Question

Two-thirds of a number is five-sixths. How do you find the number?

Answer 1

The number is `15/12`

Explanation:

Let the unknown value be `x`

`2/3xx x=5/6`

Multiply both sides by `color(red)(3/2)`

`color(green)(2/3color(red)(xx3/2)xx x=5/6color(red)(xx3/2)`

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Consider just the left hand side

Using the principle (by example) that `2xx6=6xx2` we have:

`(2xx3)/(3xx2)xx x`

`(3xx2)/(3xx2)xx x`

`3/3xx2/2xx x`

`1xx1xx x" "->" "x`
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Back to the question

So `x=5/6xx3/2`

`x=15/12`
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Check

left hand side `->" "(cancel(2)^1)/(cancel(3)^1)xx(cancel(15)^5)/(cancel(12)^6) = 5/6` as per the right hand side.

Answer 2

`5/4`

Explanation:

`"let x represent the number"`

`"then " 2/3" of " x =5/6`

`"replace of by multiplication"`

`rArr2/3 xx x=5/6`

`rArr(2x)/3=5/6`

`color(blue)"cross multiplying"`

`rArr2x xx6=5xx3`

`rArr12x=15`

`"divide both sides by 12"`

`(cancel(12) x)/cancel(12)=15/12`

`rArrx=15/12=5/4larr" in simplest form"`