We will have limited time in Osaka, and on one of those days--a very long day-- we would like to visit Hiroshima. Does anyone have a ballpark figure for the round trip fare?

There is a website that lets you put in the destinations and times and it will give you all the fares for different types, times and platforms.

See below

http://www.hyperdia.com/cgi-english/

The JR site shows a ticket cost of 9950 yen for the Hikari shinkansen from Shin-Osaka to Hiroshima. It's 10710 for the Nozomi train.
http://www.jreast.co.jp/e_charge/index.asp

The hyperdia site gives 10150 for the Hikari and 10910 for the Nozomi.

Ballpark is 20,000 JPY round-trip. Children for 10,000 JPY.

That means that you should look at a JR pass. The JR West Sanyo Area Pass for 20,000 yen which includes the Nozomi (the fastest of the shinkansen that can get you between those stations in 79 minutes).

That Sanyo Area pass is good for 4 days and you can take the shinkansen all the way to Hakata.

You mentioned a long day, but I checked the schedule. You can have breakfast in Osaka and at 0600 catch (at Shin-Osaka) the Hikari that arrives 0735. Spend 3 hours visiting and be back in Osaka for lunch.

The hyperdia site shows distance travelled in addition to time. The Nozomi train to Hiroshima averages about 150 mi/hr. The train takes 44 minutes to go from Hiroshima to Kokura, around 180 mph, I think, the fastest in Japan. The train to Kokura goes through a couple of tunnels and across some nice country.

It would be less than 20 min from Kokura to Hakata (Fukuoka).

Thank you, all. You've been very helpful, and I appreciate the time and effort you spent in answering my question.

The JR limited express or express trains, instead of the shinkansen would cost between 11000 and 15000 yen round trip and take SIX HOURS EACH WAY!

From Osaka eki to Hiroshima
5460 yen, 2 transfers, 5 hr 49 min
6700 yen, 1 transfer, 5 hr 49 min
7600 yen, 4 transfers, 6 hr 14 min

The JR passes are a good deal.