Questions tagged [arithmetic]

Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag. Questions about number theory (sometimes called "arithmetic") should not use this tag and should instead use (number-theory) or (elementary-number-theory).

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1answer
26 views

In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d.

Q: In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d. My solution: $S_5$ =$\frac{1}{4}$*{$S_{10}$- $S_5$}. Q says that it is ...
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1answer
7 views

Q regarding finding total sum at end of year with an increment till n-1th year.

Q: The income of a person is 3,00,000, in the first year and he receives an increase of 10,000 to his income per year for the next 19 years. Find the total mount, he received in 20 years. Currency = ...
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0answers
74 views

The Plausibility of Cavemen "Inevitably Discovering" Mathematical Principles

Recently, I had the following idea about the inevitableness of a prehistoric caveman "discovering" fundamental principles of mathematics based on the laws of physics and nature of the ...
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13 views

How do I interpret weighted and non-weighted gains on a stock portfolio?

I have the following stock gains and losses for a stock portfolio. Overall, the portfolio has lost $40 or 28%. But when I calculate weights based on market value, and then calculate the gain/loss ...
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1answer
78 views

Why can multiplication be done in any order?

I have proof of $ab=ba$ and $abc=acb=bac=cab=cba=bca$ when the terms are all positive. But what about when they are not positive? What about when any number of terms multiplied together? I can't ...
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57 views

Who discovered the method of finding square root by division method? [closed]

We learn the method of finding the square root by division method in middle school.But who discovered it? Is there any such method of finding cube root?
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19 views

What is Mean - Difference ratio called? [closed]

Let's assume a rectangular chamber having two different temperatures at its walls, then if $\Delta T=T_2 – T_1$ is called the temperature difference across the chamber, what is $$\frac{\frac{...
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26 views

Arithmetic and common fixed point

Does there exist a relation between arithmetic and fixed points? I prove a new common fixed point theorem. I search for an application for it. I think that I was able to apply this theorem on common ...
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34 views
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1answer
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if we have a magnitude of 3/4 of one unit and me multiply by 3/5, where exactly did the idea come from that this would equal 9/20? [duplicate]

I'm wondering how the method for multiplying fractions was conceived and why it became the norm for math?
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1answer
70 views

Is the meaning (semantics) of an arithmetic expression its value? Eg, is "$2$" the meaning of "$1+1$"?

I think that in some math book I read long ago it said that the meaning -- semantics -- of an arithmetic expression is the value of the expression. For example, the meaning of this expression: ...
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1answer
127 views

Hidden patterns in the natural numbers

Consider the matrix $\{a_{ij}\}_{i,j\in\mathbb{N}}$ with $a_{ij} = 1$ iff $i \equiv 0\,\text{mod}\,j$. Drawing it with stretched y-axis by a factor of $5$ reveals some "waves" which I ...
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50 views

Interesting set problem involving cardinality of sets, intersection and union.

I'm stucked on the following problem of an Algebra book in Spanish I've been working on. The problem goes as following: If #A = 4 and #B = 7 and #A ∩ B = 3, then #A ∪ B is: (4), (7), (8), (9), (11) ...
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2answers
70 views

Is there a knock down argument showing that , in arithmetics, equality means identity? [closed]

If I am correct, addition on the set of natural numbers is a function that maps all ordered pairs of natural numbers to a single natural number( i.e. a function from $\mathbb N \times \mathbb N $ to $\...
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2answers
46 views

How many adjacent edges in an $n \times n$ grid of squares?

I'm trying to find a general expression for the number of adjacent edges in a $n \times n$ grid of uniform squares. A square can have adjacent edges above, below, to the left, or to the right, if ...
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18 views

How to find number of items per page and offset

I want to know number of items in a page and offset. For example, I have 20 items in an array and per page 9 items. Page 1, the number of items is 9 and offset is 0-8. Page 2, the number of items is 9 ...
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54 views

Using Karatsuba algorithm to multiply [Edit: Answered]

I've been looking over this, and haven't seen anything wrong in my procedure. Take 2468 *3162 [Using Karatsuba multiplication]: x=2468 y=3162 x={(a)24}{(b)68} y={(c)31}{(d)62} Step 1: a*c=(24)(31)=744 ...
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18 views

How to understand why the alligation technique works well here?

Question: "The ratio of expenditure and savings is 3:2. If the income increases by 15 % and the savings increases by 6 %, then by how much per cent should his expenditure increases?" Doubt: ...
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1answer
174 views

How much is 93 more than 47?

This was a question that i came across in my 7 year old daughters book. My wife and i have argued at length whether the two numbers should be added and subtracted. I think the statement is ambiguous ...
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0answers
37 views

Sum of fractions in lowest terms

Let $a, b, c, d$ be natural numbers. Suppose we want to find the sum $\frac{p}{q}$ of the fractions $\frac{a}{b}$ and $\frac{c}{d}$ in lowest terms, that is, $p$ and $q$ are relatively prime. Suppose ...
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1answer
28 views

How to solve this problem on alligations with a precision?

Problem: " A milkman brings 100 litres pure milk from a daily farmer and he sells 10 litres of it to the first customer, then he refills his vessel by adding 10 litres water. After this, he ...
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0answers
47 views

Find an integer $x$ such that $x \equiv 3$ (mod $4$) and $x \equiv 5$ (mod $9$)

For this question I basically used the Chinese Remainder Theorem from my book. I just want to make sure my reasoning and logic was correct: Notice that $4$ and $9$ are relatively prime so the ...
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1answer
24 views

Definition of a congruence in Chinese Remainder Theorem

I'm learning modular arithmetic right now and having a small hang-up in the application of the definition of a congruence. The book states that a congruence $c \equiv d$ (mod $n$) can be written if $(...
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62 views

Order of operations general

I have a question regarding the history of the hierarchy of operations. Why do multiplication and division have the same hierarchy? And even more so why operate from left to right, if they have the ...
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21 views

How do we add multiple binary bits with carry using boolean operators XOR and AND.

My question is similar to How do I add multiple binary numbers without using a partial sum?. For example, if we add two bits, a and b, then sum bit = a XOR b and carry bit = a AND b. Is there a way to ...
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1answer
45 views

Evaluating $3\lfloor{\frac{n}{2}}\rfloor + 6\lfloor{\frac{n}{2}}\rfloor (n - \lfloor{\frac{n}{2}}\rfloor -1)$ [closed]

$$3\left\lfloor{\frac{n}{2}}\right\rfloor + 6\left\lfloor{\frac{n}{2}}\right\rfloor \left(n - \left\lfloor{\frac{n}{2}}\right\rfloor -1\right) = \frac{3}{2}n(n-1)$$ How do you get the RHS? Do you just ...
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1answer
32 views

How to find the RHS of $\sum_{1 \leq a \leq \lfloor{\frac{n}{2}}\rfloor} (n-2a) $ [closed]

How does $$\sum_{1 \leq a \leq \lfloor{\frac{n}{2}}\rfloor} (n-2a) = \lfloor{\frac{n}{2}}\rfloor (n - \lfloor{\frac{n}{2}}\rfloor - 1)$$ I cannot see how you get to the RHS. This looks close to number ...
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45 views

Is a $\mathbb{Q}$-linear combination of square-roots of distinct square-free integers always irrational? [duplicate]

I want to know whether the following statements are true, the second one being a particular case of the first one. Let $n_1$, $n_2$, $\dots$, $n_k$ be distinct square-free positive integers. Then: $\...
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14 views

Measuring the error-rate

I have a theoretical function which returns positive rational results. Its implementation over a limited-resource platform returns positive integer results. I need to measure the error-rate of the ...
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1answer
28 views

the first 4 terms of an arithmetic are x,9,3x,3x+y find the sum of first 100 terms of this sequence [closed]

the first 4 terms of an arithmetic are x,9,3x,3x+y Find the sum of first 100 terms of this sequence.
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1answer
135 views

Find $n$ given average distance between $0 \le a \le n-1$ and $1 \le b \le n-1$

Given a natural number $n>1$, natural numbers $a$ are chosen uniform randomly between $0$ and $n-1$ i.e.: $$0 \le a \le n - 1$$ and natural numbers $b$ are chosen uniform randomly between $1$ and $...
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38 views

How to write an integer into a product [duplicate]

of two nontrivial rational numbers ? Given an integer $n$, suppose that it has two decompositions : $p^2 +q^2 = s^2 +t^2 \ $ ($p,q,s,t$ are all positive and $(s,t) \not \in \{(p,q),(q,p)\} \ $). We ...
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21 views

Quotient Property of Square Roots: why does GeoGebra seem not to care about it?

I was arguing with my friend about the Quotient Property of Square Roots and it resulted in an exchange of results between my knowledge and WolframAlpha's, and that of GeoGebra; the latter "...
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0answers
37 views

Algorithm to add continued fractions

Lets say I have two continued fractions: $$a=[a_0; \overline{a_1, a_2,a_3...}]$$ $$b=[b_0; \overline{b_1, b_2,b_3...}]$$ How do I add them together?
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37 views

Prove the following properties of $\mathbb{Z}_n$ [duplicate]

I am looking to prove the following properties of $\mathbb{Z}_n$ Commutativity and associativity of addition $[0]$ is identity element for addition and existence of additive inverses Commutativity ...
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3answers
93 views

When to know when it's possible to square an equation

I do understand that if: $a=b \Rightarrow a^2 = b^2 $ But clearly, the graph representing these two equations won't be the same. So, (correct me if I'm wrong) this would suggest that if you square ...
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40 views

Why are we using GCD here in the given problems?

There are two questions where I am having difficulty to solve. One of the question asks us the calculate the minimum number of square tiles required to cover a floor of dimension 247 ft x 209 ft. The ...
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0answers
65 views

Addition and subtraction by natural algorithms

We compute additions and subtractions to model phenomena in Nature. That's perfect. In our first learnings, we play with stones, by arranging them on one side to add and by eliminating pairs, common ...
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24 views

Why does the Lehmer's conjecture imply the Short Geodesic Conjecture?

I need some translation help. In this article (https://homeweb.unifr.ch/kellerha/pub/IML-2013summer4-01.pdf), on page 15 of the pdf, it says that the "short geodesic conjecture" is a ...
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45 views

How to convert a numerical value to a percentage of 100?

So, consider that are three tests a student has taken. The overall score is 100 points. The test has three sections. Section 1 - 30 points Section 2 - 30 points Section 4 - 40 points Student scores ...
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1answer
44 views

Series to Sum Equation

I have this series and I want to convert it to the Summation form. $1+(1+\alpha _{1})+(1+\alpha _{1})(1+\alpha _{2})+(1+\alpha _{1})(1+\alpha _{2})(1+\alpha _{3})+...$ I know that the next series is ...
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0answers
29 views

Fractional part sequence

I am having one question that ask finding one sequence defined by $b(n) = \{a_n\} = a_n - [a_n]$ , where $\{x\} = x - [x]$ is a fractional part of a real number $x$ and $[x]$ is the largest integer ...
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33 views

Frequency of Digits Appearing in the Decimal Part of $\sqrt 2$

There's no doubt that the decimal part of $\sqrt 2$ is determined and unique, but do the digits 1~9 appear evenly? More precisely, suppose we take the first $N$ numbers, count the number of times ...
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0answers
26 views

Maximum number of composite numbers that can be produced for a set of numbers

Assume a set of numbers {2, 3, 4} where n is 3, or the size of the set. Is there a way I can calculate the total unique composites that could be produced from pairs in that set? The total number of ...
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2answers
35 views

what is the exact requirement in the given problem? [closed]

Question : "In my office the average age of all the female employees is 21 years and that of male employees is 32 years, where the average age of all male and female employees is 28 years. The ...
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1answer
4k views

Help with a quantitative aptitude question.

In a factory packing red and green apples, 6% of the apples were found to be defective. Also, 90% of the non-defective apples were red apples. If 4700 non-defective apples were all green, then how ...
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0answers
38 views

In what proportion should water be mixed with spirit to gain 12.5% by selling it at cost price?

How to solve this problem by using laws of proportions analytically? In what proportion should water be mixed with spirit to gain 12.5% by selling it at cost price? Doubt: I feel that the problem is ...
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0answers
33 views

Why is the unemployment ratio considered to be double in this context?

Question : "If 98 out of 100 children were in school before, but now 96 in 100 are enrolled, that is a fall in enrollment rate of 2% (98%-96%=2%). You could also say the number of unenrolled ...
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1answer
36 views

Dividing Rs.69 among 115 students, each girl gets 50 paise less than a boy; each boy gets twice the paise as each girl. How many girls in the class?

Question: Rs.69 were divided among 115 students so that each girl gets 50 paise less than a boy. Thus each boy received twice the paise as each girl received. The number of girls in the class is? ...
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2answers
52 views

Prove that if $k\mid q_1...q_n$ then we can find $k_i$ such that $k=k_1...k_n$ and for every $i$, $k_i\mid q_i$

Let $k=8$ and $q_1=12$, $q_2=2, q_3=5$. We note that $k\mid q_1q_2q_3$, but $k\nmid q_1$ and $k\nmid q_2$ and $k\nmid q_3$. Also $$ 8=k_1k_2k_3=4*2*1, $$ where $k_1\mid q_1$, $k_2\mid q_2$, and $k_3\...

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