gvaldytist

2022-06-29

Simple fraction calculation
I'm trying to write this expression as one fraction:
$\begin{array}{rl}\frac{5}{y+1}\cdot \frac{1}{5}+\frac{y}{\frac{y+1}{3}}& =\frac{5y}{5y+5}+\frac{3y}{y+1}\\ & =\frac{5{y}^{2}+5+15{y}^{2}+15y}{5{y}^{2}+10y+5}\\ & =\frac{20{y}^{2}+15y+5}{5{y}^{2}+10y+5}\\ & =\frac{4{y}^{2}+3y+1}{2y}.\end{array}$
Am I doing this in the right way? Any hints about what to do next, please?

Sawyer Day

Key idea to remember is that
$\frac{1}{\frac{a}{b}}=\frac{b}{a}.$
So you get
$\frac{5}{y+1}\cdot \frac{1}{5}+\frac{y}{\frac{y+1}{3}}=\frac{1}{y+1}+\frac{3y}{y+1}=\frac{1+3y}{y+1}.$

Eden Solomon

Your mistake was in simplifying from step 3 to 4. You cannot simplify each term like that. You can only simplify factors.
For example $\frac{5}{13}$ could be rewritten as $\frac{2+3}{4+9}$ Using your method this would simplify to $\frac{1+1}{2+3}$ or $\frac{2}{5}$ But we know that $\frac{5}{13}$ isn't equal to $\frac{2}{5}$
You can only simplify by factoring and canceling out factors. YOu should factor out the extra (y+1) that you have in the numerator and denominator. Alternatively you could simplify the first fraction before you add it making your life so much simpler.

Do you have a similar question?