Can you really multiply any two-digit numbers in seconds with this trick?

Question

This trick is all over Instagram:

To multiply $23 \times 47$ :

20 \times 40 = 800

20 \times 7 = 140

3 \times 40 = 120

3 \times 7 = 21

Total: $800 + 140 + 120 + 21 = 1081$

But isnt this just FOIL? $(20+3)(40+7)$ expanded? So its not really a "trick" but just basic algebra?

I showed this to my friend and she was amazed but I was like "thats just the distributive property."

Actually theres another trick: to square a number ending in 5 like $75^2$ :

7 \times 8 = 56

Append 25: $5625$

This one actually IS surprising. Why does this work? Prove it with algebra.

Emily Thompson · Accepted Answer

Most two-digit multiplication tricks are place value in disguise. If the numbers are ( 10 a + b ) ( 10 c + d ) , then expanding gives 100 a c + 10 a d + 10 b c + b d = 100 a c + 10 ( a d + b c ) + b d . For 23 ⋅ 47, the hundreds contribution is 2 ⋅ 4, the cross contribution is 2 ⋅ 7 + 3 ⋅ 4, and the ones contribution is 3 ⋅ 7. Handling the carries reconstructs the ordinary product. It is the distributive property, not a separate rule. The shortcut for squaring a number that ends in 5 follows from the same algebra. Write the number as 10 a + 5: ( 10 a + 5 ) 2 = 100 a 2 + 100 a + 25 = 100 a ( a + 1 ) + 25. Therefore, multiply the part before the final 5 by the next integer, then append 25. For 75 2, use a = 7: 7 ⋅ 8 = 56 , so 75 2 = 5625.

Can you really multiply any two-digit numbers in seconds with this trick?

1 Answer

Can you really multiply any two-digit numbers in seconds with this trick?

1 Answer