How do I install Ubuntu 14.04 without losing data on a Windows 7 PC (physically)?

You should install Ubuntu on a separate partition so that you won't lose any data. The most important thing is you should create a separate partition for Ubuntu manually, and you should select it while installing Ubuntu.

First create a separate partition for Ubuntu while running Windows (like a partition with more than 10 GB). Also create a small partition for the swap partition (size is equal to your RAM size).

• You should know the partition sizes you created so that you can identify them easily while installing Ubuntu.

See the below images - I picked them by googling so they won't be same as yours (like partition sizes and number of partitions).

The second important thing is while installing Ubuntu select "Something else".

It will allow you to select your manually created partition to install Ubuntu onto it.

Now select and edit the partition you created for Ubuntu. Be careful in selecting. Check twice that you selected the correct partition, not some other partition as you may lose data like before.

Select the format partition, use as mount point as shown in the image. The size is up to you.

After that, select the swap partition, and edit it as shown in the image (size is your choice).

After that select the "Device for boot loader installation" at the bottom and select the one with /dev/sda at the beginning.

Now you can select "install now" at the right bottom.

It seems lengthy, but it's really easy once you know this installation.

I hope that helps.