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Sometimes we can't be equal, but we can aim for fair. It's a great opportunity for your younger daughter, she shouldn't miss out. For your older daughter, while she may be a little annoyed, hopefully she'll see the opportunity her little sister has & will encourage her. I think evening things with your older daughter will be positively received.
I would send my child to the private school. She's obviously very smart and has great potential. Im sure there are ways to give your other daughter something extra
I went to a public school with a friend who’s sister (only 1 year younger) went to a private school. Both girls were happy with their schools and didn’t mind going to different schools. I don’t think it was about money or scholarships for them- they chose the school they wanted to attend before year seven and happened to choose differently. They also rarely had pupil free school days on the same days so had no distractions at home to be able to study or allowed their mother to have some one on one time with each daughter.
Ask your older daughter how she feels about it. You could let your older daughter know that the students (and teachers) at the private school aren’t better than public school students and their fancy uniforms are just for show. They have to promote themselves as better because they’re a business and need numbers.
Have you considered a government advanced school or program ? They get better results than private without the cost.
I would take the opportunity, firstly because they don't come along often and secondly because if you don't give your daughter this opptunity she might hold it against you down the track. The older child might not care. You do need to look at the extra costs as uniforms and extra-curricula and trips can be expensive.
I would not do this especially if they are close in age. This will also bring financial and emotional strain to your family and will have lifelong repercussions for everyone. Your child can still do well in any school.
Keeping up with the extra activities at the private school will also be a point of contention for your older daughter and will be a cause of ruining the relationship between sisters.
There are always ways to make up the equality for the other child, even later in life. But be aware there will be costs such as "overseas trips to Japan", if the child is studying Japanese, which you would not face in a public school. The fees are the tip of the iceberg. The networking for future job opportunities is one advantage of private school education.