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Sweet mental math trick

I was in my math class talking about something (probably a difference of two squares) and my math teacher showed us a cool trick to multiplying certain numbers. One example would be 35,45. First take the number In between these (40). Then get the amount needed for each number to equal 40 (5). Then multiply 40x40=1600 and get 5x5=25. Then 1600-25=1575. Pretty cool? Yes it is. Please tell me if you already knew that. I would love to catch up on these simple tricks.

Note by Robert Fritz
3 years, 5 months ago

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Let the inbetween number be \(\hat{r} = \frac{x+y}{2}\) so that

\[\begin{align} x &= \hat{r} + \frac{x-y}{2} \\ y &= \hat{r} - \frac{x-y}{2} \end{align} \]

where the difference between \(x\) and the inbetween number (and also from \(y\) to the inbetween number) is \(\frac{x-y}{2}\)

Then \(\displaystyle xy = \left(\hat{r} + \frac{x-y}{2}\right)\left(\hat{r} - \frac{x-y}{2}\right) = \hat{r}^2 - \left(\frac{x-y}{2}\right)^2\), just like you say! Josh Silverman Staff · 3 years, 5 months ago

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This is based on the identity \( (a+b)(a-b) = a^2 - b^2 \). I learnt this identity and started applying it to use some 2 years ago. Shabarish Ch · 3 years, 5 months ago

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Yeah pretty cool right, i learned it in 5 grade, but then I couldnot use algebra to justify it lack of Algebra Mardokay Mosazghi · 3 years, 5 months ago

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@Mardokay Mosazghi It seems I'm far behind sweet mental math tricks. Robert Fritz · 3 years, 5 months ago

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Yep,I already knew about that.Our Maths teacher introduced us to it. Abdur Rehman Zahid · 2 years, 9 months ago

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We could have also multiplied in 10 s (3040)=1200 nd den (55)= 25 finally add 1200 +25 =1225 Sujatha Suvarna · 2 years, 10 months ago

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The following is a trick to square a number ending with 5. For example take 35 then ignore 5 take rest of the number 3 in this case, now 3(3+1)=12 and 55=25 so the answer is 1225. Try it out with other cases Rakvi · 3 years, 5 months ago

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@Rakvi Thank you! I am forever in your gratitude. Robert Fritz · 3 years, 5 months ago

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\((\frac{x+y}{2})^{2}-(\frac{x-y}{2})^{2} \Longrightarrow (\frac{x+y}{2}+\frac{x-y}{2})(\frac{x+y}{2}-\frac{x-y}{2}) \Longrightarrow (\frac{2x}{2})(\frac{2y}{2}) \Longrightarrow \boxed{xy}\) which is the desired result! Pretty awesome! Finn Hulse · 3 years, 5 months ago

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