Round and fix
Turn an awkward number into a round one, do the easy sum, then pay back the difference. This one habit handles every 98, 99, 197 and 499 you will ever meet.
THE IDEA
Borrow a bit, then pay it back
Adding 98 is the same as adding 100 and then taking 2 off, because 98 = 100 − 2. Nothing was lost: you added 2 too many on purpose, so you remove exactly 2 at the end. Mathematicians call this compensation. The round number does the heavy work and the small fix keeps you honest.
347 + 98. Go to 347 + 100 = 447, then hand back the 2: 445.
347 + 98
round up: 98 → 100 (2 too many)
easy sum: 347 + 100 = 447
pay back: 447 − 2 = 445445Write the size of the fix down in your head before you start: 'two too many'. People lose the answer by forgetting which way the fix goes, not by adding wrong.
Check yourself
A hotel night is 256 and breakfast is 99. In your head: 256 + 99 = ?
- 354
- 356
- 355
- 357
Show the answer
355
Yes. 256 + 100 = 356, and 99 is one less than 100, so give one back: 355.
SUBTRACTING
Take away too much, then give it back
Subtracting 97 is subtracting 100 and then adding 3 back, because you took away 3 more than you should have. The fix flips direction when the sum does. That flip is where nearly every mistake in this lesson lives.
423 − 97. Take 100: 323. You removed 3 too many, so put them back: 326.
Check yourself
To work out 512 − 198 in your head, you take away 200 and then take away 2 more.
Show the answer
False
False, and this is the classic trap. 512 − 200 = 312, but 198 is 2 less than 200, so you removed 2 too many and must add them back: 314. Taking 2 more gives 310, four away from the truth.
The same move on small numbers: making tens
- Look at the gap to the next ten
68 + 25. From 68 the next round stop is 70, and the gap is 2.
- Split the other number at that gap
25 becomes 2 + 23. You are not changing anything, just cutting 25 where it is useful.
- Jump to the round number
68 + 2 = 70. Round numbers are where your head stops working hard.
- Add what's left
70 + 23 = 93. Done, and you never held an awkward number.
Check yourself
Sara's basket has an item at 68 and another at 25. Bridge through 70 in your head. What's the total?
- 83
- 93
- 95
- 92
Show the answer
93
Right. 68 + 2 = 70, and 25 had 23 left in it, so 70 + 23 = 93.
When to round, and when not to bother
Worth rounding
One number sits within 1, 2 or 3 of a round number: 98, 199, 497, 52. The fix is tiny and you can hold it while you work. This is most prices, because shops write 999 instead of 1,000 on purpose.
Leave it alone
Neither number is near anything: 43 + 36. Rounding 36 to 40 means remembering a fix of 4 for no gain. Just go left to right, as in lesson 1: 70, then 9, so 79.
A jacket at 399,000 comes off a 1,250,000 total
1,250,000 − 400,000 = 850,000
took away 1,000 too many
850,000 + 1,000 = 851,000851,000Prices ending in 9 are built for this. 399,000 is 400,000 minus a single thousand, and 400,000 is a number you can subtract without thinking.
Check yourself
Round the awkward number, then fix. Match each one to its answer.
Show the answer
- 46 + 99 → 145
- 75 − 48 → 27
- 238 + 97 → 335
- 604 − 199 → 405
Make it a habit
- Say the fix before you do the sum: 'two too many', 'three short'. A fix you named is a fix you won't reverse.
- Adding a rounded-up number means subtracting at the end. Subtracting a rounded-up number means adding.
- Anything ending in 8 or 9 is asking to be rounded. Anything ending in 3, 4, 6 usually is not.
- You can round the first number instead if it's the friendlier one: 98 + 347 is the same problem.
Check yourself
The hard one. Omid has 2,000,000 and pays a bill of 1,197,000. What's left?
- 797,000
- 800,000
- 803,000
Show the answer
803,000
Exactly. 2,000,000 − 1,200,000 = 800,000, and you took 3,000 too many, so add them back: 803,000.
Lesson recap
- Replace an awkward number with a round one, do the easy sum, then undo the difference.
- It works because 98 is literally 100 − 2: you only rearranged the problem, you didn't change it.
- The fix flips direction: round up while adding, subtract at the end; round up while subtracting, add at the end.
- On small numbers the same move is making tens: 68 + 25 becomes 70 + 23.