How to Multiply Two Numbers Ending in 5 That Are 10 Apart
Multiply two ...5 numbers that are 10 apart (35×45, 65×75).
35 × 45 = 1575
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35 and 45 sit 5 either side of 40, so this is (40 − 5)(40 + 5).
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That is the difference of squares: , or 1600 − 25.
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A multiple of ten squared ends in 00, and 00 − 25 is why it always ends 75.
How do you multiply two numbers ending in 5 that are 10 apart?
Two numbers ending in 5 that are 10 apart sit 5 either side of a multiple of ten, so their product is that multiple of ten squared, minus 25. For 35 × 45, the middle is 40, so 35 × 45 = 40² − 5² = 1600 − 25 = 1575.
It works because (m − 5)(m + 5) = m² − 25, since the two middle terms cancel.
A multiple of ten squared ends in 00, so taking 25 away always leaves an ending of 75. That gives a shortcut for two-digit numbers: square the tens digit of the middle, take away 1, and write 75 after it. For 65 × 75, the middle is 70, so 7 × 7 = 49 and 49 − 1 = 48, and the answer is 4875. For 95 × 105, the middle is 100, so 10000 − 25 = 9975. Check with the ending, because an answer that does not end in 75 has gone wrong.
Your turn
Three to try in your head. Tap what you get.
35 × 45 = ?
25 × 35 = ?
55 × 65 = ?
All 36 shortcuts: the Magic Tricks library.