Questions tagged [mental-arithmetic]

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

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66 views

Is there a reasonably accurate and easy way to approximate lbs-stones or kg-stones (and vice versa)?

For example, when I need to talk with my American friends about body weight: $\mathrm{KG}\times2+10\%$ ($100\times2=200$, $200+10\%=220$... Pretty accurate and easy to do mentally, both ways) However, ...
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1answer
113 views

prove that $5<\sqrt{5}+\sqrt[3]{5}+\sqrt[4]{5}$ [duplicate]

prove that $$5<\sqrt{5}+\sqrt[3]{5}+\sqrt[4]{5}$$ .A little use of calculator shows that $\sqrt{5}+\sqrt[3]{5}+\sqrt[4]{5}=5.44$.Thus the inequality is indeed true. Generalising this result with $...
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1answer
72 views

Prove that $e^\pi > 21$. Without a proof, you can use the two facts: $e > 2.71$ and $\pi > 3.14$.

I heard that the following problem is for high school students. Prove that $e^\pi > 21$. Without a proof, you can use the two facts: $e > 2.71$ and $\pi > 3.14$. My solution is the ...
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4answers
125 views

Multiplication of decimal numbers

I was wondering if anyone knows any good resources to use or tricks to be able to solve these kinds of mental-arithmetic questions(See image below)? Would be really grateful for any help ! Edit: I ...
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Average and growth rate of multiple observations

A really simple question about averages, and the pitfalls of calculating them wrongly. We have observations between 2 periods: a and b periods. In Each period there are two miles of per gallon for ...
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5answers
69 views

Mental arithmetic: Add numbers by combining numbers that add up to 10

I'm reading a book on mental arithmetic, and came across this paragraph: You were taught - or should have been taught - at school that speed in addition is acquired by combining pairs of successive ...
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67 views

I want to learn math do I need to memorise the times tables?

As a front end developer, I do not need Math, however, I want to seriously learn it so that, maybe, one day, I might be able to get so good at it as to get into AI. I started with Algebra 1 and ...
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1answer
41 views

Whats best method to calculate multiplication mentally?

I would like to know which method is better for mentally multiplication calculation between a) Cross-Multiplication and b) Left-to-Right Multiplication Thank you!
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180 views

How many digits are there in $99^{99}$? [duplicate]

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....
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9answers
291 views

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 \...
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1answer
75 views

Determine the number of digits of the product a.b

If $ a = 3,643,712,546,890,623,517 $ and $ b = 179,563,128 $, determine the number of digits of the product $ a.b $. I went searching the site and found this post where advise to use logarithm ...
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256 views

Are there techniques for quickly seeing patterns in numbers?

Recently I was watching 8 out of 10 cats does countdown (S18E01 - 26 July 2019), for which the math question there was the following numbers: if you haven't seen countdown, you are provided 6 numbers,...
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1answer
64 views

are combinatorics-leaning mathematicians very good at mental math? [closed]

a common cliché in movies is that mathematicians who specialize in combinatorics are very adept at mental math (arithmetic), is that a real thing? if yes why?
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1answer
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Can a mental calculator plot any function in seconds? [closed]

As we learnt at school, to draw the graph of a function in details it may be necessary to perform some steps of calculation such as finding domain and codomain, limits, asymptotes and sign of ...
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295 views

How do you calculate bitwise XOR in your head?

Subtraction has an easy method: you can literally count down until you get it. So does addition, multiplication, division, etc. However, does bitwise XOR have a method like this that you can do in ...
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1answer
86 views

Mental techniques for large arithmetic and decimals

Consider the following problems: $$\frac{14}{4.6\dot{6}}$$ $$20 \cdot 45$$ $$\frac{7}{6}$$ $$28 \cdot 0.28$$ $$\frac{42}{168}$$ I'm looking for some techniques to solve these problems fast without ...
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Estimating square numbers

A large dice has a side length of 9.2 cm. Estimate the surface area of the cube. What I did: 6× $9^2$ = 6 × 81 = 6 × 80 = 480 But the answer says that $9.2^2$ is 85 as an estimate. How do I get ...
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What would be the best way to memorize the 10 by 10 multiplication table?

Hear me out before you start downvoting please. I have a learning disability so no matter how hard I try I can’t memorize the table. Please give some tips/hints on how to memorize the table. Thanks in ...
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69 views

Mental approximations of 1) log, 2) non-integer power

I am currently preparing for a interview that is notorious for asking mental approximations. Two example questions came up: 1) $\ln 514$ and 2) $3^{3.6}$. What are some of the best ways to calculate ...
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4answers
427 views

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 ...
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1answer
543 views

Vinculum number of 989?

To find vinculum number I was subtracting finding the complement of 989 and cosidering the number to be 0989 I raised the 0 to 1 so my answer was 1011 , but why was the answer given 1111 ? Do I also ...
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146 views

Age questions: How to approach solving this?

Completely foxed by this question: Moses is twice as old as Methusaleh was when Methusaleh was one-third as old as Moses will be when Moses is as old as Methuselah is now. The difference in their ...
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Generating “random” mental calculation exercises?

I recently tried to come up with "random" mental calculation exercises in an attempt to fight traffic jam boredom. Unfortunately, I quickly got bored by the lack of creativity of the problems I can ...
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2answers
201 views

How many zeros will $ 3^{3}4^{4}5^{5}6^{6} - 3^{6}4^{5}5^{4}6^{3}$ ends with?

I have this GRE practice question, asking me to find the quantity $$ 3^{3}4^{4}5^{5}6^{6} - 3^{6}4^{5}5^{4}6^{3}$$ will end in how many zeros? The answer given is $4$, but I don't quite understand ...
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How to find a volume of this figure (which is $3080 \text{ cm}^3$) in a few seconds?

I was watching this Japanese game show and came across this question: The contestants were told that each small cube is 2cm on its side and were asked to find the volume of the above figure. The ...
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328 views

Multiplication tables: up to what number should I memorise? [closed]

I never memorised them and suffered immensely through school as a result. Eg. I calculated 7x8 as (7x10=70) minus 7 using my fingers (63) then minus again on my fingers (56). In Australia children ...
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3answers
236 views

Why is it so good to know that $(1+x)^n \approx 1+nx$ for $nx \ll 1$?

I’ve often heard that it’s good to memorize the fact that $(1+x)^n \approx 1+nx$ for $nx \ll 1$ (most recently here), especially for mental arithmetic or making quick approximations. But why? Could ...
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152 views

What is the method for mentally computing $3^{3.5}$ and similar calculations?

What is the method for mentally computing an approximation of $3^{3.5}$ and similar calculations? (without using any calculator) The best I did is: $3^{3.6}=e^{3.6ln(3)}=e^{3.6*1.098} \approx e^{3.6*...
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237 views

Without using a calculator how to solve $x^x = 100$? [duplicate]

Without using a calculator how to solve $x^x = 100$ ? A way of finding an approximation to 2 decimals would be good neough. I know about the Lmabert W function but one cannot compute it mentally. ...
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1answer
81 views

How can I best mentally calculate the total sum of a sequence of linear increases?

I often run into the following situation: In January, I'm producing $10$ items/month. By December, I want to be producing $65$ items/month. So assuming things improve linearly, my production is ...
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981 views

I want to learn math from basics the Indian way and am looking for a book to guide me and some workbooks to practice. Any recommendations?

I was taught math in a very stoic method during my childhood and as a result became math-phobia. Now as an adult, I wish to relearn math as a long term hobby and a cure for my phobia. I found that in ...
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1answer
292 views

Mentally generating a (pseudo)random {0,1}-sequence with uniform distribution

I want to learn of good ways by which to generate $\{0,1 \}$-sequences in my head which are (pseudo)random with uniform distribution, so that I may simulate flipping a fair two-sided, standard coin. I ...
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1answer
236 views

Proof behind this mental math multiplication?

Recently came across this technique of multiplying two $2$-digit numbers involving the same digit at the tens place and the sum of digits at units place being $10$. E.g., $73 X 77$ has same digit at ...
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6answers
542 views

Why doesn't $26\times 24 = 25\times 25?$ (I remove and $+1$ from both numbers) [closed]

I'm solving a math puzzle: "how quickly can you multiply $26$ by $24?$" I don't know the answer so I use tutorials. One tutorial say to do it quickly you can round numbers up and down to closest ...
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3k views

(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 ...
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2answers
65 views

Mental division of two fractions?

I've got a non-calc paper coming up, and when going through a test, this fraction came up: $$ \frac{8}{-0.4} \equiv \frac{8}{\big(\frac{-2}{5}\big)} $$ Going through the answers he says: $$8/2=4$$ ...
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51 views

Approximate this fraction (simple arithmetic)

If I have $1.5\cdot10^{-5}=\frac{1.5}{10^5}$ , how can I rewrite (approximate) this fraction as $$ \approx \frac{1}{66.7\cdot 10^3}\quad ? $$ My calculator gives the exact answer $\frac{3}{200000}$, ...
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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 ...
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1answer
2k views

Mixture and Alligation:::

A solution contains alcohol and water in the ratio 3:1, 16 litres of the Mixture is drawn off and 12 litres of water is added.11 litres of Mixture is replaced by 11 litres of water.The alcohol and ...
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888 views

Is there an easy way to multiply two 2-digit numbers that have flipped digits?

I am currently a sophomore in high school competing in UIL academics in Texas. I am competing in the number sense test. Among the questions is the need to quickly multiply 2 two digits numbers, but as ...
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85 views

How to multiply any number with decimal number [closed]

I can't remember multiplication tables, is there any math trick for multiplication?
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1k views

Multiplying very small numbers without the use of a calculator

As part of my college course I have to sit some math exams and I cannot use a calculator. The problem is I suck at maths with a calculator not to mind how bad I am at it without using a calculator! ...
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9answers
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How can I easily double any size number in my head?

I'm a software engineer, and I often double numbers especially when doing binary to decimal conversions. When numbers get large, I have trouble doubling a number in my head without using paper. For ...
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2answers
1k views

Why does this method for finding the square root of a number work?

Recently I read this, which teaches a trick for finding the square root of a number very quickly and was quite astounded. How/why does this method work? For those of you with linkphobia, I copied and ...
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0answers
34 views

Pandigital strings that end in $0$ and available to mental arithmetic?

I'm looking for a sequence that includes every number from $1$-$9$ and then ends with $0$. It should look random on first glance (or as random as possible given the mental arithmetic constraint) but ...
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3answers
417 views

Mentally calculate the first 9 terms of this Fibonacci sequence.

There was a question in a mental math test and it expected me to calculate the first 9 terms of the following Fibonacci sequence (Note that this is how the problem was exactly given.): The sum of ...
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4answers
80 views

Problems on Trains

A train Leaves station $x$ at $5 AM$ and reaches station $y$ at $9AM$. Another train leaves station $y$ at $7AM$ and reaches station $x$ at $10.30 AM$ . At what time do the two trains cross each other?...
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2answers
89 views

Mental Arithmetic Problems

The speed of a Motor - Boat is that of the current of water as 36:5.The boat goes along with the current in 5 hours 10 Minutes,It will Come Back in which time? I have Tried: Speed of Motor : Speed of ...
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283 views

Divison In Mental ArithMatic

The nearest number to $99548$ which is divisible by $687$ is? How can I find the answer quickly, is there any short cut to check if a number is divisible by $687$?
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Boats and Streams [closed]

A boat can travel with a speed of $13$ km/hr in still water. If the speed of the stream is $4$ km/hr, find the time taken by the boat to go $68$ km downstream. Can you Explain the Difference between ...