Large Numbers Around Us
Shortcuts in multiplication
Learn Roxie and Estu's shortcut for multiplying by numbers close to 10, 100 and 1000.
Roxie and Estu found quick ways to multiply by 5, 25 and 125: they are close relatives of 10, 100 and 1000.
| To multiply by | Do this | Because | Example |
|---|---|---|---|
| 5 | × 10, then ÷ 2 | 5 = 10 ÷ 2 | 116 × 5 = 1160 ÷ 2 = 580 |
| 25 | × 100, then ÷ 4 | 25 = 100 ÷ 4 | 824 × 25 = 82,400 ÷ 4 = 20,600 |
| 125 | × 1000, then ÷ 8 | 125 = 1000 ÷ 8 | 72 × 125 = 72,000 ÷ 8 = 9000 |
Can you explain, using the meaning of multiplication and division, why these work?
Look for pairs that make 100 or 1000:
- 2 × 1768 × 50 = 1768 × (2 × 50) = 1768 × 100 = 1,76,800
- 125 × 40 × 8 × 25 = (125 × 8) × (40 × 25) = 1000 × 1000 = 10,00,000
Some products make patterns. Work them out and continue them: 11 × 11, 111 × 111, 1111 × 1111 ...; 3 × 5, 33 × 35, 333 × 335 ...; 101 × 101, 102 × 102, 103 × 103 ...
How many digits does a product have? Roxie says two 2-digit numbers always give a 3- or 4-digit answer. Must we try every pair? No. Her reasoning: the smallest product is 10 × 10 = 100; every product is less than 100 × 100 = 10,000 (the smallest 3-digit numbers). Anything from 100 up to below 10,000 has 3 or 4 digits. Done, with no other cases to test!
The same idea gives: 1-digit × 1-digit → 1 or 2 digits; 2-digit × 2-digit → 3 or 4; 3-digit × 3-digit → 5 or 6.
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