Question

If 1/6 of a number,plus one third returns two-thirds of that number what is the number

please help!!!!

Answer 1

`{\mathsf {n=(2)/(3)}}`

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:

`\displaystyle \mathsf {(1)/(6)n}`

1/6th of this number plus one-third translates to:

`\displaystyle \mathsf {(1)/(6)n} + (1)/(3)`

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

`\displaystyle \mathsf {(2)/(3)n}`

So putting all this together we get the equation

`\displaystyle \mathsf {(1)/(6)n} + (1)/(3) = (2)/(3)\mathsf n`

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

`\displaystyle \mathsf{ Subtract\:}(1)/(3)\mathsf {\:from\:both\:sides}`

`\mathsf {(1)/(6)n+(1)/(3)-(1)/(3)=(2)/(3)n-(1)/(3)}`

Simplify to get

`\mathsf {(1)/(6)n=(2)/(3)n-(1)/(3)}`

Subtract
`\mathsf {(2)/(3)}` from both sides

`\mathsf {(1)/(6)n-(2)/(3)n=(2)/(3)n-(1)/(3)-(2)/(3)n}`

Simplify to get

`\mathsf{(1)/(6)n-(2)/(3)n= -(1)/(3)}`

The left side term is
`\mathsf {(1)/(6)n-(2)/(3)n}`
Factor out the common term
`n`:

`\mathsf {n\left((1)/(6)-(2)/(3)\right)}`

`\mathsf {(1)/(6)-(2)/(3)=-(1)/(2) }`

So we get our equation as

`\mathsf {-(1)/(2)n=-(1)/(3)}`

Multiply both sides by -2 to get

`\boxed {\mathsf {n=(2)/(3)}}`

Answer 2

`(2)/(3)`

Explanation:

let n be the number , then

`(1)/(6)` n +
`(1)/(3)` =
`(2)/(3)` n

multiply through by 6 ( the LCM of 6 and 3 ) to clear the fractions

n + 2 = 4n ( subtract n from both sides )

2 = 3n ( divide both sides by 3 )

`(2)/(3)` = n