Although the hints explain the puzzle fairly well, here's three ways to approach the problem

The question tells us that

a) the total cost of the camera and the case is $310

b) the camera costs $300 more than the case

LOGICALLY

While the puzzle traps many players into initially thinking that the camera costs $300 and the case $10, this doesn't satisfy the condition that the difference in price is $300

As the total price is $310 and the camera must cost over $300 (being the cost of the case plus $300), there's not many possibilities to consider

When you then consider that the cost of the case is the same as the cost of the camera less $300 (so twice this value must be $10), the cost of the case must be $5 and the cost of the camera $305

MATHEMATICALLY

camera = case + 300

camera + case = 310

so...

(case + 300) + case = 310

case + case = 10

case * 2 = 10

case = 10 / 2

case = 5

TRIAL AND ERROR

We know that the camera must cost at least $300 (as it costs $300 more than the case) and can not cost more than $310 (being the total cost)

Remembering that the camera costs $300 more than the case, we can work through the possibilities (camera cost + case cost = total cost)

300 + 0 = 300

301 + 1 = 302

302 + 2 = 304

303 + 3 = 306

304 + 4 = 308

305 + 5 = 310 **

306 + 6 = 312

307 + 7 = 314

308 + 8 = 316

309 + 9 = 318

310 + 10 = 320

The only combination where the total of both the camera and the case is $310 is where the camera costs $305 and the case costs $5

CONCLUSION

As you hand over $100 to pay for the $5 case, then you should get $95 in change

OTHER DISCUSSION

Can someone please explain to me the "The Camera and The Case" puzzle?

http://www.gamefaqs.com/boards/936050-professor-layton-and-the-curious-village/48954207

Puzzle 42 - The Camera and The Case

http://www.gamefaqs.com/boards/936050-professor-layton-and-the-curious-village/43994563

The Camera and the case

http://www.gamefaqs.com/boards/936050-professor-layton-and-the-curious-village/43259132

With the puzzle the camera and the case..

http://www.gamefaqs.com/boards/936050-professor-layton-and-the-curious-village/41671871

Someone please explain to me The Camera and Case

http://www.gamefaqs.com/boards/936050-professor-layton-and-the-curious-village/41426939

puzzle 42?

http://www.gamefaqs.com/boards/936050-professor-layton-and-the-curious-village/41342259

Professor Layton and the Curious Village DS Illustrated Walkthrough

http://professorlaytonwalkthrough.blogspot.com