Answer:
[tex]{\mathsf {n=\frac{2}{3}}}[/tex]
Step-by-step explanation:
In word problems like this, you have to break down the problem description into math form
This is an equation description so we need a variable to work with. Let's call this variable n
1/6th of this number can be translated to:
[tex]\displaystyle \mathsf {\frac{1}{6}n}[/tex]
1/6th of this number plus one-third translates to:
[tex]\displaystyle \mathsf {\frac{1}{6}n} + \frac{1}{3}[/tex]
1/6th of this number, plus one-third returns two-thirds of that number
returns is just a fancy way of saying equal to.
So returns two-thirds of that number translates to
= two-thirds of that number
Two-thirds of the number is
[tex]\displaystyle \mathsf {\frac{2}{3}n}[/tex]
So putting all this together we get the equation
[tex]\displaystyle \mathsf {\frac{1}{6}n} + \frac{1}{3} = \frac{2}{3}\mathsf n[/tex]
We have to now solve for the equation.
Bring all the n terms to the left side and the constant(1/3) to the right to solve for n
[tex]\displaystyle \mathsf{ Subtract\:}\frac{1}{3}\mathsf {\:from\:both\:sides}[/tex]
[tex]\mathsf {\frac{1}{6}n+\frac{1}{3}-\frac{1}{3}=\frac{2}{3}n-\frac{1}{3}}[/tex]
Simplify to get
[tex]\mathsf {\frac{1}{6}n=\frac{2}{3}n-\frac{1}{3}}[/tex]
Subtract [tex]\mathsf {\frac{2}{3}}[/tex] from both sides
[tex]\mathsf {\frac{1}{6}n-\frac{2}{3}n=\frac{2}{3}n-\frac{1}{3}-\frac{2}{3}n}[/tex]
Simplify to get
[tex]\mathsf{\frac{1}{6}n-\frac{2}{3}n= -\frac{1}{3}}[/tex]
The left side term is [tex]\mathsf {\frac{1}{6}n-\frac{2}{3}n}[/tex]
Factor out the common term [tex]n[/tex]:
[tex]\mathsf {n\left(\frac{1}{6}-\frac{2}{3}\right)}[/tex]
[tex]\mathsf {\frac{1}{6}-\frac{2}{3}=-\frac{1}{2} }[/tex]
So we get our equation as
[tex]\mathsf {-\frac{1}{2}n=-\frac{1}{3}}[/tex]
Multiply both sides by -2 to get
[tex]\boxed {\mathsf {n=\frac{2}{3}}}[/tex]
