Whakaoti i te (1-1/3)(1-1/5)(1-1/7)/(1+frac{1{3})(1+frac{1}{5})(1+frac{1}{7})}

$$\frac { ( 1 - \frac { 1 } { 3 } ) ( 1 - \frac { 1 } { 5 } ) ( 1 - \frac { 1 } { 7 } ) } { ( 1 + \frac { 1 } { 3 } ) ( 1 + \frac { 1 } { 5 } ) ( 1 + \frac { 1 } { 7 } ) }$$

Aromātai

$$\frac{1}{4}=0.25$$

Ngā Upane Otinga

Me tahuri te 1 ki te hautau \frac{3}{3}.

$$\frac{\left(\frac{3}{3}-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tā te mea he rite te tauraro o \frac{3}{3} me \frac{1}{3}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{\frac{3-1}{3}\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tangohia te 1 i te 3, ka 2.

$$\frac{\frac{2}{3}\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{\frac{2}{3}\left(\frac{5}{5}-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{1}{5}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{\frac{2}{3}\times \frac{5-1}{5}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tangohia te 1 i te 5, ka 4.

$$\frac{\frac{2}{3}\times \frac{4}{5}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Me whakarea te \frac{2}{3} ki te \frac{4}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\frac{2\times 4}{3\times 5}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Mahia ngā whakarea i roto i te hautanga \frac{2\times 4}{3\times 5}.

$$\frac{\frac{8}{15}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Me tahuri te 1 ki te hautau \frac{7}{7}.

$$\frac{\frac{8}{15}\left(\frac{7}{7}-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tā te mea he rite te tauraro o \frac{7}{7} me \frac{1}{7}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{\frac{8}{15}\times \frac{7-1}{7}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tangohia te 1 i te 7, ka 6.

$$\frac{\frac{8}{15}\times \frac{6}{7}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Me whakarea te \frac{8}{15} ki te \frac{6}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\frac{8\times 6}{15\times 7}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Mahia ngā whakarea i roto i te hautanga \frac{8\times 6}{15\times 7}.

$$\frac{\frac{48}{105}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Whakahekea te hautanga \frac{48}{105} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$\frac{\frac{16}{35}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Me tahuri te 1 ki te hautau \frac{3}{3}.

$$\frac{\frac{16}{35}}{\left(\frac{3}{3}+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tā te mea he rite te tauraro o \frac{3}{3} me \frac{1}{3}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{\frac{16}{35}}{\frac{3+1}{3}\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tāpirihia te 3 ki te 1, ka 4.

$$\frac{\frac{16}{35}}{\frac{4}{3}\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{\frac{16}{35}}{\frac{4}{3}\left(\frac{5}{5}+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{1}{5}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{\frac{16}{35}}{\frac{4}{3}\times \frac{5+1}{5}\left(1+\frac{1}{7}\right)}$$

Tāpirihia te 5 ki te 1, ka 6.

$$\frac{\frac{16}{35}}{\frac{4}{3}\times \frac{6}{5}\left(1+\frac{1}{7}\right)}$$

Me whakarea te \frac{4}{3} ki te \frac{6}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\frac{16}{35}}{\frac{4\times 6}{3\times 5}\left(1+\frac{1}{7}\right)}$$

Mahia ngā whakarea i roto i te hautanga \frac{4\times 6}{3\times 5}.

$$\frac{\frac{16}{35}}{\frac{24}{15}\left(1+\frac{1}{7}\right)}$$

Whakahekea te hautanga \frac{24}{15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$\frac{\frac{16}{35}}{\frac{8}{5}\left(1+\frac{1}{7}\right)}$$

Me tahuri te 1 ki te hautau \frac{7}{7}.

$$\frac{\frac{16}{35}}{\frac{8}{5}\left(\frac{7}{7}+\frac{1}{7}\right)}$$

Tā te mea he rite te tauraro o \frac{7}{7} me \frac{1}{7}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{\frac{16}{35}}{\frac{8}{5}\times \frac{7+1}{7}}$$

Tāpirihia te 7 ki te 1, ka 8.

$$\frac{\frac{16}{35}}{\frac{8}{5}\times \frac{8}{7}}$$

Me whakarea te \frac{8}{5} ki te \frac{8}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\frac{16}{35}}{\frac{8\times 8}{5\times 7}}$$

Mahia ngā whakarea i roto i te hautanga \frac{8\times 8}{5\times 7}.

$$\frac{\frac{16}{35}}{\frac{64}{35}}$$

Whakawehe \frac{16}{35} ki te \frac{64}{35} mā te whakarea \frac{16}{35} ki te tau huripoki o \frac{64}{35}.

$$\frac{16}{35}\times \frac{35}{64}$$

Me whakarea te \frac{16}{35} ki te \frac{35}{64} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{16\times 35}{35\times 64}$$

Me whakakore tahi te 35 i te taurunga me te tauraro.

$$\frac{16}{64}$$

Whakahekea te hautanga \frac{16}{64} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$\frac{1}{4}$$

Tauwehe

$$\frac{1}{2 ^ {2}} = 0.25$$