Problem 4 · 1993 AJHSME
Medium
Arithmetic & Operations
regrouping
1000 × 1993 × 0.1993 × 10 =
Show answer
Answer: E — (1993)².
Show hints
Hint 1 of 2
Don't multiply left-to-right — rearrange first. Multiplication lets you reorder freely, so pull the 'easy' factors (the powers of ten and the decimal) together and leave the lonely 1993 aside.
Still stuck? Show hint 2 →
Hint 2 of 2
Powers of ten and a decimal undo each other: ×1000 and ×10 push the decimal point right, and 0.1993 moves it back. Watch 0.1993 turn into a whole number.
Show solution
Approach: regroup so the powers of ten cancel the decimal
- Reorder: (1000 × 10 × 0.1993) × 1993. The 1000 and 10 multiply 0.1993 by 10000, sliding the decimal point 4 places right: 0.1993 → 1993. So the bracket is just 1993.
- That leaves 1993 × 1993 = (1993)².
- Intuition: a decimal like 0.1993 is secretly '1993 with the point moved.' Hunting for powers of ten that snap it back to a whole number is the whole trick — and it dodges any messy long multiplication.
Mark:
· log in to save