Question

a part has broken on a machine and needs to be replaced .The replacement part must be between 71/16 cm and 71/6 cm long in order to fit.

Answer 1

Yes, it will fit

Explanation:

The red part of the detail is
`4(7)/(9)\ cm` long, the yellow part is
`2(1)/(3)\ cm` long.

In order to fit the conditions, both parts together should be no longer than
`7(1)/(6)\ cm` and no less than
`7(1)/(18)\ cm.`

Find the combined length of red and yellow parts:

`4(7)/(9)+2(1)/(3)\\ \\=(4+2)+\left((7)/(9)+(1)/(3)\right)\\ \\=6+\left((7)/(9)+(3)/(9)\right)\\ \\=6+(10)/(9)\\ \\=6+1(1)/(9)\\ \\=7(1)/(9)\ cm`

Note that

`7(1)/(9)=7(2)/(18)>7(1)/(18),`

so the combined length is no less than
`7(1)/(18)\ cm`

and

`7(1)/(9)=7(2)/(18)\\ \\7(1)/(6)=7(3)/(18)\\ \\7(2)/(18)<7(3)/(1)8\Rightarrow 7(1)/(9)<7(1)/(6),`

so the combined length is no more than
`7(1)/(6)\ cm`