Fractions are Hard!

3.7 Algebra connections

In an advanced algebra course students are often asked to work with complicated expressions of the following ilk

.

\(\dfrac{\dfrac{1}{x}+1}{\dfrac{3}{x}}\).

 

We can make it look friendlier by following exactly the same technique of our fraction work. In this example, let’s multiply the numerator and denominator each by \(x\). (Do you see why this is a good choice?) We obtain

 

\(\dfrac{\left(\dfrac{1}{x}+1\right)\times x}{\left(\dfrac{3}{x}\right)\times x}=\dfrac{1+x}{3}\).

 

and \(\dfrac{1+x}{3}\) is much less scary.

 

As another example, given

 

\(\dfrac{\dfrac{1}{a}-\dfrac{1}{b}}{ab}\)

 

one might find it helpful to multiply the numerator and the denominator each by \(a\) and then each by \(b\).

 

\(\dfrac{\left(\dfrac{1}{a}-\dfrac{1}{b}\right)\times a\times b}{ab\times a \times b}=\dfrac{b-a}{a^{2}b^{2}}\),

 

 

and for

 

\(\dfrac{\dfrac{1}{\left(w+1\right)^{2}-2}}{\dfrac{1}{\left(w+1\right)^{2}+5}}\)

 

 

it might be helpful to multiply top and bottom each by \(\left(w+1\right)^{2}\)

 

 

\(\dfrac{\dfrac{1}{\left(w+1\right)^{2}-2}}{\dfrac{1}{\left(w+1\right)^{2}+5}}=\dfrac{1-2\left(w+1\right)^{2}}{1+5\left(w+1\right)^{2}}\).

 

 

 

Question: Make each of the following expressions look friendlier.

 

a) \(\dfrac{2-\dfrac{1}{x}}{1+\dfrac{1}{x}}\).

 

b) \(\dfrac{\dfrac{1}{x+h}+5}{\dfrac{1}{x+h}}\)

 

c)\(\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{b}}\)

 

d) \(\dfrac{\dfrac{1}{x+a}-\dfrac{1}{x}}{a}\)

 

e) \(\dfrac{1}{s^{-2}}\)

 

 

Question:

 

a)    Find the two numbers \(x\) that satisfy the equation \(x=1+\dfrac{1}{x}\).

The larger of these two numbers is called “the golden ratio.”

 

b)    Let \(x\) be the golden ratio. Show that \(x\) also satisfies the equation \(x=1+\dfrac{1}{1+\dfrac{1}{x}}\).

 

c)    Show that \(x\) also satisfies \(x=1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{x}}}}}}\).

 

It seems that the golden ratio is connected to the sequence of fractions:

 

\(1\);   \(1+\dfrac{1}{1}=2\);   \(1+\dfrac{1}{1+\dfrac{1}{1}}=\dfrac{3}{2}\);   \(1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1}}}=\dfrac{5}{3}\); …

 

d)    Compute the next five fractions in this sequence.

 

e)    Explain why the numerator of one fraction is sure to be the denominator of the next.

 

f)    Explain why the numerator of each fraction is sure to be the sum of the denominator and numerator of its previous fraction.

 

g)    (INTERNET RESEARCH): What famous sequence of numbers seems to be appearing within these fractions?

 

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