Technique explanation
Teaching Explanation
When multiplying numbers near a round base like 100, we can use their "distance" from that base to simplify the calculation. If a number is 103, its distance is +3 (above); if it is 98, its distance is -2 (below). When we multiply one of each, we call this an "across the base" problem. We find the answer in two parts. First, we cross-add: take one number and add the distance of the other (98 + 3 = 101, or 103 - 2 = 101). This gives us the left part of our answer. Second, we multiply the distances (+3 × -2 = -6). Because this second part is negative, we must "borrow" 1 from the left part (turning 101 into 100) and subtract the negative value from the base (100 - 6 = 94). The final result is 10094.
Method Condition
This method is applicable when: Both numbers must be within 20% of a common round base (10, 100, 1000).
Standard Fallback
If this method is not suitable, use: Standard column multiplication.
Misconception & Correction
Misconception: Treating the negative product of differences as a positive number (e.g., 101 | -12 becoming 10112). Correction: A negative right-hand side indicates a subtraction from the place-value column to its left. Always subtract the absolute value from the base after borrowing.
Worked examples
Worked Examples
| Step | Example 1: 97 × 104 | Example 2: 12 × 8 | ||
|---|---|---|---|---|
| 1. Identify Base | 100 | 10 | ||
| 2. Find Differences | 97 is -3; 104 is +4 | 12 is +2; 8 is -2 | ||
| 3. Cross-Add | 97 + 4 = 101 | 12 - 2 = 10 | ||
| 4. Multiply Diffs | -3 × 4 = -12 | +2 × -2 = -4 | ||
| 5. Structure | 101 | -12 | 10 | -4 |
| 6. Adjust | (101 - 1) | (100 - 12) = 10088 | (10 - 1) | (10 - 4) = 96 | ||
| 7. Independent Check | 97 × 100 + 97 × 4 = 9700 + 388 = 10088 | 12 × 8 = 96 (Multiplication table) |
Always perform an independent check to verify your result.
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