0
votes
2answers
34 views

How many boys, girls, men and women are there?

In a village, there are exactly $10$% more boys than girls; $15$% more women than men; $20$% more children than adults. The population is less than $6000$. Solution: $b = g + 0.1g$--------(i), ...
14
votes
3answers
192 views

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. ...
6
votes
3answers
101 views

Confused about the $\pm$ sign?

I have multiple questions about the $\pm$ sign, since it seems to confuse me in general... Question 1: Say I have $15=\pm(a+x)$, Can I use the distributive property so it becomes $15=\pm a \pm x$? ...
0
votes
1answer
31 views

Quadratics help please [closed]

Does it matter when you are solving quadratics and you get an or would it matter if you said does x always go first $x<9$ or is it the same as $9<x$?
2
votes
6answers
76 views

Why is the result of $-2^2 = -4$ but $(-2)^2 =4$?

I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $?
5
votes
5answers
184 views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
2
votes
3answers
56 views

Ambigous question regarding how to view surds with numbers infront

Say I want to multiply 2 by 5$\sqrt3$ . Do I firstly do 2 * 5, then 2 * 3? I'm not sure about the order of operations here. Such a dumb question, I know. Edit - can someone show me the systematic ...
-1
votes
2answers
51 views

Squared binomial paradox?

When you square this $$(5-2)^2$$ you will get 49 $$ 5^2 - 2 * 5 * (-2) + (-2)^2$$ $$25 + 20 + 4 = 49$$ but if you do it like this (5-2) * (5-2) you will get 9 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
5
votes
5answers
114 views

How $\sqrt{2}=1+\frac{1}{\sqrt{2}+1}$?

I have found it in the chapter about chain fractionals. I am unable to transform it to such state. $$\sqrt{2}=1+\sqrt{2}-1=?=1+\frac{1}{\sqrt{2}+1}$$
0
votes
3answers
52 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
2
votes
2answers
152 views

Why we can't define $\frac{1}{0}$ to be $1$ (or anything else), but we can define $1^0$ to be $1$?

We know that we can't define division by zero "in any mathematical system that obeys the axioms of a field", because it would be inconsistent with such axioms. (1) Why can we define $a^0$ ($a\neq 0$) ...
0
votes
1answer
32 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
0
votes
1answer
80 views

Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
0
votes
0answers
30 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
3
votes
2answers
145 views

Simple subtraction that I can't figure out. [duplicate]

A bat and a ball cost £1.10 in total. The bat costs £1 more than the ball. How much does the ball cost? The answer to this question is somehow 5p. How?!! Should it not be 10p?
3
votes
1answer
53 views

Simplify $\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$

I am trying to find the value of: $$\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$$ I have the key with the answer $\sqrt 5$ but am wondering how I can easily get to that ...
4
votes
2answers
79 views

How prove that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $

How check that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $?
0
votes
2answers
52 views

Simplifying Surd Fractions

can someone show me how to simple surd fractions such as: $$\frac{{8\sqrt 3 }}{2}$$ Can someone please help me here?
2
votes
3answers
164 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
4
votes
2answers
67 views

How to turn arbitrary fractions into arbitrary egyptian fractions?

I am reading Stillwell's Numbers and Geometry. There is an exercise about Egyptian fractions which is the following: I've tried to do it in the following way - Expressing an arbitrary fraction ...
0
votes
3answers
47 views

When a fraction is raised to a negative exponent, do you normally transform it to 1 over the fraction, or invert the fraction?

My text shows that $$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$ It also shows that $$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$ In the first equation, it seems ...
1
vote
2answers
79 views

Why does $(3^{1/2})(10^{1/2})=30^{1/2}$ but $(3a^2)(10a^2)=30a^4$?

$(3a^2)(10a^2)=30a^4$? In that equation the exponents are added. Why does $(3^{1/2})(10^{1/2})=30^{1/2}$. In that equation the exponents are not added. Why?
2
votes
2answers
45 views

Square root each term (clarification on polynomials?)

So I'm in Algebra 2, and right now we're learning about conic sections (circles/ellipse/etc). I thought some problems in the workbook looked weird, like this one: ...
2
votes
3answers
63 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
0
votes
1answer
43 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
0
votes
2answers
34 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
2
votes
1answer
93 views

How to find the integer part of big number?

How to calculate the integer part $$\left \lfloor10^{10^{10^{10^{10^{-10^{10}}}}}} \right \rfloor ?$$ Does this equal $$10^{10^{10}}? $$ Both Maple and Mathematica fail with it. PS. Unmotivated votes ...
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
3
votes
8answers
357 views

$\left(\sqrt{8}+\sqrt{2}\right)^2$ = 18 why??

I would like to now why $$\left(\sqrt{8}+\sqrt{2}\right)^2$$ is equal to $18$? Please provide me the proccess. Thank you.
0
votes
1answer
21 views

Really Simple Re Weighting Question

I am having a major brain cramp and cannot remember how to to this. In a perfect world I have 7 types of activities that I could have participated in. These 7 activities each have a weight that ...
0
votes
2answers
68 views

A man invests an amount of $ ₹ 15860 $ in the names of his three sons A, B & C in such a way that they get the same interest..

A man invests an amount of $ ₹ 15860 $ in the names of his three sons A, B & C in such a way that they get the same interest after $2, 3$ & $4$ years respectively. If the rate of simple ...
6
votes
5answers
210 views

Simple solving Skanavi book exercise: $\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}$

Simple way to solve this exercise $$ x = \sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}} $$
1
vote
3answers
61 views

Given a rational, how to find the integers whose quotient it is?

I haven't found an answer to this anywhere. Excluding brute force, given a rational $q$ in its decimal form ($1.47$, for example), is there a good algorithm to find integers $m$ and $n$ such that ...
5
votes
1answer
59 views

Halmos on Definability and Luzin on Division by 0

For a successful introduction of a new symbol (e.g. '$\emptyset$') into a mathematical discourse it is necessary and sufficient that the symbol refer to something (e.g. Existence + Specification in ...
0
votes
2answers
39 views

How do you get from this to this formula?

I have the formula : $$3×4^{n-1}×1×\left({1\over 3}\right)^{n-1}$$ And I would like to know how to get to this one (which is equal) : $$3× \left({4\over 3}\right)^{n-1}$$ How can I do that ?
1
vote
3answers
55 views

Inequality involving conjugate numerator/denominator pairs

Question is to solve: $$\frac{(x-2)(x-4)(x-7)}{(x+2)(x+4)(x+7)} > 1$$ I thought I could negate terms to make them equal (i.e. $-(x-2)$), but that does not happen. I could subtract $1$ from ...
2
votes
3answers
45 views

Solving this inequality

Question: Solve: $$\frac{5x-6}{x+6}<1$$ My attempt: $$\frac{5x-6-x-6}{x+6}<0$$ $$\Rightarrow \frac{4x-12}{x+6}<0$$ $$\Rightarrow \frac{x-3}{x+6}<0$$ $$\Rightarrow (x-3)(x+6) < ...
1
vote
2answers
56 views

Isn't this wrong?

This worksheet This question: $$w^2 - w \leq 0$$ This answer: $$(-\infty, -1] \cup [0, 1]$$ Isn't this wrong ? At $w = -2$, it becomes: $(-2)^2 - (-2)$, which is $4 + 2$, which is $\geq 0$. But ...
1
vote
3answers
42 views

How to solve this inequality question without manual checking?

Question: Find the maximum integral value which satisfies: $$\frac{x-2}{x^2-9}<0$$ I know that this means either of the following: #1. $x-2<0$ and $x^2-9>0$. Implies that $x \in (3, ...
3
votes
3answers
242 views

Which is greater: $1000^{1000}$ or $1001^{999}$

Question: Find the greater number: $1000^{1000}$ or $1001^{999}$ My Attempt: I know that: $(a+b)^n \geq a^n + a^{n-1}bn$. Thus, $(1+999)^{1000} \geq 999001$ And $(1+1000)^{999} \geq ...
0
votes
2answers
34 views

Solving two systems with two unknown?

Let's say if we are giving the following two equations: $$ 1= X/(X^2 +Y^2) $$ $$ 2= Y/(X^2 +Y^2) $$ How are we going to solve for X and Y [ by HAND ] ? Why would Summing the squares of the two ...
1
vote
1answer
21 views

Comparing powers with different bases without logarithms

I want to compare : $17^{31}$ and $31^{17}$ , this is a solution but I want another one and without using logarithms, only using the fact that $17=16+1=(2^4)+1$ and $31=(2^5)-1$ how could it ...
-5
votes
2answers
52 views

PEMDAS: Why do I get different values when “trying” to use PEMDAS? [closed]

The real question is why is $(2 + 2)^2 = 16$ and $(a + b)^2 = (a + b) (a + b)$?
1
vote
0answers
30 views

Arithmetic and Algebra exercises on latex source code.

I´m currently writing a little book for two student that I teach. The book covers school arithmetics and algebra, and it include theory and examples. Since I don´t have time to prepare a good sets of ...
0
votes
1answer
33 views

What expression represents the total cost?

A customer calculated the cost of a new jacket , c, including a 7% sales tax, by multiplying 0.07 times the cost of the jacket and adding the product to the cost of the jacket. What is another way to ...
1
vote
5answers
164 views

Algebra problem stumping me

I have recently run into an algebra problem that goes as follows. Using the digits $1$ to $9$, $$ \left\{ \begin{align} A + B + C + D &= EF \\ E + F + G + H &= CJ \\ B + G + J ...
2
votes
1answer
52 views

Formula for this pattern

I am trying to develop a computer program to compute the tax for a given base salary, I believe given the format of the income tax table that I have there should be a formula to calculate the tax for ...
1
vote
1answer
59 views

Why ${(a^2)}^{\frac 12}=\sqrt {a^2}=|a| \neq a$?

Let $a\in \mathbb R$. It should be true that $\sqrt {a^2}=|a|$, since $\sqrt {(-2)^2}=\sqrt{2^2}=2$ and so on. But, it is also true that ${(a^2)}^{\frac 12}=a$, and by definition, ${(a^2)}^{\frac ...
0
votes
4answers
84 views

A seemingly basic PEMDAS problem… [duplicate]

There's one of those meme-type images posted on Facebook with the equation 6/2(1+2), challenging you to solve it. So, parenthesis first, ...
4
votes
1answer
85 views

Calculate the summation of double continued fractions

A few month ago, my brother had given me this question: \begin{equation} \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{\cdots+\frac{1}{2005}} } } }+\cfrac{1}{1 + ...