The question you opened
A backpack and a bottle for $46
A backpack and a water bottle cost $46 together. The backpack costs $40 more than the bottle.
What does the bottle cost?
$3
Take the $40 gap out of $46 and $6 is left for the two items to share. Half each gives a $3 bottle and a $43 backpack.
A backpack and a water bottle cost $46 together, and the backpack costs $40 more than the bottle. Most people answer $6 in about a second, and most people are wrong. Here is where the $6 comes from. The sentence hands you two numbers and one obvious operation, so your head subtracts 40 from 46 and stops. The answer feels finished because it used every number in the question. Test it against both sentences.
If the bottle is $6 and the backpack costs $40 more, the backpack is $46. Together they come to $52, which is six dollars over the total the question gave you. The gap is right and the total is broken. Now work it properly. Pull the $40 difference out of the $46 first. What remains is $6, and that $6 is the part the two items share equally, because once the gap is removed they cost the same.
Half of it goes to each. The bottle is $3, the backpack is $43, the difference is $40 and the total is $46. Both sentences hold. The general move is worth keeping in your pocket. When two things are described by a total and a difference, take the difference out of the total, halve what is left to get the smaller item, then add the difference back for the larger one.
It works for ages, salaries, two rents, two distances, any pair described that way. This is the shape of the classic bat and ball question, and it survives because the wrong answer arrives before you have decided to think about it. The repair is one habit: put your answer back into every sentence of the problem and see whether all of them are still true. That habit costs about three seconds and catches almost every version of this trap you will meet.
Technique: Remove the gap, then halve the rest