Multiply around a base

Choose a nearby base B. Write the numbers as B + a and B + b. A difference above the base is positive; a difference below it is negative. Expanding the product gives (B + a)(B + b) = B(B + a + b) + ab.

For 103 × 107, use B = 100, a = +3 and b = +7. Add the differences to the base: 100 + 3 + 7 = 110. Multiply by the base: 110 × 100 = 11000. Add the correction 3 × 7 = 21 to get 11021.

For 113 × 117, the differences are +13 and +17. The base part is (100 + 13 + 17) × 100 = 13000. The correction is 13 × 17 = 221. Adding it gives 13221: 221 adds two hundreds and 21 more. Do the final addition; writing two blocks of digits side by side can fail.

For 97 × 104, the differences are −3 and +4. The base part is (100 − 3 + 4) × 100 = 10100. The correction is (−3) × 4 = −12. Subtract 12: 10100 − 12 = 10088. Borrow one hundred: 10100 = 10000 + 100, and 100 − 12 = 88.

The same rule works at 1000. For 998 × 1003, use −2 and +3: (1000 − 2 + 3) × 1000 = 1001000. The correction is −6, so the answer is 1000994. Borrow from the base part: 1001000 = 1000000 + 1000, and 1000 − 6 = 994.

Try 96 × 108 around 100. The differences are −4 and +8. The base part is (100 − 4 + 8) × 100 = 10400. Decide the sign of (−4) × 8 before finishing.

The next practice opens Extended, Level 2, Ten questions for two numbers above 100. Level 3 puts the numbers on opposite sides of 100. Level 4 uses opposite sides of 1000, with larger gaps than the small teaching example above.