Question

How do you find the quotient `-9/10div3`?

Answer 1

`-3/10`

Explaining it takes a lot longer than doing the maths.

Explanation:

`color(blue)("Shortcut method")`

.............................................................
`color(magenta)("Using 2 examples I am explaining a principle I am about to adopt:")`

`2xx3" is the same as "3xx2`

In the same way we have:

`" "color(red)(2/3)xx color(blue)(16/32)`

`= (color(red)(2)xx color(blue)(16))/(color(red)(3)xx color(blue)(32))`

`= (color(red)(2)xx color(blue)(16))/(color(blue)(32)xx color(red)(3))`

`= (color(red)(2))/(color(blue)(32))xx(color(blue)(16))/(color(red)(3))`
........................................................................

Write the question as:`" "-(9/10-:3/1)`

Invert the divisor (that which you are dividing by) and change divide to multiply.

`-(9/10xx1/3)`

This is the same as:

`-(9/3xx1/10) " "->" "-((cancel(9)^3)/(cancel(3)^1) xx 1/10)`

`=-3/10`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("First principle method")`

A fraction consists of `("count")/("size indicator") ->("numerator")/("denominator")`

You can not directly divide the counts unless the size indicators are the same.

Multiply by 1 and you do not change the overall value. However, 1 comes in many forms so you can change the way a value looks without changing its overall value.

`color(green)(-(9/10-:[3/1color(magenta)(xx1)])" "=" "-(9/10-:[3/1color(magenta)(xx10/10)])`

Giving:

`-(9/10-:30/10)`

This gives the same answer as:

`-(9-:30) = -9/30 = -(9-:3)/(30-:3)`

`-(cancel(9)^3)/(cancel(30)^10) = -3/10`