3
votes
1answer
48 views

Arithmetic progression with common difference 2061

If there are 30 consequent members of an arithmetic progression with CD of 2061, show that among them are at most 20 squares of natural numbers. I wrote out $a_1$ through $a_{30}$ and tried to find ...
1
vote
4answers
795 views

Solved to be 7 after arithmetic

I recently made a blunder while trying to explain a question asked to me in an interview, The question was Think of $X$ Add $X$ to itself ($X+X = y$) Times the result by $3$ ($y\times 3 = z$) ...
6
votes
3answers
88 views

Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that: $ 6 \cdot 3.7 = 22.2$ than it is to see that $ 22.2 \div 6 = 3.7 $. Thoughts about the roots of this asymmetry? An analogous ...
0
votes
3answers
49 views

How to simplify $\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$

I in trouble simplifying this: $$\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$$ couldn't find a solution. Can you help?
1
vote
3answers
147 views

Find the distance between two towns given train timings

While practicing maths and starting to learning it, I found question this question: A train running between two towns arrives at its destination 10 minutes late when it goes 40 miles per hour and ...
0
votes
1answer
34 views

How to show that$\ \sqrt[3]{ \sqrt{y^2-x}+y}+ \sqrt[3]{-\sqrt{y^2-x}+y} = k \implies y = \frac{k\left(k^2-3 \sqrt[3]{x}\right)}{2}$?

We also have $\ x \ne y $, $\ y > 1$, $\ 0<x<1$,$\ k \ne 0$. I have tried on my own, by canceling out the roots, but they keep on appearing. I guess that is not the right way. Thanks in ...
0
votes
1answer
35 views

how many questions did D answer correctly

Each of A, B, C, and D took a test. Each of them answered at least one question correctly, and altogether they answered 67 questions correctly. A had more correct answers than anyone else. B and C ...
0
votes
1answer
27 views

Elementary arithmetic question

2 groups of people $A$ and $B$ are trying to build a road. For the first 40 days, only one group was working at any time. At first, only group $A$ worked. They worked for an unknown amount of days, ...
1
vote
1answer
99 views

Arithmetic sequence of natural numbers

Consider an arithmetic progression of natural numbers with a non-zero common difference. For each of the members of the progression its square root is taken, and if the square root is not an integer, ...
4
votes
1answer
59 views

Algebra problem solve for a,b,c and d?

Can anyone find the values of these integers: a,b,c and d? $$1+\sqrt{2}+\sqrt{3}+\sqrt{6} = \sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$ a+b+c+d = ? Thank you.
2
votes
1answer
62 views

Finding the integer solutions of the equation $3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14$

$ 3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14 $ . I already solved this using the Cauchy–Schwarz inequality and got $x=4$ and $y=5$. But I'm sure there is a prettier, simpler solution ...
-1
votes
1answer
22 views

Biking uphill and downhill

During an interview, I was asked "If you can bike 20 mph uphill and 30mph downhill, and you have 1 hour to bike, how far or how long should you ride uphill before turning back." While a very ...
0
votes
1answer
22 views

a factor in the numerator is the opposite of the denominator - simplifies to -1

I'm working on a little khan academy problem, finding the limit as x -> 36 in the solution the program explains in the last step that since there are opposite ...
0
votes
1answer
30 views

From an expression raised in a power of 2 to an expression raised in the power or 10

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent. Example: I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from ...
23
votes
5answers
4k views

In primary school I was showed this. Why does it work?

When I was in primary school a teacher showed us the following exercise in arithmetic. Take any 3 digit number between 201 and 998 provided that the hundreds digit is bigger than the ones digit and ...
4
votes
4answers
72 views

Rationalize $\left(\sqrt{3x+5}-\sqrt{5x+11} -\sqrt{x+9}\right)^{-1}$

I was trying to find if there a method similar to multiplying and dividing by the conjugate $$\frac{1}{\sqrt{3x+5}-\sqrt{5x+11} - \sqrt{x+9}},$$ but that doesn't seem to work here. Also, is there a ...
0
votes
1answer
18 views

SAT Math Problem about decimal

In the decimal representation of $\frac{1}{k}$, where $0 < \frac{1}{k} < 1$. the tenths digit is $1$, hundredths digit is $3$ and at least one other digit is nonzero. What is the tenths digit ...
6
votes
4answers
1k views

Why exactly does the distributive property work?

Suppose I have this expression that needs to be simplified: $$4(2x + 4)$$ It can be simplified down to this: $$8x + 16$$ In this case, this expression has been simplified down using the ...
0
votes
3answers
45 views

How to find the total investment from interest received

Dave Horn invested half of his money at $5$%, one-third of his money at $4$%, and the rest of his money at $3.5$%. If his total annual investment income was $\$530$, how much had he invested? I found ...
0
votes
2answers
52 views

Is there a general way to do arithmetic involving binomials more quickly?

I'm talking about exercises like these for example: $ (a+2b)^3 - (a-2b)^3 $ $(a+b+c)(a+b-c)(a-b+c)(a-b+c)(-a+b+c)$ Of course these can be done the time-consuming and mentally easy way, but are ...
0
votes
2answers
31 views

Simplifying $0.300 (1 \pm 0.0633)$

This problem had to do with finding area with uncertainty, I got this far but I'm not sure how to go on. The answer to the next step is $(0.300 \pm 0.0190)$. How do they get this? What do we do with ...
1
vote
5answers
46 views

Not clear on what we mean with numbers with infinite digits

I am confused on a rather simplistic question. 1/3 = 0.333333333333 to infinity. So it has infinite digits. How is it possible to multiply such a number with another one and get a finite number? 6/3 = ...
0
votes
2answers
46 views

How do break down this addition?

I've been given the following expression: $2(a + b) + (n + 1)(2a + c) + 2n(2a + d + b) + (a + r)$ And I've been told that it can be simplified to: $n(6a + 2b + c + 2d) + (5a + 2b + c + r)$ I've ...
1
vote
2answers
84 views

How to simplify $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1}$?

This is the original problem: $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1} = x$. I'm really confused about how to solve this problem, I come as far as saying this: $\sqrt[4]{5} + \sqrt{1}\cdot ...
0
votes
2answers
40 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), ...
16
votes
3answers
257 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
110 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 $15=\pm a \pm x$? ...
2
votes
6answers
79 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
195 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
58 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
58 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
130 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
69 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 ...
3
votes
2answers
160 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
51 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
81 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
1answer
35 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
154 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
60 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
85 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
56 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
74 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
52 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
88 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
49 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
67 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
50 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
43 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
99 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 ...