Questions tagged [mental-arithmetic]

Mental arithmetic comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper.

Mental arithmetic comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper.

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(Quickly) finding the smallest fraction

Please see these fractions: (A) $\frac{33}{128}$ (B) $\frac{45}{138}$ (C) $\frac{53}{216}$ (D) $\frac{83}{324}$ (E) $\frac{15}{59}$. I need to find out quickly (in about a minute) the smallest of these fractions. I am not allowed to use a…
Masroor
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Mentally calculating trigonometric function values such as $\sin(47^\circ)$

This may sound dumb, but does such a way exist to mentally (and quickly) determine the values of trigonometric functions such as $\sin(47^\circ)$ and so forth--quickly being a mere matter of seconds? My physics teacher suggested to our class that it…
Avant Guard
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How many digits are there in $99^{99}$?

Question: How many digits are there in $99^{99}$? My attempt: Observe that $$99^{99} = (100\times 0.99)^{99} = 100^{99}\times 0.99^{99}.$$ Note that $100^{99} = 10^{198}$ has $199$ digits and $$0.99^{99} = \left(1-\frac{1}{100}\right)^{99}…
Idonknow
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Show that $\left(\frac{1}{2}\left(x+\frac{2}{x}\right)\right)^2 > 2$ if $x^2 > 2$

Okay, I'm really sick and tired of this problem. Have been at it for an hour now and we all know the drill: if you don't get to the solution of a simple problem, you won't, so ... I'm working on a proof for the convergence of the Babylonian method…
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How to obtain an approximate expression for $\sqrt{\varepsilon}$ where $\varepsilon \ll 1$?

Is there a way to obtain an approximate expression for the square root $\sqrt{\varepsilon}$ of a small number $\varepsilon \ll 1$? To be more precise, I would like to have an expression which (1) I can easily handle by a mental calculation and (2)…
shark.dp
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Soroban Abacus - how to memorise?

I saw a video where Indian kids where fiddling fingers in the air and working out big sums. WOW I thought. I researched loads and have figured out that it's all based on using the Soroban Abacus (a Japanese abacus if you don't know). I got a little…
Omar
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Fastest way to multiply numbers mentally?

I'm wondering what is the fastest way to multiply numbers? For now, let's focus on 2-digit numbers and were one cannot use scrap paper. I've come across 3 fast methods: 64x43 1) 60x43 + 4x43 (note that 60x43 is actually a 1-dig. times…
Cindy88
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How to factor 5671?

The other day I wanted to factor 5671 in my head. (It turns out to be $53\cdot107$, but I did not know this at the time.) I quickly ruled out the easy divisors, 2, 3, 5, 7, 11, and 13. At this point I saw no obvious way to proceed short of a very…
MJD
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Is there a quick way of finding the coefficients in an expression like $(ax^3+bx^2+cx+d)^3$?

We can raise a sum to the power of $n$ quickly and easily using Pascal's triangle, due to the binomial theorem: $$(a+b)^n = \sum_{i=0}^n {n \choose i} a^i b^i$$ For sums of more than one term, we can still do this kind of thing, using multinomial…
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How can I mentally calculate $\cos(x), x∈(0.7, 1.2)$

I'm trying to learn how to calculate trig functions in my head. I'm planning on learning $\cos(x), x∈[0,π/2]$ and then using symmetry to calculate the others. I think the quadratic Maclaurin series at $0$ and the linear at $π/2$ could be calculated…
Zaz
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Doing simple math in your head

I thought I was pretty good at doing math in my head until yesterday I saw someone do $17.4/4$ in their head, without writing anything down and it took them less than 20 seconds. What do you do to do this in your head? Even with the answer of 4.35 I…
user138246
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Fastest way to multiply small numbers with decimals mentally

Is there a fast way to multiply these numbers mentally? Some examples: $0.85 \times 1.15$ $0.5 \times 1.5$ $0.2 \times 1.4$
Math is fun
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Remembering numbers while performing calculations mentally

Specific question: how can I improve on mentally performing calculations like these : $$(78-59)\times23-8\%\times(270+130)$$ $$[2\text{digit}+-2\text{digit}]\times2\text{digit}+-x\%\text{of}[3\text{digit}+-3\text{digit}]$$ Without writing anything…
Cindy88
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Calculate $2^{5104} \bmod 10$ using mental arithmetic

I am practising interview for Jane Street's trader internship and I found the following question. Question: Calculate $2^{5104} \bmod 10$ using mental arithmetic. I know that $2^5 \bmod 10 \equiv 2 \bmod 10.$ So, \begin{align*} 2^{5104} & =…
Idonknow
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What method for mentally computing 2-digit multiplication problems, minimizes the amount of mental steps?

So I've been practicing alot of mental math recently and ofcourse as a part of that, multiplying a double-digit number by another double-digit number. I have been doing some research into what the quickest way to computing the outcome of such a…