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Problem: Not for all apps, but for some apps or others you are facing "Package file has a bad manifest" when app is downloaded and is being installed.
Solution 1 (Memory Card): If you have a memory card, take it out from your mobile. Now, try to install your app. (If this solves your problem then your memory card might be corrupted). If your app is installed this time then simply plug back your memory card and format it.
Solution 2 (Phone): If memory card did not solve your problem then it is probably in fine condition. The problem might be in your phone itself. Simply do Factory Reset your phone and try installing the apps now.
Note: For some reason, I couldn't install some of my favorite apps which was working before on my Samsung Galaxy Duos 2 (Android 4.2). I searched a lot and spent lot of time to find its solution but nothing worked and I did not find its solution. After that, I prepared myself to battle this and find its solution because it was driving me crazy for not getting my favorite apps installed which was working fine before.
Reason & Solution for Me: I did factory reset several times, updated my account, removed google play store services and some other tricks but nothing worked. At the end, I figured it out that the app which was giving me error "Package file has a bad manifest" is getting installed in my memory card after downloading and memory card has some problem. I simply formatted the card and then it solved the problem like a charm.