Simplifying (an equation with a fraction within a fraction)

I find it difficult to simplify an equation which has a fraction within a fraction. For example, I do not understand how to simplify this:

\frac{1}{\frac{1}{x+1}+1}

Apparently, this can be rewritten as this:

\frac{x+1}{x+2}

But I do not understand the process used to achieve this. I would be grateful if someone could take me through the process of simplifying this equation and perhaps help me to understand a general method for simplifying equations involving a fraction within a fraction.

Reply 1

crazyflight

Stikki

I find it difficult to simplify an equation which has a fraction within a fraction. For example, I do not understand how to simplify this:

\frac{1}{\frac{1}{x+1}+1}

Apparently, this can be rewritten as this:

\frac{x+1}{x+2}

But I do not understand the process used to achieve this. I would be grateful if someone could take me through the process of simplifying this equation and perhaps help me to understand a general method for simplifying equations involving a fraction within a fraction.

If you multiply the entire fraction by (x+1)/(x+1) [which does not change the value since it is equal to 1] you find that the numerator becomes x+1 and the denominator becomes 1+x+1=x+2

Hence we can write it as

\frac{x+1}{x+2}

Reply 2

LittleBill

10

Another way is

Find a common denominator for the bottom

\frac{1}{\frac{1+x+1}{x+1}}

Simplify

\frac{1}{\frac{x+2}{x+1}}

Then just remembering division of fractions, 1 divided by a fraction is the same as 1 times the reciprocal.

Reply 3

Pheylan

take the denominator:

(1/(x+1)) + 1

you can multiply the 1 by (x+1)/(x+1) to make it:

(1/(x+1)) + (x+1)/(x+1)

now both fractions in the denominator have a common denominator of x+1, so turn it into one fraction:

(1/(x+1)) + (x+1)/(x+1) = (1+(x+1))/(x+1)

which simplifies to (x+2)/(x+1)

so the fraction is now 1/((x+2)/(x+1))

as i assume you know, you can take the denominator of the denominator and stick it on the numerator to turn this into:

(x+1)/(x+2)

voila

Reply 4