This answer demonstrates how to install the driver for your printer using the Mac operating Big Sur. The version of macOS used in the answer is 11.6.1. This answer was also tested using macOS Monterey 12.1 with success. However, the procedures and results were slightly different.
When a HP printer is added to the list shown in Printers & Scanners pain of the System Preferences application, the driver chosen by default may not be the newest of the available versions. A newer driver version may be obtained by selecting the driver while manually adding the printer.
I do not have a HP Ink Tank 115. To provide this answer, I will instead substitute an HP Office Pro 8600. Normally the printer would be directly connected to my Mac. However, this would require moving the printer, which I am not will to do in order to answer this question. You can connect your HP Ink Tank 115 to your Mac through a USB cable. I will access the HP Office Pro 8600 through printer sharing with a 2011 iMac named
Marlin. The 2011 iMac is using macOS High Sierra 10.13.6
I started by opening the Printer & Scanners pane of System Preferences. Next I selected the
+ button to add a printer, then highlighted the
Officejet Pro 8600 @ Marlin, as shown below.
Select Software... for "Use:" to cause the Printer Software popup to appear, as shown below. Note, the
Office Pro 8600 was not listed. Therefore, if I added this printer, the driver from 2011 iMac would have been used. I desired instead to get the driver from HP. At this point, I closed all windows.
By going to HP's websites, I can determined that I needed download and execute the HP Easy Start application version 220.127.116.11124. Conveniently, the same download is specified for the HP Ink Tank 115. You can confirm the download is the same by going to the link in your question and changing the selection from macOS 10.13 to macOS 11". The link to download is titled "Install HP Easy Start".
The following list of items pertain to the execution of the HP Easy Start application.
When asked to "Select the device you want to use and then click Coninue", the
Office Pro 8600 Marlin was already selected, as shown below.
The window below shows the software which was found for the HP Office Pro 8600. Note, installing Essential Software also installs the HP drivers. I chose the default to install the Essential Software and HP Easy Scan. I assume the HP Ink Tank 115 does not scan, so I would assume the HP Easy Scan application would not appear when a HP Ink Tank 115 is selected as the device.
The following popup appeared twice, which I allowed.
As shown below, there were popups related to system extension(s) from HP, which a I allowed.
When the popup below appeared, I selected the
Not Now button, because the HP Easy Start application had not completed. After this application completed, I restarted the Mac.
When the window below appeared, I selected the
Add Device... button.
When the window below appeared, I did not print a test page. In other words, I did not select the
Print Test Page button.
Using the Print Driver Downloaded from HP
The printer driver version can be determined by first opening the Printer & Scanners pane of System Preferences. Next, highlight
Officejet Pro 8600 @ Marlin in the Printers list, then select the
Open Print Queue... button. Finally, select the setting (gear) button to view the driver version under the
General tab, as shown below. The print driver version is 4.1.1.
To switch to the print driver version which was downloaded and installed when the HP Easy Start application was executed, the
Officejet Pro 8600 @ Marlin printer would have to be removed from the list, then manually added back in. To insure the print driver downloaded from HP is used, you have select the driver when manually adding the printer, as shown below.
The image below shows the version of the print driver is now 4.7.0.