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2 votes
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Easy way to convert from one fiat to another without using a calculator

First you have to understand the quote you are given. If you are told the conversion is $1.6$ to $1$, is that $1.6$ of $A$ is the same as $1$ of $B$ or the other way around? That tells you whether ...
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3 votes

Easy way to convert from one fiat to another without using a calculator

I'm trying to convert from one fiat to another without using a calculator. In my case, I was trying to convert 25 TRY to BGN. Let's assume that the daily rate for 1 BGN is 13 TRY. That means 25 * 0.13 ...
1 vote

Prove $2^n + 2^m ≠ 2^p$

You might enjoy this view of the question, which provides an alternative proof. Writing $2^m$ in binary, we have a $1$ followed by $m$ $0$'s. For example: $2^1: 10 \\ 2^5: 100000$ How can we add two ...
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2 votes
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How can multiplication by 0 be defined as a many to one mapping to, or compactification of a point at, origin.

As I understand it the plan comes in two versions, which are the same except for when we do this or that. One version, as far as I can make out, is this: The Plan: Start with the line, $\Bbb R$. ...
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-1 votes

Finding sum borrowed when two unequal annual instalments are given

We need the equality of the present value of the borrowed money $C_0$ and the present value of the two installments $I_1$ and $I_2$. I assume that the first installment is paid at the end of the first ...
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1 vote
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Find $n$ given average distance between $0 \le a \le n-1$ and $1 \le b \le n-1$

I will start with a few points aimed at the original poster, which they may find useful for the future; the actual problem at hand is going to be dealt with at the very end. The original problem we ...
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1 vote

How to "model" $\frac{6}{x-9}$

You have to drive $6$ km, but you forgot to change the car's summer tires to winter tires, so you have to drive $9$ km per hour slower than usual as a safety precaution. How long will it take you to ...
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1 vote

How to "model" $\frac{6}{x-9}$

There are $x$ people in a life boat. There are $6$ gallons of drinking water in the boat. $9$ of the people get swept off the boat by a big wave and drown (sorry about that). How much drinking water ...
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0 votes
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Fraction substraction arithmetic

First, expanding the fractions $\left(3+\frac{3}{8}\right)-\frac{5}{6}-\left(1+\frac{2}{3}\right) = \frac{24+3}{8}-\frac{5}{6}-\frac{3+2}{3}=\frac{27}{8}-\frac{5}{6}-\frac{5}{3}$ Next, finding a ...
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  • 52
0 votes

Is it possible to calculate the arithmetic mean from the geometric mean?

Since you want a estmated relationship between AM and GM, and there's no further constraint on the data, I can present a widely used formula for approximation in finance. Hope this will light you up a ...
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  • 1
2 votes

How get the cost price?

One way is $C=46200-B$. I am sure that is not what you are looking for, but it gets all the right numbers. Added: with your addition that you want to understand $$X+3\%=24200$$ that should really be ...
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0 votes

Show that $a^3+5a$ is an integer

From the Binomial Theirem: $(u+v)^3=u^3+3u^2v+3uv^2+v^3=(u^3+v^3)+3uv(u+v).$ So with $u=\sqrt[3]{1+\sqrt{\dfrac{152}{27}}}$ $v=\sqrt[3]{1-\sqrt{\dfrac{152}{27}}}$ (where in the expression for $v$ we ...
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4 votes

Show that $a^3+5a$ is an integer

Let $x=1+\sqrt{\frac{152}{27}},~~y=-1+\sqrt{\frac{152}{27}}$ $$x^{\frac{1}3}y^{\frac{1}3}=\frac{5}{3}$$ $$a=x^{\frac{1}3}-y^{\frac{1}3}$$ $$\begin{align} a\left(x^{\frac{2}3}+x^{\frac{1}3}y^{\frac{1}3}...
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7 votes
Accepted

Show that $a^3+5a$ is an integer

Let $x=\sqrt[3]{1+\sqrt{\frac{152}{27}}}$ and $y=\sqrt[3]{-1+\sqrt{\frac{152}{27}}}$ . Then $$\begin{aligned}a^3+5a&=(x-y)^3+5(x-y)\\ &=x^3-3xy(x-y)-y^3+5(x-y)\\ &=x^3-y^3+(5-3xy)(x-y) \...
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  • 9,717
0 votes

Fraction substraction pre algebra

To rephrase lulu's excellent answer in the comments: $3 \frac{15}{20}$ is the same as $3 + \frac{15}{20}$. Now we can convert $1$ into the fraction $\frac{20}{20}$, so we have $2 + 1 + \frac{15}{20} = ...
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  • 15.9k
0 votes
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Ways to split water bill based on person and days

You have to calculate the person hdays. so multiply the number of persons with the number of days the stay. keep the information for all apartments, then add all numbers up and divide the cost by this ...
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10 votes
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Why counting the number of 1 digits that appear in all integers in 0-9, 0-99, 0-999, 0-9999 follow an arithemtic-geometric sequence?

For the purpose of this question, let's consider that a number between $0$ and $999...9$ = $10^n-1$ ($n$ digits $9$) is always written with $n$ digits, by adding leading zeros when necessary. Then all ...
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1 vote

Is absolute value an algebraic operation?

The absolute value (modulus) is a function that takes in any real number and makes it its non-negative value without regard to its sign. The definition of the absolute value of any real number $a$ is ...
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0 votes

Is there an algebra equivalent to the matrix cookbook, IE a collection of properties or formulas that you can look at for reference?

The CRC Standard Mathematical Tables and Formulas has many formulas like these -- factorizations, hyperbolic trigonometric identities, etc. (In some earlier editions the title said "formulae"...
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1 vote

Finding the ages of father and son without using algebraic equations

It's the same as dividing $60$ (cookies, for example) into the ratio $3:1$. The sum of the parts, which are all equal, is $60$. Thus $4 \ \text{parts} = 60 \implies 1 \ \text{part} = 15$, and one ...
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  • 15.9k
0 votes

Can you solve this stolen money riddle?

OK, first people are trying to answer this riddle like it's a bottom-line question. It's a top-line question - don't complicate a simple riddle. Second, people seem to be looking at this linearly. The ...
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  • 1
1 vote
Accepted

Show that $\forall k,j,m,n \in \mathbb{N}^: 3kn^2 +3jn +3m+2 \notin S=\{x^2 \mid x\in \mathbb{N}\}$

Move the LHS $2$ to the RHS, we get a multiple of $3$ on LHS. RHS is $x^2-2 = x^2-1-1 = (x-1)(x+1)-1$ If $x$ is $1$ more than or $1$ less than a multiple of $3$ , then this RHS $(x-1)(x+1)-1$ is not a ...
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  • 750
3 votes
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If $1/16$ of $x$ is $120$, what is the $1/4$ of $x$?

That's certainly one way to look at it, yeah. Just multiply the first equation by $4$: $$\frac{1}{16} x = 120 \implies \frac{1}{16} x \cdot 4 = \frac 1 4 x= 120 \cdot 4 = 480$$ You could also solve ...
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  • 31.8k
1 vote

Explain the mechanism behind the "1" in the Present value calculation of money

So many people answer this question by stating the formula and not the calculation behind it; they expiate on the contents of the formula,$P, N, R$, etc, but never refer to the "$1.$" An ...
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  • 11
3 votes

How to determine what integer value of $j$ returns an integer when plugged in to $\sqrt{5j + 1}$ without trial and error?

What characterizes numbers which are one more than a multiple of $5$? Answer: Such numbers have a $1$ or a $6$ in the ones digit. What characterizes numbers whose squares have either a $1$ or a $6$ in ...
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3 votes

How to determine what integer value of $j$ returns an integer when plugged in to $\sqrt{5j + 1}$ without trial and error?

If we write $n^2=5j+1$, we can see that every value of $n^2 \equiv 1 \pmod 5$. That means $n^2$ must end in $1$ or $6$, which implies that $n$ ends in $1,4,6,$ or $9$. But our choice of $n$ is free; ...
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  • 1,534
6 votes

Express $|\pi - \dfrac{23}{7}|$ without the absolute value symbol

Here's an idea that, like insipidintegrator's answer, avoids an approximation. Use a circumscribing polygon to upper-bound the circumference of the circle. I use a dodecagon, whose perimeter is $$ P = ...
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  • 31.7k
19 votes

Express $|\pi - \dfrac{23}{7}|$ without the absolute value symbol

I will not use any approximations in this answer. Consider the definite integral $$\int_0^1\frac{t^4(1-t)^4}{1+t^2}dt.$$ Simply expand the numerator using binomial formula and reduce the numerator in ...
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2 votes

Express $|\pi - \dfrac{23}{7}|$ without the absolute value symbol

It has been known for more than 22 centuries that $\pi<22/7$ (Archimedes). But if you don't know it, you can do primary school division: \begin{array}{cccc|l} 2&3&0&0&\underline{7~~~...
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  • 230k
4 votes

Express $|\pi - \dfrac{23}{7}|$ without the absolute value symbol

You have already said that $\frac{21}{7}$ is 3, and that both $\pi$ and $\frac{23}{7}$ are greater than 3. But, $\frac{23}{7}$ would be $3\frac{2}{7}$, and we know that the first digit of $\frac{2}{7}$...
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1 vote

Why is -4 x -4 = +16 and when added its -16

If you can agree that $(-4) = -(4)$, we can do the following: \begin{align} (-4)\cdot(-4) = -(4)\cdot(-4) = -\left[4\cdot(-4) \right]. \end{align} If you know the addition of $4\cdot(-4)$, you can ...
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1 vote

Billing distribution

The answer depends on how the billable service is charged. If the service has a fixed daily price independent of use (for example - unlimited internet at price 3.87 per day), then the first method is ...
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  • 1,101
4 votes
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Why does $-\frac{1}{17-x}$ equal $\frac{1}{x-17}$?

It comes from the following algebra facts: $$-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}$$ and $$\begin{align} -(c-d) &= (-c)-(-d)\\ &=-c+d\\ &=d-c \end{align}$$ (For this second fact we first &...
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3 votes

Why does $-\frac{1}{17-x}$ equal $\frac{1}{x-17}$?

You can break this down into a couple of basic facts. First of all, we have $\frac{1}{-1}=-1$. Thus, if you want to see the opposite, or negative, of a fraction, we can multiply either the numerator ...
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1 vote
Accepted

Calculating a discount when getting better currency rates

Shop B offers a $\dfrac{13-10}{13}=23\%$ discount over Shop A. (For example, a $130€$ items costs you $13C$ in Shop A but only $10C$ in Shop B.) If Shop A's exchange rate remains unchanged, then Shop ...
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  • 28.5k
1 vote

Calculating a discount when getting better currency rates

This is an application of dimensional analysis. If the item is listed as $10€$ in both shops, then in terms of $C$ that works out to \begin{eqnarray} shopA:&&10€ = 10€ \frac{1C}{10€} = 1 C\\ ...
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0 votes

Solving $-1=e^a-2e^{av}$ as part of an equation system

1) $$f_2(x)=e^{ax-b}+c$$ Your constraints yield the equation system $$\left\{e^{-b}+c=0,e^{a-b}+c=1,e^{av-b}+c=\frac{1}{2}\right\}.$$ This equation system has solutions if $$b=\ln(e^{a}-1)+2k_1\pi i\ \...
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  • 4,509
2 votes

Mercari says something like, "Your purchase is actually half price". But I don't think so. (Keyword: Shopping)

It is completely acceptable for the company to say that. I read the example that you provided. Mercari says that it the discount is 50% off the original price of the product. I believe that the ...
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  • 21
1 vote

A question on compound interest?

The way compound interest rates are normally defined, an interest rate of $r$ compounded $n$ times per year means that after one year, you have $(1 + \frac{r}{n})^n$ times your original principal. ...
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