The question you opened
Splitting 87 dollars three ways
Dinner comes to 87 dollars and the three of you split it evenly.
What does each person pay, in dollars?
$29
Cut 87 into 60 and 27, since both divide by three cleanly. That gives 20 plus 9, so 29 dollars each.
Eighty seven divided by three is 29. The fast route is to break the number before dividing it. Sixty and 27 both divide by three, giving 20 and 9, which add to 29. Nothing has to be held in memory except two small results. Choosing where to cut is the whole skill. Look for the largest multiple of the divisor that sits below the number and is easy to see. For 87 and 3 that is 60.
For 128 and 4 it is 120, leaving 8, so the answer is 30 plus 2. For 96 and 6 it is 60, leaving 36, so 10 plus 6 is 16. The parts should be chosen for friendliness rather than for size. The same splitting works for multiplication, where it is usually called distribution. Twenty three times 6 becomes 20 times 6 plus 3 times 6, so 120 plus 18 is 138.
Every mental arithmetic method that survives contact with a real bill is some version of this: turn one hard operation into two easy ones and add. For restaurant bills there is a practical extra. Split the total including tax and tip rather than splitting the food and adding tip at the end. Tip on 87 at 20 percent is 17.40, so the true total is 104.40, and a third of that is 34.80.
Doing it in the other order makes people argue about whose dish was more expensive. The last piece is knowing when to stop. If the split does not land clean, round up to the nearest dollar for each person and let the extra ride on the tip. Nobody has ever improved a dinner by insisting on the third decimal place.
Technique: Split into friendly parts