Nearest neighbours, and estimating

Every big number has nearest neighbours: the closest round thousand, ten thousand, lakh, ten lakh and crore. To find one, look at the digit just after that place: 5 or more → round up, less than 5 → round down.

For 6,72,85,183:

Try the same for 3,87,69,957 and 29,05,32,481. Puzzle: a number has all five nearest neighbours equal to 5,00,00,000. What could it be? How many such numbers are there?

Neighbours help you estimate. For 4,63,128 + 4,19,682, Roxie says near 8,00,000 and more than it; Estu says near 9,00,000 and less than it. Both are right. The exact sum is 8,82,810, so Estu's is closer.

You can even find bounds without adding. Is the sum more than 8,50,000? Both numbers are over 4,00,000 and the first is over 4,60,000, so yes. Is it more than 8,83,128? The second number is under 4,20,000, so the sum is just under.

For 14,63,128 − 4,90,020, Roxie says near 10,00,000 and less; Estu says near 9,00,000 and more. The exact answer is 9,73,108, so this time Roxie is closer.

Estimating reads tables too. City populations in 2001 and 2011: Mumbai 1,19,78,450 → 1,24,42,373; Pune 25,38,473 → 31,15,431; Bengaluru 43,01,326 → 84,25,970. Without exact sums you can see that Bengaluru almost doubled, and that Patna's 16,84,222 times about 7 comes close to Mumbai.