Whakaoti i te 1/c_{T}=1/2+1/9+1/2+frac{1}{6}+frac{1}{12}

$$\frac { 1 } { c _ { T } } = \frac { 1 } { 2 } + \frac { 1 } { 9 } + \frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 }$$

Whakaoti mō c_T

$$c_{T}=\frac{36}{49}\approx 0.734693878$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe c_{T} ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 36c_{T}, arā, te tauraro pātahi he tino iti rawa te kitea o c_{T},2,9,6,12.

$$36=36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{9}+36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakareatia te 36 ki te \frac{1}{2}, ka \frac{36}{2}.

$$36=\frac{36}{2}c_{T}+36c_{T}\times \frac{1}{9}+36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakawehea te 36 ki te 2, kia riro ko 18.

$$36=18c_{T}+36c_{T}\times \frac{1}{9}+36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakareatia te 36 ki te \frac{1}{9}, ka \frac{36}{9}.

$$36=18c_{T}+\frac{36}{9}c_{T}+36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakawehea te 36 ki te 9, kia riro ko 4.

$$36=18c_{T}+4c_{T}+36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Pahekotia te 18c_{T} me 4c_{T}, ka 22c_{T}.

$$36=22c_{T}+36c_{T}\times \frac{1}{2}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakareatia te 36 ki te \frac{1}{2}, ka \frac{36}{2}.

$$36=22c_{T}+\frac{36}{2}c_{T}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakawehea te 36 ki te 2, kia riro ko 18.

$$36=22c_{T}+18c_{T}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Pahekotia te 22c_{T} me 18c_{T}, ka 40c_{T}.

$$36=40c_{T}+36c_{T}\times \frac{1}{6}+36c_{T}\times \frac{1}{12}$$

Whakareatia te 36 ki te \frac{1}{6}, ka \frac{36}{6}.

$$36=40c_{T}+\frac{36}{6}c_{T}+36c_{T}\times \frac{1}{12}$$

Whakawehea te 36 ki te 6, kia riro ko 6.

$$36=40c_{T}+6c_{T}+36c_{T}\times \frac{1}{12}$$

Pahekotia te 40c_{T} me 6c_{T}, ka 46c_{T}.

$$36=46c_{T}+36c_{T}\times \frac{1}{12}$$

Whakareatia te 36 ki te \frac{1}{12}, ka \frac{36}{12}.

$$36=46c_{T}+\frac{36}{12}c_{T}$$

Whakawehea te 36 ki te 12, kia riro ko 3.

$$36=46c_{T}+3c_{T}$$

Pahekotia te 46c_{T} me 3c_{T}, ka 49c_{T}.

$$36=49c_{T}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$49c_{T}=36$$

Whakawehea ngā taha e rua ki te 49.

$$c_{T}=\frac{36}{49}$$