You are right - well, mostly right. Apps that are listed in the multi-task bar are generally not running. The iPad's operating system, iOS, is very good at 'memory management'. Only the most recent apps are even kept in memory - the rest are removed and only their 'context' saved - for example, what page you were surfing in Safari or what document you had open in Pages. When you open up the multi-task bar and select an app that hasn't been used for a while, the iPad reloads the app and uses the context information to re-instate the app into exactly the same condition as when you last quit it.
However, some limited background activity is allowed by the iPad so, for example, music apps can continue to play music even when they've been 'unloaded' and the email app Mail can periodically check the email - again, even though it's been off loaded.
But the iPad operating system is not really a multi-tasking operating system. If you have an app that's performing a complex calculation and you switch to another app, that calculation will be frozen and will not continue while the app is unload. The iPad's operating system is more like a task switcher - it allows the user to seamlessly and quickly switch between apps.
Why did Apple do this? Precisely for the reason that your daughter mentioned. If all the apps were left running in the background they would consume system resource - battery power, CPU performance - and the iPad would not appear as wonderfully responsive as it does and it wouldn't have the great battery life that it does.
So, in a way, both of you are right - but you are 'more right' than she is!! Generally, apps that have been swapped out, consume little or no battery power or CPU resource.
Tim
