Multiply 103 × 107 with the base-100 trick

Question

Both factors sit a little above 100, and that structure hands you the answer in two chunks. Solve in your head. What is 103 	imes 107? Aim: under 10 seconds.

Accepted Answer

The surpluses over 100 are +3 and +7. Add either number to the other's surplus for the leading part, then multiply the surpluses for the trailing part: 103 + 7 = 110 \Rightarrow 11000, \qquad 3 	imes 7 = 21. 103 	imes 107 = 11000 + 21 = 11021. So 103 	imes 107 = \mathbf{11021}. Check: 103 	imes 107 = 103 	imes 100 + 103 	imes 7 = 10300 + 721 = 11021. Matches. Takeaway. For numbers near 100, (100+a)(100+b) = 100(100 + a + b) + ab. Cross-add for the front, multiply the small excesses for the back.

Multiply 103 × 107 with the base-100 trick

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