), which allows for cancellation in subsequent steps. This methodical approach helps break down complex problems into manageable parts. The discussion emphasizes the importance of systematic simplification techniques.
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Hey guys,

I know it's pretty simple but after an hour of thinking on it i just can't get how to simplify this. Can someone help me?

Thank you in advance!
 

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Canzy said:
Hey guys,

I know it's pretty simple but after an hour of thinking on it i just can't get how to simplify this. Can someone help me?

Thank you in advance!
Start by clearing the complex fractions. For example:
[math]\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}}[/math]

[math]= \frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}} \cdot \frac{xy}{xy}[/math]

[math]= \frac{y - x}{y + x}[/math]

Now we pull a trick. Look at the fraction just above. I'm going to multiply the fraction by (y - x)/(y - x) to get the denominator to be of the form y^2 - x^2. That will cancel out the y^2 - x^2 in the last factor.

Can you finish this?

-Dan
 
Yes, i can. Thank you!
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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