Problem 1 · 1991 AJHSME
Easy
Arithmetic & Operations
subtraction
1,000,000,000,000 − 777,777,777,777 =
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Answer: B — 222,222,222,223.
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Hint 1 of 3
Subtracting from a row of 0's forces borrowing all the way across — twelve times. Is there a nearby number you'd MUCH rather subtract from?
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Hint 2 of 3
The number just below, 999,999,999,999, is all 9's: every column subtracts with no borrowing. Solve that easy problem first, then fix up the difference of 1.
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Hint 3 of 3
Notice four of the five choices begin with 2's; only the last digit decides between them, so most of the work is just nailing down that ones place.
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Approach: swap the all-0's number for the all-9's number, then add back the 1
- Subtracting from 1,000,000,000,000 means borrowing across every column — slow and error-prone. Instead use the number one less, 999,999,999,999: now every digit is 9, so each column gives 9 − 7 = 2 with no borrowing. That clean subtraction is 222,222,222,222.
- We replaced the top number by one that was 1 smaller, so the true difference is 1 bigger: 222,222,222,222 + 1 = 222,222,222,223.
- Why this transfers: when subtracting from a power of ten (a 1 followed by 0's), shift to the all-9's number right below it to dodge every borrow, then adjust by 1 at the end. The same trick turns 1000 − 638 into 999 − 638 = 361, then +1 = 362.
- Sanity check: 222,222,222,223 + 777,777,777,777 should rebuild the top number — the 2's-and-7's add to 9's and the final 3 + 7 carries up to make 1,000,000,000,000. ✓
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