Question

The length of a rectangle is 2/3 mile. If the area is 1/2 square mile, what is the width?

Answer 1

Using first principles: `" area "=1/2 " square mile"`

Once you are used to these you should be doing some of the work in your head and write just a few lines.

Explanation:

If any doubt and the question lends itself to it draw a little sketch. This helps in understanding what to do. Does not need to all neat and fancy like mine. A very quick and rough sketch will do.

`"area " ="length "xx" width" color(magenta)( larr" Important to remember this")`
`color(white)("d")1/2color(white)("dd")= color(white)("d")2/3color(white)("dd.d")xx color(white)("dd")wcolor(white)("dd") larr " Given in the question"`
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`color(blue)("Determine the value of "w)`

`color(green)(1/2=2/3w)`

This is the same as:

`color(green)(1/2=2xx1/3xxw)`

Multiply both sides by `color(red)(1/2)`

`color(green)(ubrace(1/2color(red)(xx1/2))=ubrace(2color(red)(xx1/2))xx1/3xxw)`

`color(green)(color(white)("dd")1/4color(white)("dd")=color(white)("d..")ubrace(2/color(red)(2)color(white)("dd")xx1/3)xxw)`

`color(green)(color(white)("dd")1/4color(white)("dd")=color(white)("ddddd")1/3color(white)("ddd")xxw)`

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

`color(green)( color(red)(3)/4 = ubrace(color(red)(3)/3xx w)`

`color(green)(3/4 = color(white)("dd")w )`
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`color(blue)("Determine the area")`

`"area = length x width"`

`color(white)("d")a= 2/3xx3/4color(white)("dd") =color(white)("dd") (2xx3)/(3xx4) color(white)("dd")=color(white)("dd")6/12`

`a = (6-:6)/(12-:6)color(white)("dd") =color(white)("dd") 1/2` square mile