Patrick Reany
Abstract: This page presents mathematics diversions, which are basically just
math problems of varying difficulty, which one can try to solve for fun or for
testing one's math skills, or for learning new methods and concepts.
Disclaimer: Many of the following problems were taken from SATs or old
Olympiad tests. My alternative solutions may not follow Olympiad rules, however.
The techniques I use may not be fit for use in Olympiads or entrance exams or
the like. I believe that it's good to see alternative solutions to better fill out one's
mathematical abilities. Four decades ago, I quipped that one cannot truly
understand a theorem until one is able to prove it in essentially two different ways.
It's meant to be a thought-provoking claim, perhaps containing a morsel of truth.
Announcement: If I'm to continue posting on this page, I need to broaden
the scope of the mathematics I cover here. So, I intend to include topics on
groups, rings, matrices, abstract algebra, partial differentiation, word problems,
integration, chemistry, physics, and others. I also intend to include some
theory and proofs.
Note: The comments 'Unipodal' or 'Not Unipodal' placed besides the problems
is there to indicate whether or not the problem solution has incorporated the
Unipodal algebra or not. If it has, the proof will be 'nonstandard', though
(hopefully) still correct.
Note: Here are some short monographs on standard math that I use in my papers:
Link to my write-up on Basic Complex Numbers.
Link to my write-up on FlowCharting Math Proofs.
Link to my write-up on Basic Geometric Algebra.
Link to my write-up on Geometric Series.
Link to my first write-up on Group Theory 1 (very basic).
Link to my write-up on Hyperbolic trig functions over the real numbers.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Link to my write-up on A History of Scheme for solving word problems and stoichiometry problems.
Link to my write-up on Pascal's Triangle.
Link to my write-up on An introduction to stoichiometry (long version).
Link to my write-up on An introduction to stoichiometry (short version).
Link to my write-up on Trigonometric functions.
Link to my write-up on Word Problem solving.
Link to my write-up on Basic Matrix Algebra.
Link to my write-up on Mathematical Induction.
Link to my write-up on GCD & LCM.
Link to my write-up on Virtual Emplacement.
Link to my write-up on the Fibonacci sequence.
Link to my write-up on the Method of Partial Fractions.
Link to my write-up on Set Theory Basics.
Link to my write-up on Basic Ring Theory, 1.
Link to my write-up on Basic Gibbs Vector Calculus.
Link to my write-up on Basic Integration Techniques. (Short Paper)
Link to my write-up on Basic Probability.
Link to my write-up on The Unipodal Algebra.
To go to Diversions1 page Link to Diversions1
To go to Diversions2 page Link to Diversions2
To go to Diversions3 page Link to Diversions3
To go to Diversions4 page Link to Diversions4
To go to Diversions6 page Link to Diversions6
Problem 800 [Not Unipodal: Stoichiometry]
Source: Chemical Principles: The Quest for Insight
Title: The Stoichiometric Limit
Presenters: P. Atkins and L. Jones
A tablet of vitamin C was analyzed to determine whether it did
in fact contain, as the manufacturer claimed, 1.0 g of the vitamin.
One tablet was dissolved in water to form 100.00 mL of solution,
and 10.0 mL of solution was titrated with iodine (as potassium
triiodide). It required 10.1 mL of 0.0521 M I3- aq to reach the
stoichiometric point reacts with 1 mol vitamin C in the reaction,
is the manufacturer's claim correct?
Solution to the problem.
Link to my write-up on An introduction to stoichiometry (short version).
Problem 801 [Not Unipodal: Calculus: Quotient Rule]
Source: The Ether of Great Mathematical Ideas
Title: The Quotient Rule
Presenter: Patrick
Given the relation \begin{equation} \phi(x) = \frac{f(x)}{g(x)}\,, \end{equation} where $f(x)\ne0, g(x)\ne0$ are differentiable functions, show that \begin{equation} \phi'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\,. \end{equation}
Problem 802 [Not Unipodal: Algebra]
Source: The Ether of Great Mathematical Ideas
Title: The Hidden Quadratic Equation
Presenter: Patrick
Given the relation \begin{equation} 2^x + 12(2)^{-x} = 7\,, \end{equation} solve for the real values of $x$.
Problem 803 [Not Unipodal: Algebra: Logarithms]
Source: https://www.youtube.com/watch?v=i-Q74u_tPHo
Title: Stanford math tournament algebra tiebreaker
Presenter: blackpenredpen
Given the relation \begin{equation} \frac{1}{\log_8 n} + \frac{1}{\log_n \fourth} = - \frac{5}{2}\,, \end{equation} solve for the integer values of $n$.
Problem 804 [Not Unipodal: Algebra: Linear Algebra: Fibonacci]
Source: The Ether of Great Mathematical Ideas
Title: Prooof of the Binet Formula and the Cassini Identity
Presenter: Patrick
We are returning to the Fibonacci numbers that we already met in Problem 796.
Given
\begin{equation}
F_n = \frac{\varphi_{+}^n-\varphi_{-}^n}{\sqrt{5}} \,,
\end{equation}
prove the Binet Formula and after that, prove the Cassini Identity using linear algebra.
Problem 805 [Not Unipodal: Algebra]
Source: The Ether of Great Mathematical Ideas
Title: Attack of the square roots
Presenter: Patrick
Given the relation \begin{equation} x = \sqrt{x}^{\sqrt{x}}\quad\mbox{where}\quad x>0\,, \end{equation} find the solutions for $x$.
Problem 806 [Unipodal]
Source: https://www.youtube.com/watch?v=aX4IHM0VcXk
Title: A fancier way to solve this radical equation
Presenter: blackpenredpen
Given the relation \begin{equation} (x-40)^{1/3} + (-x +3)^{1/3} = -1\,, \end{equation} find the real values of $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 807 [Not Unipodal: Logarithms: Complex Numbers]
Source: https://www.youtube.com/watch?v=0eWSViFOqFI
Title: Olympiads PROBLEM || n + 9 = n
Presenter: Learn with Christian Ekpo
Given the relation \begin{equation} (-9)^n =9\,, \end{equation} find the complex solutions for $n$.
Problem 808 [Not Unipodal: Group Theory]
Source: The Ether of Great Mathematical Ideas
Title: The Daily Commute
Presenter: Patrick
Let $G$ be a group in which \begin{equation} g^2 = e, \ \forall\ g \in G\,.\label{eq:Given} \end{equation} Show that $G$ is an abelian group, which means that for all $g,a \in G$, \begin{equation} ga = ag\,.\label{eq:ShowThat} \end{equation}
Solution to the problem.
Link to my first write-up on Group Theory 1 (very basic).
Problem 809 [Not Unipodal: Geometry: Circles]
Source: https://www.youtube.com/watch?v=TtbUbU8-BA0
Title: 98% Students FAILED to Solve this Math Problem
Presenter: Lines & Logic
In the figure below is depicted a circle of radius $R$, centered at
point $\mathbf0$. Use the given information to determine the radius.
Problem 810 [Not Unipodal: Category Theory: Commutative Diagram]
Source: The Ether of Great Mathematical Ideas
Title: The Commutative Diagram
Presenter: Patrick
For a discussion of commutative verses path-independent diagrams,
see
A chat with Copilot about path-independent diagrams..
The Commutative Diagram is a subclass of path-independent diagrams.
In the figure below, think of the vertices as objects of some sort, and the
arrows between them as representing morphisms between them. So,
$x,x',y,y'$ are the objects and $f,g,\eta_x,\eta_y$ are morphisms between
the objects.
Problem 811 [Not Unipodal: Lambert W function: Logarithms]
Source: https://www.youtube.com/watch?v=oN_iZB-K-Go
Title: International Math Olympiads PROBLEM
Presenter: Learn with Christian Ekpo
Given the relation \begin{equation} t^5 =9^t\,, \end{equation} find the values for $t$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 812 [Not Unipodal: Lambert W function: Logarithms]
Source: https://www.youtube.com/watch?v=-tqcf36BWO0
Title: 99% Students FAILED to Solve this Math Problem
Presenter: Lines & Logic
Displayed below is a rectangle inscribed within a right triangle.
Find the area of the rectangle.
Problem 813 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=cJaxM9nvQrA
Title: A logarithm equation - Viewer submitted problem
Presenter: Math Out Loud
Given the relation
\begin{equation}
\log_x \alpha + \log_{ax} \alpha^2
+ \log_{ax^2}\alpha^3 = 0\,,
\end{equation}
where $a>1$ and $\alpha>1$, solve for $x$ in terms of $a$ and $\alpha$.
Note: I made both $a$ and $\alpha$ different than originally posed.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 814 [Not Unipodal: Stoichiometry]
Source: https://www.dsd.k12.wi.us/faculty/SBAXTER/Unit
%205%20Practice%20Problems%20(answers).pdf
Title: Molecules-to-Molecules
Presenter: Patrick
Given the balanced equation 2H2 + O2 $\rightarrow$ 2H2O, how many
molecules of water are produced from $2.0 \times 10^{23}$ molecules of oxygen?
Solution to the problem.
Link to my write-up on An introduction to stoichiometry (short version).
Problem 815 [Not Unipodal: Quantum Mechanics]
Source: The Ether of Great Mathematical Ideas
Title: Schrodinger Equation's to the Hamilton-Jacobi equation and Bohmian Mechanics
Presenter: Patrick
Investigate the relation of Schrodinger Equation's to the Hamilton-Jacobi
equation and Bohmian Mechanics.
This note explores what happens to the Schrodinger Equation
when the usual form of the complex function $\psi$ is presented in polar
form $\psi = R\, e^{iS/\hbar}$, and we find a surprising connection to
Hamilton-Jacobi theory and its further connection to Bohmian Mechanics.
Problem 816 [Not Unipodal: Integration]
Source: The Ether of Great Mathematical Ideas
Title: An exponential Integration
Presenter: Patrick
Find the integral \begin{equation} I = \int\! a^x\,dx\,. \end{equation}
Problem 817 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: An interesting word problem
Presenter: Patrick
A vendor wishes to buy fruit concentrate at \$10 per quart and mix it with water
in two different proportions to make two different drinks, at two different sales
prices, to make two different profits off each mix. The high concentrate mix is
in ratio $1:3$ concentrate to water, and the low concentrate mix is in ratio $1:4$
concentrate to water. The vendor sells the high concentrate for $6 per quart, and
the low concentrate for \$4 per quart. If on a given evening, he sells twice as much
of the low concentrate mix as high concentrate mix, what profit does the vendor
make per quart?
Problem 818 [Not Unipodal: Geometry]
Source: The Ether of Great Mathematical Ideas
Title: The tangent vector to a line in the plane
Presenter: Patrick
Given the equation of a line $L$ in the plane in standard form as
\begin{equation}
ax + by + c = 0 \,,
\end{equation}
find a vector t tangent to this line.
Solution to the problem.
Problem 819 [Not Unipodal: Geometry: Geometric Algebra]]
Source: The Ether of Great Mathematical Ideas
Title: The tangent vector to a line in the plane
Presenter: Patrick
We are given the equation of a line $L$ in the plane in standard form as
\begin{equation}
ax + by + c = 0 \,.
\end{equation}
Let $ \mathbf{p}$ be any point not on $L$. Our task is use geometric algebra to find
the reflection on $ \mathbf{p}$ through the line $L$ We'll call this point $ \mathbf{p}'$. Refer
to Fig. 1. Show that $ \mathbf{p}'$ can be found from the given information as
\begin{equation}
\mathbf{p}'= \mathbf{p} - 2( \mathbf{p} - \mathbf{x}_0)\cdot \mathbf{n}\, \mathbf{n}\,,
\end{equation}
where $\bx_0$ is any point on $L$.
Solution to the problem.
Link to my write-up on Basic Geometric Algebra.
Problem 820 [Not Unipodal: Geometry: Geometric Algebra]
Source: The Ether of Great Mathematical Ideas
Title: Where a line intersects a plane
Presenter: Patrick
Use Geometric Algebra to solve for intersection point
of a line and a plane in 3D.
(Knowledge of geometric algebra is assumed.)
Solution to the problem.
Link to my write-up on Basic Geometric Algebra.
Problem 821 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: Mixed-Rate word problem
Presenter: Patrick
A job takes 4 hours for two people ($A$ and $B$) to perform. $A$, working alone,
can do the job in 6 hours. How long would it take $B$ to perform the job alone?
Problem 822 [Not Unipodal: Lambert W Function: Logarithms]
Source: https://www.youtube.com/watch?v=_4S6bl_G2akGiven the relation \begin{equation} z\, \ln z = -\frac{\pi}{2}\,,\label{eq:TheGiven} \end{equation} find the most general solution for $z$.
Title: This Equation Breaks Your Brain! | P580
Presenter: aplusbi
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 823 [Not Unipodal: Sets and Functions]
Source: Conceptual Mathematics: a first introductionFunction $f: B \to C$ has an inverse. Show that if \begin{equation} f \circ h = f\circ k\,, \end{equation} then \begin{equation} h = k\,. \end{equation}
to categories, 2nd Ed
Title: Isomorphims, Exercise 3a
Presenter: F. W. Lawvere & S. H. Schanuel
Problem 824 [Not Unipodal: Sets and Functions]
Source: Conceptual Mathematics: a first introductionFunction $f: A \to B$ has an inverse. Show that if \begin{equation} h\circ f = k\circ f\,, \end{equation} then \begin{equation} h = k\,. \end{equation}
to categories, 2nd Ed
Title: Isomorphims, Exercise 3b
Presenter: F. W. Lawvere & S. H. Schanuel
Problem 825 [Not Unipodal: Logarithms: Lambert W function]
Source:https://www.youtube.com/watch?v=I063-OrNNlYGiven the relation \begin{equation} z^\frac{1}{z} = i\,, \end{equation} solve for $z$.
Title: A Viewer Suggested Equation | Problem 293
Presenter: aplusbi
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 826 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical IdeasA pump can fill a tank in 3 hours. This time, the tank a sprung a
Title: A word problem, mixed rate
Presenter: Patrick
leak and now it take 3.5 hours to fill the tank. How long does it
take the leak to empty the tank?
Problem 827 [Not Unipodal: Algebra: Stoichiometry]
Source: http://www.msduncanchem.com/Unit_9/unit_9_ws_reg.pdfTitanium is a transition metal used in many alloys because it is extremely
Title: Moles-to-Grams
Presenter: Web authors
strong and lightweight. Titanium tetrachloride (TiCl4) is extracted from
titanium oxide using chlorine and carbon. \begin{equation} \underline{\hskip.15in} \mbox{TiO}_2 +\underline{\hskip.15in}\text{C} +\underline{\hskip.15in} \mbox{Cl}_2 \longrightarrow \underline{\hskip.15in} \mbox{TiCl}_4 +\underline{\hskip.15in} \mbox{CO}_2\,. \end{equation} If you begin with 1.25 moles of TiO2, what mass of Cl2 gas
is needed? (Ans: 178 g Cl2.)
Problem 828 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical IdeasTwenty workers can do a job in 35 days. Beginning with 20 workers,
Title: A word problem with a timeline
Presenter: Patrick
work proceeds as normal for 11 days. After that time, 5 workers quit
but are not replaced for the next 4 days. How many workers should
be hired on day sixteen so that the job will be finished on the 35th day?
Problem 829 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the first part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 1 - Functional Equation
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 830 [Not Unipodal: Complex numbers: Logarithms]
Source: https://www.youtube.com/watch?v=psS7fG_kFd8Given the relation \begin{equation} i^{z+i} =1 \,, \end{equation} solve for $z$.
Title: An Imaginary Exponential Equation | Problem 346
Presenter: aplusbi
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on logarithms over the real numbers.
Problem 831 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the second part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 2 - The Gauss Representation
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 832 [Not Unipodal: Complex numbers: Logarithms]
Source: https://www.youtube.com/watch?v=LHyg9-fBv_UGiven the relation \begin{equation} \cos\, (e^{z}) = i \,, \end{equation} solve for $z$.
Title: A Trigonometric Exponential Equation | Problem #103
Presenter: aplusbi
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on logarithms over the real numbers.
Problem 833 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the third part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 3 - Weierstrass Representation
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 834 [Not Unipodal: Integration]
Source: The Ether of Great Mathematical IdeasFind the integral \begin{equation} I = \int\! \frac{9^x-4^x}{3^x+2^x}\,dx\,. \end{equation}
Title: An Integral
Presenter: Patrick
Problem 835 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the fourth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 4 - Gamma and sine functions together
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 836 [Not Unipodal: Complex: Logarithms]
Source: https://www.youtube.com/watch?v=zUBKDZQy0p8Given the relation \begin{equation} \cos z - i\sin z = \frac{1}{e}\,, \end{equation} solve for $z$.
Title: The Equation That Connects Everything | P582
Presenter: aplusbi
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on logarithms over the real numbers.
Problem 837 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the fifth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 5 - Gamma of 1/2
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 838 [Not Unipodal: Complex: Logarithms]
Source: The Ether of Great Mathematical IdeasA widow received $\frac{1}{3}$rd of her husband's estate, and each of her three
Title: A Word Problem
Presenter: Patrick
sons received $\frac{1}{3}$rd of the balance. If the sum the widow received is
added to that of one of her sons, the total is \$60,000. What was the
value of the estate?
Problem 839 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the sixth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 6 - Stirling's Approximation
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 840 [Not Unipodal: Stoichiometry]
Source: https://www.youtube.com/watch?v=qvOVXg24NpoIf 85.3 g of NaN3 decomposes at 75$^\circ$ C and 2.30 atm, what volume of N2
Title: Finding Volume of Evolved Gas not at STP
Presenter: Tyler DeWitt
will be made? (Ans: 24.5 L)
Problem 841 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the seventh part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 7 - The Euler Integral I
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 842 [Not Unipodal: Algebra: Word Problem]
Source: https://www.algebra.com/An ultralight plane has been flying for 40 minutes when a change in wind
Title: An ultralight plane had been flying
Presenter: Patrick
direction doubles its ground speed. If the entire trip of a 160 miles took 2
hours, how far did the plane travel during the first 40 minutes?
Problem 843 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the eighth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 8 - The Euler Integral II
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Problem 844 [Not Unipodal: Plane Geometry]
Source: The Ether of Great Mathematical Ideas\begin{equation} \frac{AC}{AD} = \frac{BC}{BD}\,. \end{equation}
Title: Prove the Angle Bisector Theorem
Presenter: Patrick
Problem 845 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the ninth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 9 - The Euler Integral III
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Establish the Fresnel Integrals \begin{align} \int_0^\infty \sin u^2\, du &= \frac{\sqrt{2\pi}}{4}\,,\\ \int_0^\infty \cos u^2\,du &= \frac{\sqrt{2\pi}}{4} \,. \end{align}
Problem 846 [Not Unipodal: Complex Analysis: Cauchy-Riemann Equations]
Source: The Ether of Great Mathematical Ideas\begin{align} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y}\,,\\ \frac{\partial v}{\partial x} &= -\frac{\partial u}{\partial y}\,. \end{align}
Title: Establish the Cauchy-Riemann Equations
Presenter: Patrick
See also my chat with Copilot, if you want.
Cauchy-Riemann in more detail.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 847 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the tenth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 10 - The Beta Function
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Beta Function:
\begin{equation} B(s_1,s_2) = \frac{\Gamma (s_1) \Gamma (s_2)}{ \Gamma (s_1 + s_2)} = \int_0^1 u^{s_1-1} (1-u)^{s_2-1}\, du \,. \end{equation}
Problem 848 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: Divvying up the bowls of food
Presenter: Patrick
How many people $N$ can eat a meal containing soup, salad, and spaghetti, given the following constraints:
- All of the food will be served from from 55 bowls, each containing exactly one of soup, salad, or spaghetti.
- Each person gets his or her own bowl of soup.
- When the people are paired off, each pair gets only one bowl of spaghetti.
- When the people are tripled off, each triple gets only one bowl of salad.
Problem 849 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the eleventh part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 11 - The Legendre Duplication Formula
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Legendre Duplication Formula:
\begin{equation} \Gamma(2s) = \frac{ 2^{2s-1}}{\sqrt{\pi} } \, \Gamma(s)\,\Gamma(\half+s) \,. \end{equation}
Problem 850 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: How Many Correct
Presenter: Patrick
Ralph scored 61 points on a 10-question test that scores +10 points for every
correct answer and -3 points for every incorrect answer. How many questions
did Ralph get correct?
Problem 851 [Not Unipodal: Analytic Number Theory: Gamma Function]
Source: https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBEThis is the twelfth part of a 12-part series on the Gamma function.
Title: Gamma Function - Part 12 - Relation to the Zeta Function
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Relation to the Zeta Function:
\begin{equation} \Gamma (s)\zeta(s) = \int_0^\infty u^{s-1}\frac{du}{e^{u} - 1} \,. \end{equation}
Problem 852 [Not Unipodal: Algebra]
Source: YouTube
Title: A Table Problem
Presenter: Math Olympiad
Given the relation \begin{equation} 3^z - 2^z = 65\,, \end{equation} find the integer solutions of $z$.
Problem 853 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=ZlYfEqdlhk0This is the first part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 1 - Convergence
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Euler Zeta Function:
\begin{equation} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s}+ \frac{1}{2^s}+ \frac{1}{3^s} + \cdots\,. \end{equation}
Problem 854 [Not Unipodal: Logarithms: Lambert W function]
Source: YouTubeGiven the relation \begin{equation} 3^k = k^9\,, \end{equation} find the solutions of $k$.
Title: A Log-Lambert Problem
Presenter: Math Olympiad
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 855 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=ZlYfEqdlhk0This is the second part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 2 - The Euler Product
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Euler Product:
\begin{equation} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \in P}\frac{1}{1 - p^{-s}}\,. \end{equation}
Problem 856 [Not Unipodal: Algebra: Word Problem]
Source: Intermediate Algebra for College Students, 3rd Ed.A heat-loss survey by an electrical company indicated that a wall of a house
Prentice-Hall (2002), p. 169.
Title: A Mixed-Rate Problem
Presenter: R. Blitzer
containing 40 ft$^2$ of glass and 60 ft$^2$ of plaster lost 1920 BTU of heat (in a
given time period). A second wall containing 10 ft$^2$ of glass and 100 ft$^2$ of
plaster lost 1160 BTU of heat. Determine the heat lost per square foot of glass
and plaster in that house.
Solution to the problem.
Link to my write-up on Word Problem solving.
Problem 857 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=TDdGisWD5OUThis is the third part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 3 - Euler Product Revisited
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Euler Product produced another way:
\begin{equation} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \in P}\frac{1}{1 - p^{-s}}\,. \end{equation}
Problem 858 [Not Unipodal: Ring Theory]
Source: The Ether of Great Mathematical IdeasLet $R$ be a ring without a unity element. Show that the set $S$ defined as
Title: Embedding a Ring into a Ring with Unity
Presenter: Patrick
follows is a ring with unity:
\begin{equation} S \definedas \{(r,n)\, |\, r \in R, n \in \Integers\} \,, \end{equation} where
1) $(r,n) + (s,m) = (r+s, n+m)$,
2) $(r,n) (s,m) = (rs + ns+mr, nm)$.
Problem 859 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=SKa7b-3C32AThis is the fourth part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 4 - Infinitude of Primes
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Euler Product produced to be used for a special corollary:
\begin{equation} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \in P}\frac{1}{1 - p^{-s}}\,. \end{equation}
Problem 860 [Not Unipodal: Complex numbers]
Source: https://www.youtube.com/watch?v=z9pzZ1qjMXA
Title: This Expression Breaks Reality! | P585
Presenter: aplusbi
Given the expression \begin{equation} \phi \definedas i^{i^{i^i}} \,, \end{equation} express $\phi$ in a more conventional form.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 861 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=3eN9tQX3JJ4This is the fifth part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 5
- Divergence of reciprocal sum of Prime Numbers.
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Show that \begin{equation} \sum_{p\, \in P}\,\frac{1}{p}\rightarrow \infty\,. \end{equation}
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 862 [Not Unipodal: Exponentials & Logarithms]
Source: https://www.youtube.com/watch?v=nLaIpO8YnpE
Title: The Weirdest Equation Yet
Presenter: SyberMath
Given the relation \begin{equation} y+\ln y = x + e^x \,, \end{equation} find a simpler relation between $x$ and $y>0$ over the reals.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 863 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=3eN9tQX3JJ4This is the sixth part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 6
- - The Prime Counting Function.
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
Show that \begin{equation} \log \zeta(s) = \int_2^{\infty}\! \frac{ s\pi(x)}{x( x^{s} - 1)}\, dx \,. \end{equation}
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 864 [Not Unipodal: Algebra: Polynomials]
Source: https://www.youtube.com/watch?v=4EeKtTY1GwI
Title: A Polynomial Riddle | Can You Solve?
Presenter: SyberMath
Given the relation, where $P(x)$ is a polynomial, \begin{equation} P(x) - P'(x) = x^2 - 2 \,, \end{equation} solve for $P(x)$.
Problem 865 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=GeKDmoYHiAkThis is the seventh part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 7: $\zeta(2)$ is calculated.
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
\begin{equation} \zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \,. \end{equation}
Problem 866 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: It can get confusing
Presenter: Patrick
Nine liters are drawn from a tank full of wine. Then 9 liters of pure water
are added to the tank and the mix is allowed to homogenize. After that,
9 more liters are drawn off and again replaced by 9 liters of pure water
and allowed to homogenize. If the final wine-to-water mix is in ratio 16:9,
how much does the full tank hold?
Problem 867 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=axQqExF7NsU (1st)This is the eighth part of a 14-part series on the Zeta function.
https://www.youtube.com/watch?v=XHQ0OzqTjd0 (2nd)
https://www.youtube.com/watch?v=1f24RZfP6m8 (3rd)
Title: Zeta Function - Part 8:$\zeta(2n)$ and the Bernoulli Numbers
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
The Bernoulli Numbers: \begin{equation} \frac{s}{e^s - 1} = \sum_{n=0}^\infty \frac{\beta_n}{n!} s^n \,. \end{equation}
Problem 868 [Not Unipodal: Conformal Field Theory: Tobias Osborne]
Source: https://www.youtube.com/watch?v=NGYX6gtObec
Title: Introduction to conformal field theory, Lecture 1
Presenter: Tobias Osborne
(Read-along notes and a problem to solve.)
We have been working with the following relation from conformal field theory
for some time now:
\begin{equation}
\partial_\mu \epsilon_\nu + \partial_\nu \epsilon_\mu
=\frac{2}{d} (\partial \cdot\epsilon)\,\eta_{\mu\nu}\,,
\end{equation}
By restricting the underlying vector space to $\Reals^{p,q} = \Reals^{2,0}$, show
that the resulting coordinates can be interpreted as satisfying the
Cauchy-Riemann equations. [See also Problem 846.]
Problem 869 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=UEZ4ClCdog8This is the ninth part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 9: The Gamma Function
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
\begin{equation} \Gamma (s)\zeta(s) = \int_0^\infty u^{s-1}\left[\frac{1}{1 - e^{-u}} - 1\right] \, du = \int_0^\infty u^{s-1}\frac{du}{e^{u} - 1} \,. \end{equation}
Problem 870 [Not Unipodal: Complex numbers: Cauchy-Riemann equations]
Source: The Ether of Great Mathematical Ideas
Title: Show that f(z) = z^n is analytic
Presenter: Patrick
This continues our short series on the Cauchy-Riemann equations.
This time we use them and mathematical induction to show that
$f(z)=z^n$ is analytic.
Problem 871 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=-GQFljOVZ7IThis is the tenth part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 10: The Jacobi Theta Function
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
We define the Jacobi Theta Function as \begin{equation} \theta(x) \definedas \sum_{n \in \Integers} e^{-\pi n^2 x}\,. \end{equation} Then we have the result: \begin{equation} \theta(x) = \frac{1}{\sqrt{x}}\, \theta(\frac{1}{x})\,. \end{equation}
Problem 872 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://talk.collegeconfidential.com/ap-tests-preparation/
233538-hard-stoichiometry-problem.html
Title: How Long Will the Lithium Hydroxide Last?
Presenter: collegeconfidential
The space shuttle environmental control system handles carbon dioxide (4% by mass
exhaled air) by reacting it with Lithium Hydroxide pellets to form lithium carbonate
and water. If there are 7 astronauts on board the shuttle, and each exhales 20 liters
of air per minute, how long could clean air be generated if there were 25,000 g
of lithium hydroxide pellets available for each shuttle mission? Assume the density
of air is 0.0010 g/mL. (Proposed Ans: 4109 minutes.)
Problem 873 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=-GQFljOVZ7IThis is the eleventh part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 11: The Riemann Functional Equation I
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
\begin{equation} \pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s) = \pi^{\textstyle-\frac{1-s}{2}}\, \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s) \,. \end{equation}
Problem 874 [Not Unipodal: Algebra: Word Problem]
Source: https://www.algebra.com/algebra
Title: Question 147255.
Presenter: Patrick
A distillate flows into an empty 64-gallon drum at spout $A$ and out
of the drum at spout $B$. If the influx at $A$ is 2 gallons per hour,
what is the outflux rate at $B$ so that the drum is full in 96 hours?
Problem 875 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=-GQFljOVZ7IThis is the twelfth and thirteenth parts of a 14-part series on the Zeta function.
Title: Zeta Function - Parts 12--13: The Riemann Functional Equation II
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
We need to establish the alternative form of the Riemann Functional Equation: \begin{equation} \zeta(s) = 2^s\, \pi^{s-1}\, \sin \frac{\pi s}{2}\,\Gamma(1-s)\,\zeta(1-s) \,. \end{equation} We'll also establish the trivial zeroes to the Zeta function.
Problem 876 [Not Unipodal: Algebra: Percentage Problem]
Source: https://www.youtube.com/shorts/uSYDKq7ojQw
Title: A percentage problem.
Presenter: SAFALATA Ek Target
If $x$% of $a$ is the same as $y$% of $b$,
then $z$% of $b$ will be what?
Problem 877 [Not Unipodal: Analytic Number Theory: Zeta Function]
Source: https://www.youtube.com/watch?v=QfDbF_qlp58This is the fourteenth part of a 14-part series on the Zeta function.
Title: Zeta Function - Part 14: The Riemann Xi Function
Presenter: MrYouMath
What I'm presenting here is what I refer to as the `read-a-long notes'
to the videos. They are brief on explanations. For better explanations,
please see the videos by MrYouMath, as listed above.
If we define $\xi(s)$ by \begin{equation} \xi(s) \definedas \half s(s-1) \pi^{-s/2}\Gamma\big(\frac{s}{2}\big)\zeta(s) \,, \end{equation} show that \begin{equation} \xi( \half + it) =\xi( \half - it)\,. \end{equation}
Problem 878 [Not Unipodal: Self-similarity: Lambert W Function]
Source: The Ether of Great Mathematical Ideas
Title: A Self-similarity problem.
Presenter: Patrick
Given the relation \begin{equation} \phi = \sqrt{3}^{\sqrt{3}^{.\cdot{}^{.}}} \,, \end{equation} find the value of $\phi$ in some finite form.
Problem 879 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://answers.yahoo.com/question/index?qid
=20081230083424AA2lfMV
Title: Hard stoichiometry problem
Presenter: Yahoo Answer site
You are given a mixture of three hydrated salts: \begin{equation} \mbox{Na}_2 \mbox{CO}_3\cdot 10\ \mbox{H}_2 \mbox{O},\quad \mbox{MgSO}_4\cdot7\mbox{H}_2 \mbox{O},\quad \text{and}\quad \mbox{CuSO}_4\cdot5 \mbox{H${}_2$O}\,. \end{equation} The total mass of the mixture is 12.123 grams. When the mixture is heated gently, the following two reactions occur: \begin{align} \mbox{Na}_2\mbox{CO}_3 \cdot10\mbox{H}_2 \mbox{O}\mbox{(s)} &\longrightarrow \mbox{Na}_2\mbox{CO}_3\cdot 7 \mbox{H}_2 \mbox{O}\mbox{(s)} + 3\mbox{H}_2 \mbox{O}(g)\\ \mbox{MgSO}_4\cdot7\mbox{H}_2 \mbox{O}\mbox{(s)} &\longrightarrow \mbox{MgSO}_4\cdot\mbox{H${}_2$O}\mbox{(s)} + 6 \mbox{H}_2 \mbox{O}(g) \end{align} After these reactions are complete, the mass of the mixture has decreased to 9.049 grams. This mixture is then heated more strongly, and the following additional reactions occur: \begin{align} \mbox{Na}_2\mbox{CO}_3\cdot7\mbox{H${}_2$O}\mbox{(s)} &\longrightarrow \mbox{Na}_2\mbox{CO}_3\mbox{(s)} + 7\mbox{H${}_2$O}(g)\label{eq:HydrateReaction2a} \\ \mbox{MgSO}_4\cdot\mbox{H${}_2$O}\mbox{(s)} &\longrightarrow \mbox{MgSO}_4\mbox{(s)} + \mbox{H${}_2$O}(g)\label{eq:HydrateReaction2b} \\ \mbox{CuSO}_4\cdot5\mbox{H${}_2$O}\mbox{(s)} &\longrightarrow \mbox{CuSO}_4\mbox{(s)} + 5 \mbox{H${}_2$O}(g) \label{eq:HydrateReaction2c} \end{align}
After this final heating, the mass of the mixture has decreased to 6.412 grams. From this information, calculate the masses of each of the three compounds in the original mixture.
The provided answers are:
mass of Na${}_2$CO${}_3\,\cdot$ 10\mbox{H${}_2$O}: 1.374 g
mass of MgSO${}_4\cdot7\mbox{H}_2$O: 6.418 g
mass of CuSO${}_4\cdot5$ H$_2$O: 4.331 g
Problem 880 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: Finite Mathematics, 5th Ed. Brooks/Cole (2002), p. 57.
Title: A Mixed-Rate problem.
Presenter: H. Rolf
A woman must control her diet. She selects milk and bagel for breakfast.
How much of each should she serve in order to consume 700 calories and
28 grams of protein? Each cup of milk contains 170 calories and 8 grams
of protein. Each bagel contains 138 calories and 4 grams of protein.
Problem 881 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.alvinisd.net/cms/lib03/TX01001897/Centricity
/Domain/4240/practice%20test%20stoich.pdf
Title: From an online stoichiometry practice test
Presenter: Patrick
How many grams of nitric acid, HNO3, can be prepared from the reaction
of 138 g of NO2 with 54.0 g $\mbox{H}_2 \mbox{O}$ according to the equation below?
\begin{equation}
3\mbox{NO}_2 + \mbox{H}_2 \mbox{O} \longrightarrow 2\mbox{HNO}_3 + \text{NO}
\end{equation}
a. 92$\quad$ b. 108$\quad$ c. 126$\quad$ d. 189$\quad$ e. 279 .
Problem 882 [Not Unipodal: Algebra: Sum of the Parts = Total]
Source: https://www.algebra.com/algebra
Title: Question 174684
Presenter: Patrick
8-year-old Samantha visited Santa at a local department store. He gave her
this riddle: ``I started working at 15. I spent 1/4 of my working life in a
factory. I spent 1/5 of my working life in an office, and I spent 1/3 of my
working life as a school caretaker. For the last 13 years of my working
life I've been Santa Claus. How old am I?"
Problem 883 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.youtube.com/watch?v=e7M0JiJZSiQ
Title: A Gravimetric Analysis Problem
Presenter: Patrick
A 10.500 gram mixture contains calcium nitrate and potassium chloride.
Excess lead (II) nitrate solution, Pb(NO3)2 aq, is added to precipitate
out 4.227 grams of lead(II) chloride, PbCl2 sol. What percent by
mass of potassium chloride is in the mixture?
Problem 884 [Not Unipodal: Algebra: Mathematical Induction]
Source: The Ether of Great Mathematical Ideas
Title: An Induction Problem
Presenter: Patrick
Using mathematical induction, show that \begin{equation} \sum_{i=1}^n \frac{1}{i(i+1)} = \frac{n}{n+1}\,. \end{equation}
Solution to the problem.
Link to my write-up on Mathematical Induction.
Problem 885 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.calstatela.edu/sites/default/files/dept/chem/
07winter/201-lec/201-l-4-gravimetric-analysis.pdf
Title: A Gravimetric Analysis Problem 2
Presenter: Patrick
Consider a 1.0000 g sample containing 75% potassium sulfate
(FW 174.25) and 25% MSO4. The sample is dissolved and
the sulfate is precipitated as BaSO4
(FW 233.39). If the BaSO4 ppt
weighs 1.4900, what is the atomic weight of
M2+ in MSO4?
Problem 886 [Not Unipodal: Algebra: Complex Numbers]
Source: https://www.youtube.com/watch?v=y3PJxw30MIs
Title: A Simple Problem For Competitive Exams
Presenter: J Educational Tutorials
Given the following relations \begin{align} a^2 - b^2 &= 27\,,\\ ab&= 18\,, \end{align} find the real values of \begin{equation} \phi = a + b\,.\label{eq:a+b} \end{equation}
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 887 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.calstatela.edu/sites/default/files/dept/chem/
07winter/201-lec/201-l-4-gravimetric-analysis.pdf
Title: A Gravimetric Analysis Problem 3
Presenter: Patrick
A mixture of mercurous chloride (FW 472.09) and mercurous bromide
(FW 560.99) weighs 2.00 g. The mixture is quantitatively reduced to
mercury metal (At wt 200.59) which weighs 1.50 g. Calculate the
quantities of mercurous chloride and mercurous bromide in the original
mixture. (I modified the question to fit the answer given.)
$\quad$ ANS: 0.5182 g
Problem 888 [Not Unipodal: Algebra: Word Problems]
Source: http://regentsprep.org/Regents/math/ALGEBRA/AE3/PracWo
Title: A Mixed-Rate Problem
Presenter: Patrick
Two small pitchers and one large pitcher can hold 8 cups of water.
One large pitcher minus one small pitcher constitutes 2 cups of water.
How many cups of water can each pitcher hold?
Problem 889 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.youtube.com/watch?v=e7M0JiJZSiQ
Title: A Gravimetric Analysis Problem 4
Presenter: Patrick
An unknown metal cation has a +1 charge (M$^+$). The bromide of the
unknown metal, MBr, is dissolved in enough water to make 100.0 mL
of solution. The solution is then mixed with an excess of AgNO3
solution to precipitate AgBr (molar mass = 188 g/mol). The precipitate
is collected by filtration, dried, and the following data was obtained:
Mass of MBr = 1.38 g
Mass of filter paper = 0.98 g
Mass of filter paper and AgBr = 2.86 g\newline
What is the identity of the metal chloride?
$\quad$ a) KBr$\quad$ b) NiBr$\quad$ c) LiBr$\quad$ d) NaBr
Problem 890 [Not Unipodal: Algebra: Word Problem: Mixed-Rate]
Source: https://answers.yahoo.com/question/index?qid=
20080309104615AAMKboY
Title: Hard algebra word problems?
Presenter: Patrick
Hockey teams receive 2 points when they win and 1 point when
they tie. One season, a team won a championship with 56 points.
They won 10 more games than they tied. How many wins and how
many ties did the team have?
Problem 891 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.quia.com/files/quia/users/lockarm/stoichiometry
/activity---Stoichiometry-Word-Problems-2-SOLUTIONS.pdf
Title: A Combustion Problem
Presenter: Patrick
Nitroglycerin, C3H5(ONO2)3, was invented in 1846 by an Italian chemist
named Ascanio Sobrero. Nitroglycerin contains both an oxidant and a fuel.
When it detonates, it decomposes to form carbon dioxide, water, nitrogen,
and oxygen, all in a gaseous state. Every mole of the explosive that decomposes
in this way generates a tremendous amount of energy -- approximately 1.5 MJ
(1 MJ = 1 megajoule = $1 \times 10^6$ J = 1 MJ).
a. If 1.135 kilograms of nitroglycerin detonates, how many total liters of gas
(assuming STP) are produced?
Balanced Equation: \begin{equation} 4 \mbox{C$_3$H$_5$(ONO$_2$)$_3$} \rightarrow 10 \mbox{H$_2$O} + 12 \mbox{CO$_2$} + 6 \mbox{N$_2$} + \mbox{O$_2$} \end{equation}
Ans: 812 liter.
b. How much energy is produced by the explosion?
Ans: 7.5 MJ.
Problem 892 [Not Unipodal: Algebra: Word Problem: Mixed-Rate]
Source: https://www.youtube.com/watch?v=8cp_ij0APW4
Title: Everyone Misses This Simple Algebra Trick!
Presenter: Mental Math
Given the relations \begin{equation} x+ y = 1 \,,\label{eq:Given1} \end{equation} and \begin{equation} x^3 +y^3 = 7\,,\label{eq:Given2} \end{equation} find the value of \begin{equation} \phi = xy\,. \end{equation}
Problem 893 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.youtube.com/watch?v=9fSiy7-JurA
video time stamp 46:05
Title: Working with the amu (atomic mass unit)
Presenter: The Organic Chemistry Tutor
Calculate the [average] mass of 25 carbon atoms in amu (atomic
mass units) and in grams.
Problem 894 [Not Unipodal: Algebra: Lambert W Function]
Source: https://www.youtube.com/watch?v=q2XyddOgPjo&t=2s
Title: The Seemingly Impossible Equation That Has a
Beautiful Solution
Presenter: Mental Math
Given the relation \begin{equation} \pi^x = x^\pi \end{equation} solve for real values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 895 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.alvinisd.net/cms/lib03/TX01001897/Centricity/
Domain/4240/practice%20test%20stoich.pdf
Title: Mystery Metal, PROBLEM 13
Presenter: Patrick
11.2 g of metal carbonate, containing an unknown metal, M, were heated
to give
the metal oxide and 4.4 g CO2.
\begin{equation}
\mbox{MCO}_3\,\mbox{(sol)} \rightarrow \mbox{MO (sol)} + \mbox{CO}_2\,\mbox{(gas)}
\end{equation}
What is the identity of the metal M?
\begin{matrix} &\text{a. Mg} &&&&\text{b. Pb} &&&&\text{c. Ca}\\ &\text{d. Ba} &&&&\text{e. Cr} &&&& \end{matrix}
Problem 896 [Unipodal: Complex Numbers]
Source: https://www.youtube.com/watch?v=wgd6Gp-Kfbk
Title: This Question Tricked Thousands of Students
Presenter: J Educational Tutorials
Given the relation \begin{equation} x^2 - 5x + 7 = 0\,, \end{equation} find the value of $\phi$ given by \begin{equation} \phi \definedas (x-2)^{90} + (3-x)^{90}\,. \end{equation}
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on The Unipodal Algebra.
Problem 897 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.alvinisd.net/cms/lib03/TX01001897/Centricity/
Domain/4240/practice%20test%20stoich.pdf
Title: Mystery Hydrocarbon C_xH_y, Prob. 14, p.2
Presenter: Patrick
A given sample of some hydrocarbon is burned completely and
it produces 0.44 g of CO2 and 0.27 g of H2O. Determine the
empirical formula of the compound.
$\hskip.3in$ a. CH $\hskip.3in$ b. C2H3$\hskip.3in$ c. CH2
$\hskip.3in$ d. C2H5$\hskip.2in$ e. CH3
Ans: e.
Problem 898 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: What is really meant by 2% milk?
Presenter: Patrick
This is a word problem I made up, but probably someone else did also:
I have long been curious what is really meant by 2% milk. According
to one online source, it means that the milk fat content of 2% milk is
2% by weight. According to another source, whole milk is approximately
3.5% milk fat. Therefore, whole milk is about 96.5% skim milk. A few
years ago I read on a 2% milk label that the milk inside the container
was 35% less fat than whole milk. So, given just the information on that
2% milk label (and assuming it's true), what is the fat content of whole milk?
Perhaps you can improve on this problem.
Solution to the problem.
Link to my write-up on Word Problem solving.
Problem 899 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.alvinisd.net/cms/lib03/TX01001897/Centricity/
Domain/4240/practice%20test%20stoich.pdf
Title: Employing the Stoichiometric Proportion, Prob. 24, p.3
Presenter: Patrick
\begin{equation}
6\mbox{H$^+$} + 5\mbox{H$_2$O$_2$} + 2\mbox{MnO$_4^-$} \rightarrow 5\mbox{O$_2$} + 2\mbox{Mn$_2^+$} + 8\mbox{H$_2$O}
\end{equation}
According to the balanced equation above, how many moles of the
permanganate ion are required to react completely with 25.0 ml of
0.100 M hydrogen peroxide?
\begin{matrix} &\text{a. 0.000500 mol} &&&&\text{b. 0.00100 mol} &&&&\text{c. 0.00500 mol}\\ &\text{d. 0.00625 mol} &&&&\text{e. 0.0100 mol} &&&& \end{matrix}
Problem 900 [Not Unipodal: Algebra: Word Problem: Mixed Rate]
Source: http://www2.math.umd.edu/$\sim$jnd/Algebraic
\_word\_problems.pdf
Title: Ratio of gold to silver in a crown
Presenter: Patrick
A royal crown is an alloy of gold and solver. The crown weighs 3000 grams
and has a volume of 200cc. If the density of gold is 20 grams/cc and of silver
is 10 grams/cc, what is the ratio of gold to silver by volume in the crown?
Solution to the problem.
Link to my write-up on Word Problem solving.
Problem 901 [Not Unipodal: Chemistry: Stoichiometry]
Source: http://msmorrischemistry.weebly.com/uploads/3/8/9/5/38951057
/stoichiometry__ap_mc_.pdf
Title: Empirical formula oxide of nitrogen, Prob. 25
Presenter: Patrick
The simplest formula for an oxide of nitrogen that is 36.8 percent
nitrogen by weight is…
(A) N$_2$O $\hskip.2in$ (B) NO$\hskip.2in$ (C) NO$_2$
(D) N$_2$O$_3$ $\hskip.14in$ (E) N$_2$O$_5$
Ans: (D).
Problem 902 [Not Unipodal: Algebra: Word Problem: Mixed Rate]
Source: https://www.mgccc.edu/learning\_lab/
math/alg/howtomix.pdf
Title: Alcohol Dilution Problem
Presenter: Patrick
How much water must be added to 14 oz of a 20% alcohol solution
to obtain a 7% alcohol solution?
Solution to the problem.
Link to my write-up on Word Problem solving.
Problem 903 [Not Unipodal: Chemistry: Stoichiometry]
Source: http://msmorrischemistry.weebly.com/uploads/3/8/9/5/38951057
/stoichiometry__ap_mc_.pdf
Title: Empirical formula oxide of nitrogen, Prob. 25
Presenter: Patrick
The alkenes are compounds of carbon and hydrogen with the general
formula C$_n$H$_{2n}$. If 0.561 gram of any alkene is burned in excess
oxygen, what number of moles of H$_2$O is formed?
(A) 0.0400 mole$\hskip.2in$ (B) 0.0600 mole$\hskip.2in$ (C) 0.0800 mole
(D) 0.400 mole $\hskip.2in$ $\hskip.01in$ (E) 0.800 mole
Ans: (D).
Problem 904 [Not Unipodal: Lambert W Function]
Source: The Ether of Great Mathematical Ideas
Title: A Lambert Lemma
Presenter: Patrick
Lemma:
Show that the solution to \begin{equation} \log x = x\gamma\,, \end{equation} where the logarithm given is base 10 and $\gamma$ is a real number, is \begin{equation} x = e^{-W_n(-\gamma\, \ln 10)}\,. \end{equation}
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 905 [Not Unipodal: Chemistry: Stoichiometry]
Source: http://www.uh.edu/~chem1p/c3/C3F99.pdf
Title: Mystery Carbohydrate (Vit C)
Presenter: Patrick
Vitamin C (M= 176.12 \gmol) contains C, H, and O. A 1.000 g
sample was placed in a combustion apparatus [and the following
are facts about the masses of the products, derived by 'weighing'.
[Or, the mass can be calculated by use of knowledge of moles
and molar mass of the compound.]
Mass of CO2 is 1.50 grams.
Mass of H2O is 0.41 grams.
Question: What is the molecular formula of vitamin C?
Problem 906 [Not Unipodal: Lambert W Function]
Source: https://infinitemathworld.com/exploring-the-lambert
-w-function-applications-and-examples/
Title: Exploring The Lambert W Function
Presenter: Slavisa Velickovic
Maximize $\phi = x^y$ with the constraints $x + y = 4$ and $x>0$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 907 [Not Unipodal: Chemistry: Stoichiometry]
Source: The Ether of Great Mathematical Ideas
Title: Calcium in Calcium Phosphate
Presenter: Patrick
Question: How many grams of calcium are present in
890 grams of calcium phosphate?
Problem 908 [Not Unipodal: Word Problem: Mixed Rate]
Source: math.unm.edu/sites/default/files/files/
core-courses/4.95 Mixture..
Title: Counting by Feet
Presenter: Patrick
In a pen at Old MacDonald’s farm there are some sheep and some
geese. There is a total of 115 animals, and there are 424 legs. How
many sheep and how many geese are there?
Problem 909 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.chemteam.info/Mole/Determine-formula-of
-hydrate-prob1-10.html
Title: Computing an hydration formula
Presenter: Patrick
A 5.00 g sample of hydrated barium chloride, BaCl2$\,\cdot\, n\,$H2O, is heated
to drive off the water. After heating, 4.26 g of anhydrous barium chloride,
BaCl2, remains. What is the value of $n$ in the hydrate's formula?
Problem 910 [Not Unipodal: Word Problem: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Mixture-problems.lesson
Title: Dilution Problem
Presenter: Patrick
How much water should be added to 200 milliliters of a 10% salt
solution to get a 2% salt solution?
Problem 911 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.chemteam.info/Mole/Determine-formula-of
-hydrate-prob1-10.html
Title: Hydrate of LiClO4
Presenter: Patrick
Anhydrous lithium perchlorate (4.78 g) was dissolved in water and re-crystalized.
Care was taken to isolate all the lithium perchlorate as its hydrate. The mass of
the hydrated salt
obtained was 7.21 g. What hydrate is it?
Problem 912 [Not Unipodal: Algebra: Logarithms]
Source: https://www.youtube.com/watch?v=KCobtZ05fE0
Title: This Logarithm Puzzle Broke the Internet.
Presenter: Mental Math
Given the relation \begin{equation} m - n \log_3 2 = 10 \log_9 6\,, \end{equation} solve for integers $m,n$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 913 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.chemteam.info/Mole/Determine-formula-of
-hydrate-prob1-10.html
Title: Hydrate of Na2C3, problem 5a
Presenter: Patrick
A solution was made by dissolving 52.0 g of hydrated sodium carbonate
in water and making it up to 5.00 dm$^3$ of solution. The concentration of the
solution was determined to be 0.0366 M. Determine the formula of hydrated
sodium carbonate.
Problem 914 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=edcp4YnE1zA
Title: if a^2+b^2+1/b^2+1/a^2=4 then ...
Presenter: Adee Institute
Given the relation \begin{equation} a^2 + b^2 + \frac{1}{a^2} + \frac{1}{b^2}= 4 \,, \end{equation} find the value of \begin{equation} \phi = a^2 + b^2 \,. \end{equation}
Problem 915 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://www.calstatela.edu/sites/default/files/dept/
chem/07winter/201-lec/201-l-4-gravimetric
-analysis.pdf (p.\ 9)
Title: Precipitating Silver
Presenter: Patrick
How many mL of 1% potassium chloride would be needed to
precipitate all of the silver in a 0.5 g ore sample that contains
1.5 parts per thousand silver? Allow for 50% excess of the
chloride solution.
Problem 916 [Not Unipodal: Number Theory: Ring Theory]
Source: The Ring of Hyperintgers
Title:The 10-adic integers
Presenter: A. DeLugt and P. Reany
The 10-adic numbers are infinite strings of digits, which are
extensions of the integers, which have remarkable properies.
The 10-adic numbers (in HTML).
The Ring of Hyperintegers (PDF).
YouTube link to a simple introduction to the 10-adic integers
by Richard Borcherds, Fields Medal winner..
Problem 917 [Not Unipodal: Chemistry: Stoichiometry]
Source: https://answers.yahoo.com/question/index?qid
=20100413231838AAIl4qf
Title: Making Ammonia Gas
Presenter: Patrick
If water is added to magnesium nitride, ammonia gas is produced
when the mixture is heated.
Mg$_3$N$_2$ solid + 3H$_2$O liq $\rightarrow$ 3MgO solid + 2NH$_3$ gasIf 13.7 g of magnesium nitride is treated with water, what volume
of ammonia gas would be collected at 28$^\circ$ C and 737 mm Hg?
Problem 918 [Not Unipodal: Algebra]
Source:https://www.youtube.com/watch?v=Hx761RqE9Iw
Title: Netherlands | Can you solve this??
Presenter: Mr-Mathologer
Given the relation \begin{equation} \frac{1}{x} + \frac{1}{y} =\frac{1}{x+y}\,, \end{equation} solve for \begin{equation} \eta =\frac{x}{y}\,. \end{equation}
Problem 919 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem 1. p. F43 E.4
Presenter: Atkins and Jones
Given a quantity of CuSO$_4$ $\cdot$ 5H$_2$O in grams,
determine how many grams of Cu are in this compound.
Problem 920 [Not Unipodal: Number Theory]
Source: The Ether of Great Mathematical Ideas
Title: A divisor of 3 positive consecutive integers
Presenter: Patrick
Show that $3!$ is a divisor of any 3 positive consecutive integers.
Problem 921 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem Problem 2. p. F44 E.27
Presenter: Atkins and Jones
The density of sodium borohydrite (NaBH$_4$) [herafter referred to
as the compound] is 1.074 g $\cdot$ cm$^{-3}$. If 3.93 g of the compound
contains $2.50 \times 10^{23}$ H atoms, how many moles of H atoms are
present in 28.0 cm$^{-3}$ of the compound?
Problem 922 [Not Unipodal: Lambert W Problem]
Source: The Ether of Great Mathematical Ideas
Title: A Lambert W Problem
Presenter: Patrick
Given the relation \begin{equation} x^x = 18\,, \end{equation} solve for real values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 923 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem Problem 3. p. F49 F.7
Presenter: Atkins and Jones
In an experiment, 4.14 grams of phosphorus combined with
chlorine to produce 27.8 grams of a white solid compound.
What is the empirical formula of the white compound?
Problem 924 [Unipodal]
Source: The Ether of Great Mathematical Ideas
Title: Yet another example
Presenter: Patrick
Given the relation \begin{equation} \sqrt{x+1} + \sqrt{x-1} = 2\,, \end{equation} find the value of $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 925 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem Problem 4. p. F49 F.9
Presenter: Atkins and Jones
L-Dopa, a drug used for the treatment of Parkinson's disease,
is 54.46 % C, 5.62 % H, 7.10 % N, and 32.46 % O. What is
the empirical formula of the compound?
Problem 926 [Not Unipodal: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: Water is evaporated from salt water
Presenter: Patrick
How much water must be evaporated from 1000 milliliters of a
2% salt solution to get a 10% salt solution?
Problem 927 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem Problem 5. p. F54 Ex. G.2
Presenter: Atkins and Jones
Suppose we were asked to prepare 250 mL of a solution that was
approximately 0.0380 M CuSO$_4$ (aq) and we had available
only Copper (II) sulfate penta-hydrate CuSO$_4\cdot5$H$_2$O.
What mass
of the solid do we need?
Problem 928 [Not Unipodal: Word Problem]
Source: http://www.weber.edu/wsuimages/MTC/Handouts/Mixture
%20Problems%20Handout_WEllis.pdf
Title: A mixed-rate problem
Presenter: Patrick
Suppose a store keeper wants to make a mixture of cashews and peanuts. He
has on hand peanuts that cost \$3 per pound and cashews that cost \$5.50
per pound. He wants to make a 3 pound mixture that costs \$4 per pound.
Problem 929 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem Problem 6: p. F55 Ex. G.4
Presenter: Atkins and Jones
We need to prepare 250 mL of $1.50 \times 10^{-3}$ M NaOH (aq), using 0.0380 M
NaOH (aq) stock solution. How much stock solution do we need?
Problem 930 [Not Unipodal: Word Problem: Mixed Rates]
Source: https://www.algebra.com/algebra
Title: Question 702388
Presenter: Patrick
If 20 pounds of sea water contains 1.6 pounds of salt, how many
pounds of pure water must be added to produce a mixture
containing 5% salt?
Problem 931 [Not Unipodal: Chemistry: Stoichiometry]
Source: Chemical Principles: Quest for Insight, 3rd Ed
Title: Problem Problem 7. p. F61 Ex. H.17
Presenter: Atkins and Jones
Phosphorus and oxygen react to form two different phosphorus oxides.
The mass percentage of phosphorus in one of these is 43.64%; in the
other, it is 56.34%.
(a) Write the empirical formula for each phosphorus oxide.
(b) The molar mass of the former oxide is 283.33 g mol$^{-1}$ and
that of the latter is 219.88 g mol$^{-1}$. Determine the molecular
formulas and name each oxide.
Problem 932 [Not Unipodal: Algebra: Radicals]
Source: https://www.youtube.com/watch?v=hzy3yUNMPuc
Title: This Nested Radical Looks Impossible
Presenter: Mental Math
Given the relation \begin{equation} \phi = (2+\sqrt{5})^{1/3}+(2-\sqrt{5})^{1/3}\,, \end{equation} simplify $\phi$ in terms of real values.
Problem 933 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Mixture-problems.lesson
Title: Changing a Concentration Problem
Presenter: Patrick
How much salt should be added to 1000 milliliters of a 2% salt solution
to get a 4% salt solution?
Problem 934 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra
Title: Question 15639
Presenter: Patrick
[From the viewpoint of the questioner] This problem has me stumped
because I have 8 ounces already mixed and need to add to it to get 12
ounces. If I was just adding enough chocolate syrup to make it 42
percent, I would not be so confused. Here is the problem:
I want to make the perfect 12 ounce cup of chocolate milk. It requires
that the mixture is 42% chocolate syrup. What I have right now is 8
ounces of milk/syrup mixture that I know contains 30% syrup. What
must the syrup concentration be in the remaining mixture that I must
add in order to achieve perfection?
Problem 935 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Mixture-problems.lesson
Title: Changing a Concentration Problem
Presenter: Patrick
How much water should be added to 200 milliliters of a 10% acid solution
to get a 2% acid solution? (Concentrations here are volume-to-volume
concentrations, measured in [mL/mL] units, or milliliters of the acid per 1
milliliter of the solution).
Problem 936 [Not Unipodal: Algebra]
Source: https://www.algebra.com/algebra
Title: Question 22541
Presenter: Patrick
[From the viewpoint of the questioner:] Help! I can't even
think of a good equation for this problem! Brine is a solution
of salt and water. If a tub contains 50 pounds of a 5% solution
of brine, how much water must evaporate to change it to an
8% solution? Any help would be greatly appreciated!
Problem 937 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Mixture-problems.lesson
Title: Changing a Concentration Problem
Presenter: Patrick
How much of the pure acid should be added to 1000 milliliters
of a 2% acid solution to get a 4% acid solution? (Concentrations
here are volume-to-volume concentrations, measured in [mL/mL]
units.
Problem 938 [Not Unipodal: Algebra]
Source: https://www.algebra.com/algebra
Title: Question 494329
Presenter: Patrick
How many liters of an 8% solution of salt should be added to a 25%
solution in order to obtain 510 liters of an 12% solution? [Discussion only]
on this problem.
Problem 939 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Mixture-problems.lesson
Title: Changing a Concentration Problem
Presenter: Patrick
How much of a 10% acid solution should be added to 1000 milliliters
of a 2% acid solution to get a 4% acid solution? Concentrations here are
volume-to-volume, part-to-whole, measured in [mL/mL] $\times$ 100% units.
Problem 940 [Not Unipodal: Lambert W Problem]
Source: https://www.youtube.com/watch?v=u0P7uk4KdE0
Title: Can You Solve This Impossible Exponential Equation?
Presenter: Mental Math
Given the relation \begin{equation} e^{\sqrt{x}} = x^{\sqrt{e}}\,, \end{equation} find the real, positive values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 941 [Not Unipodal: Algebra: Mixed Rate]
Source: The Ether of Great Mathematical Ideas
Title: A Coin Problem
Presenter: Patrick
A jar contains a collection of less than 40 coins of two types, Type1
and Type2, valuing \$4.85. The possible coin types are penny, nickel,
dime, and quarter. Find the types of coins and the number of each
type of coin, if the following claims are true:
a) The percentage of Type1 coins by number is 41.37931%, and
b) The percentage of Type1 coins by value is 12.371134%.
Problem 942 [Not Unipodal: Algebra: Mixed-Rate]
Source: The Ether of Great Mathematical Ideas
Title: A mixed-rate problem
Presenter: Patrick
How much pure antifreeze liquid should be added to 1 gallon
of 40% antifreeze to get 60% antifreeze?
Problem 943 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Advanced-mixture-problems.lesson
Title: A Mixed-Rate Problem
Presenter: Patrick
A chemist has three different acid solutions. The first acid solution contains
15% acid, the second contains 30% acid, and the third contains 75% acid.
He wants to use all three solutions to obtain a mixture of 216 liter containing
25% acid, using 2 times as much of the 75% solution as the 30% solution.
How many liters of each solution should be used?
Problem 944 [Not Unipodal: Algebra: Mixed-Rate]
Source: Source:https://www.algebra.com/algebra/homework/word/mixtures/
Advanced-mixture-problems.lesson
Title: A mixed-rate problem
Presenter: Patrick
A sample of a compound contains two types of elements, Type1 and Type2.
Find the empirical formula of the compound if the following claims are true:
a) The percentage of Type1 element by mass is 84.80%. (These percentages
are approximate values.)
b) The percentage of Type1 element by moles of atoms in the compound
is about 27.27%.
c) When decomposed, the compound releases a gas (the Type2 element)
that is able to ignite a smoldering flint.
Problem 945 [Not Unipodal: Algebra: Mixed Rate]
Source: https://www.algebra.com/algebra/homework/word/mixtures/
Advanced-mixture-problems.lesson
Title: A Mixed-Rate Problem
Presenter: Patrick
A certain concrete mixture contains 5.00% cement and 7.00% sand.
How many kilograms of this mixture and how many kilograms of
sand should be combined with 255 kg of cement to make a batch
that is 16.0\% cement and 18.0\% sand?
Problem 946 [Not Unipodal: Algebra: Mixed-Rate]
Source: Source:https://www.algebra.com/algebra/homework/word/mixtures/
Advanced-mixture-problems.lesson
Title: A mixed-rate problem
Presenter: Patrick
A sample of a compound contains two types of elements, Type1 and Type2.
Find the empirical formula of the compound if the following claims are true:
When weighed in water, tin loses 0.137 of its weight and copper loses 0.112
of its weight. If an alloy of tin and copper weighing 18 pounds loses 2.316
pounds when weighed in water, how many pounds of each are there in the
piece [of alloy]?
Problem 947 [Not Unipodal: Algebra: Mixed Rate]
https://www.algebra.com/algebra/homework/word/mixtures
/Mixture_Word_Problems.faq.question.954102.html
Title: A Mixed-Rate Problem
Presenter: Patrick
Problem (paraphrased): Starting with 100 lbs alloy of 20% copper and
5% tin, how many pounds of copper and pounds oftin must be melted
into it to produce a new alloy that's 30% copper and 10% tin?
Problem 948 [Not Unipodal: Algebra: Tabular Solution]
Source https://www.algebra.com/algebra/homework/word/
age/Age-problems-and-their-solutions.lesson
Title: An Age problem
Presenter: Patrick
Kevin is 4 years older than Margaret. Next year Kevin will be
2 times as old as Margaret. How old is Kevin?
Problem 949 [Not Unipodal: Algebra: Logarithms]
Source https://www.youtube.com/watch?v=_WMaStdei2g
Title: An logarithm problem
Presenter: NonsoMaths
Given the relation \begin{equation} \log \left(\frac{x^{1/x}}{x^{1/(x+1)}}\right) = \frac{1}{5050}\,, \end{equation} and where the logarithm is base 10, find the integer value of $x$.
Problem 950 [Not Unipodal: Algebra: Tabular Solution]
Source: https://www.basic-mathematics.com/
hard-word-problems-in-algebra.html
Title: An hourly-wage comparison problem
Presenter: Patrick
Jacob’s hourly wage is 4 times as much as Noah. When Jacob got
a raise of 2 dollars Noah accepted a new position that pays him 2
dollars less per hour. Jacob now earns 5 times as much money as Noah.
How much money do they make per hour after Jacob
got the raise?
Problem 951 [Not Unipodal: Algebra: Word Problem]
Source: https://www.algebra.com/algebra
Title: Question 521012
Presenter: Patrick
Two Kleaning ladies company needs a 50\% bleach mixture solution.
They make 12 liter at a time from a 40% bleach solution and a 70%
bleach solution. How many liters do they need of each?
Problem 952 [Not Unipodal: Complex numbers: Logarithms]
Source: https://www.youtube.com/watch?v=UDbTLp77rfU
Title: A tricky math olympiad exam
Presenter: Higher Mathematics
Given the odd-looking relation \begin{equation} 1^x= 8\,, \end{equation} solve for $x$.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on logarithms over the real numbers.
Problem 953 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=wJ8OvYj6MUg
Title: Harvard Entrance Exam Math Question
Presenter: Math Beast
Given the relation \begin{equation} x^{\log 2x} =5\,, \end{equation} solve for $x$
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 954 [Not Unipodal: Logarithms: Lambert W functon]
Source: The Ether of Great Mathematical Ideas
Title: A Lambert Identity
Presenter: Patrick
Prove the Lambert identity:
\begin{equation}
W(x^{x+1}\ln x)= x \ln x\,.
\end{equation}
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Link to my write-up on the Lambert W function.
Problem 955 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: The case of the shrinking watermelon
Presenter: Patrick
This problem brings us some unintuitive results: A 100Kg watermelon
is estimated to be 99% water and 1% flesh part (that is, what would be
left of the watermelon if all water were removed from it). After some
days, the watermelon has dehydrated a bit and its water content is down
to 98%. How much does the dehydrated watermelon weigh?
Problem 956 [Not Unipodal: Algebra: Diophantine Problem]
Source: https://www.youtube.com/watch?v=_y-64fzCl1o
Title: A Problem from A Primer For Mathematics Competitions
Presenter: SyberMath
Given the relation
\begin{equation}
7a + 13b = 1000 \,,
\end{equation}
find a few pairs of integer values $(a,b)$ that satisfy this relation,
where we assume $a,b>0$.
Problem 957 [Not Unipodal: Algebra: Word Problem]
Source: https://www.algebra.com/algebra
Title: Question 835994
Presenter: Patrick
To produce sausage formulations, packers normally use lean beef,
pork belly, and soy concentrate. The manufacturer uses 3 percent
soy in final sausage mass. Water is added to the formulation as ice.
The final sausage must contain: 15 percent protein, 60 percent moisture,
and 25 percent fat. The ingredients contain the following:
- lean beef -- 20% protein, 67% moisture, 13% fat
- pork belly -- 10% protein, 40% moisture, 50% fat
- soy -- 90% protein, 7% moisture, 3% fat
How much of each ingredient should be combined to make 600 kg
of sausage emulsion?
Problem 958 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=caSZtP7YnFE
Title: Harvard University Interview Test
Presenter: Spencer's Academy
Given the relations \begin{align} a b &= 10\,,\\ b c &= 20\,,\\ a c &= 30\,,\\ \phi &= a+b+c\,, \end{align} solve for $\phi$, where we assume $a,b,c>0$.
Problem 959 [Not Unipodal: Algebra: Word Problem]
Source: The Ether of Great Mathematical Ideas
Title: A Mixture Problem
Presenter: Patrick
A jar containing a mixture (Mix 1) of two liquids, $A$ and $B$, in ratio $A:B::4:1$.
If 10 liters is removed and replaced by 10 liters of $B$, the ratio of $A$ to $B$ becomes
$2:3$. How much of $A$ was in Mix 1?
Problem 960 [Not Unipodal: Algebra: Cubics]
Source: https://www.youtube.com/watch?v=amu87QlQrP0
Title: Harvard University Entrance math Examination question
Presenter: JJ ONLINE MATHS CLASS
Given the relation
\begin{equation}
27^x - 3^x = \sqrt{12}\,,
\end{equation}
find the real solutions for $x$. (If you are interested in
all the possible
solutions for $x$, see WolframAlpha.)
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 961 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 10885: A Mixed-Rate Problem
Presenter: Patrick
A car gets 28 mpg on highway, 22 mpg in city. If the total trip is 627 miles
using 24 gallons of gas, how many miles were driven in city?
Problem 962 [Not Unipodal: Algebra: Alpha Substitution]
Source: https://www.youtube.com/watch?v=y_LGJI8vBYs
Title: A Challenging Exponential Problem
Presenter: Click Academics
Given the relation \begin{equation}\ x^{x^5}= 100\,, \end{equation} find the real value for $x$.
Problem 963 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 364411: A Mixed-Rate Problem
Presenter: Patrick
Two alloys contain silver and copper in the ratios $3:1$ and $5:3$,
respectively.
The alloys are mixed to get a third alloy. The possible ratio of silver to copper
in the third alloy is?
Problem 964 [Not Unipodal: Algebra: Logarithms]
Source: https://www.youtube.com/watch?v=uRtalj-oqKk
Title: Think You Know Logs?
Presenter: NonsoMaths
Given the relation \begin{equation}\ 15^{\log_5 3} = \frac{1}{x^{\log_5(9x)+1}}\,, \end{equation} solve for $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 965 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 377832: A Mixed-Rate Problem
Presenter: Patrick
A bag of peanuts is worth \$0.28 less than the same size bag of cashews.
Equal amounts of peanuts and cashews are used to make 45 bags of a
mixture that sells for \$1.25 per bag. How much is a bag of cashews worth?
(Give your answer to the nearest cent.)
Problem 966 [Unipodal]
Source: https://www.youtube.com/watch?v=aX4IHM0VcXk
Title: A fancier way to solve this radical equation
Presenter: blackpenredpen
Given the relation \begin{equation} \cuberoot{x-40} + \cuberoot{-x+3} = -1\,, \end{equation} find the real values of $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 967 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 377832: A Mixed-Rate Problem
Presenter: Patrick
As the problem was stated:
........Lead....Zinc....Copper
Alloy A..40\%...30\%...30\%
Alloy B..20\%...30\%...50\%
Alloy C..........10\%...90\%
How many grams of each alloys A, B, and C must be mixed to get
325 gm of an alloy that is 25\% lead, 13\% zinc, and 62\% copper?
I've tried this using the Matrix, and the inverse Matrix, and keep
coming up with negatives on some alloys.
Thanks for your help.
First, I also got negative numbers with the original data set, so
(after some trial and error) I changed the percentages in the final
alloy to the values in the table below:
Problem 968 [Not Unipodal: Lambert W Function: Logarithms]
Source: https://www.youtube.com/watch?v=VwNwG3ddo6k
Title: Germany | Can you solve this?
Presenter: SALogic
Given the relation \begin{equation}\ 4^{x} = x^{8}\,, \end{equation} solve for the real values of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Link to my write-up on the Lambert W function.
Problem 969 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 732982: A Mixed-Rate Problem
Presenter: Patrick
Fred is analyzing the cost of producing two different items at an
electronics company. an electrical sensing device uses 5 grams
of copper and requires 3 hours to assemble. a smaller sensing
device made by the same company uses 4 grams of copper but
requires 5 hours to assemble. the first device has a production
cost of \$27. the second device has a production cost of \$32. How
much does it cost the company for a gram of this type of copper?
What is the hourly labor cost at this company? (assuming that
production cost is obtained by adding the copper cost and the
labor cost)
Problem 970 [Not Unipodal: number theory]
Source: https://www.youtube.com/watch?v=OnvFLXVqYZQ
Title: nice solution, lil' bro
Presenter: Wrath of Math
Given the relation \begin{equation}\ 2^{x} + 2^{y} = 160\,, \end{equation} solve for the integer values of $x$ and $y$.
Problem 971 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 842253: A Mixed-Rate Problem
Presenter: Patrick
A reservoir containing 1 million gallons of water has been tainted with
arsenic. Scientists conducted several tests and found that the solution
in the reservoir is 2% arsenic. Although scientists realize that the arsenic
can't be removed from the supply, they do understand that they can
reduce the percentage to safe levels. A safe level of arsenic is .001%
How much water must be added to the reservoir to make the water
safe to drink?
Problem 972 [Unipodal]
Source: https://www.youtube.com/watch?v=euBnlvAanxY
Title: France | Can you solve?
Presenter: Math Master TV
The following relation \begin{equation}\ \phi = (\sqrt{11}+\sqrt{5}\,)^8 +(\sqrt{11}-\sqrt{5}\,)^8\,, \end{equation} has a simple (though large) integer value. Find it.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 973 [Not Unipodal: Differential Equations]
Source: https://www.youtube.com/watch?v=t6IzRCScKIc
Title: Can d^2y/dx^2=(dy/dx)^2?
Presenter: blackpenredpen
Given the relation \begin{equation} \frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2\,, \end{equation} solve for $y=y(x)$.
Problem 974 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=QFDxHYb1cDk
Title: Most Students Give Up on This Log Equation
Presenter: NonsoMaths
Given the relation \begin{equation} \log_{1/2}^2(4x) + \log_2 \left(\frac{x^2}{8}\right)=8\,, \end{equation} find $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 975 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 722538: A Mixed-Rate Problem
Presenter: Patrick
Jessica has money in two savings accounts. One rate is 10% and the other is 15%.
If she has \$450 more in the 15% account and the total interest is \$258, how much
is invested in each savings account?
Problem 976 [Not Unipodal: Integration]
Source: The Ether of Great Mathematical Ideas
Title: An easy indefinite integral
Presenter: Patrick
Perform the indefinite integral \begin{equation} I = \int x^3\, e^{x^2}\, dx\,. \end{equation}
Problem 977 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 638401: A Mixed-Rate Problem
Presenter: Patrick
A textile company has specific dyeing and drying times for its different cloths.
A roll of Cloth A requires 65 minutes of dyeing time and 50 minutes of drying
time. A roll of Cloth B requires 55 minutes of dyeing time and 30 minutes of
drying time. The production division allocates 2440 minutes of dyeing time
and 1680 minutes of drying time for the week. How many rolls of each cloth
can be dyed and dried?
Problem 978 [Not Unipodal: Matrix theory]
Source: https://www.youtube.com/watch?v=ykR7psuKHw4
Title: GRE Mathematics Subject Test
- Trace of self-inverse matrix
Presenter: Math Out Loud
Let $A$ be an element of the space of $2\times2$ matrices over the
complex numbers. Let $I$ be the identity matrix in this same
space. If $A \ne \pm I$, and $A=A^{-1}$, what is the trace of $A$?
Problem 979 [Not Unipodal: Algebra: Alpha Substitution]
Source: https://www.youtube.com/watch?v=8FmRRCuOqyo
Title: A Challenging Exponential Problem
Presenter: Click Academics
Given the relation \begin{equation} x^x= 2^{x+4}\,, \end{equation} solve for real values of $x$.
Problem 980 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 269702: A Mixed-Rate Problem
Presenter: Patrick
The time it takes to do homework includes a fixed amount of time
to prepare plus a constant amount of time per problem. If a student
can do 5 homework problems in 40 minutes, and 10 problems in
70 minutes, how many minutes will 25 problems take?
Problem 981 [Not Unipodal: Matrix theory]
Source: https://www.youtube.com/watch?v=WJfhnOMtsaY
Title: Solving a 'Harvard' University Entrance Exam Question
Presenter: Math Explorer
Given the relation \begin{equation} \log_2 x + \log_3 x= 5\,, \end{equation} solve for real values of $x$.
Problem 982 [Not Unipodal: Matrix theory]
Source: https://www.youtube.com/watch?v=zhMzgB0Mcdg
Title: GRE Mathematics Subject Test - A matrix code
Presenter: Math Out Loud
The problem is a spy-vs-spy math thriller. For details,
consult the video.
But it boils down to solving the following matrix
equation for $M$:
\begin{equation}
MC=
\begin{bmatrix}
51 & -3\\
31 & -8
\end{bmatrix}
\,,
\end{equation}
where $M$ is a $2\times2$ matrix and
\begin{equation}
C=
\begin{bmatrix}
2 & -1\\
1 & 1
\end{bmatrix}
\,.
\end{equation}
Problem 983 [Not Unipodal: Trigonometry: Triangle]
Source: https://www.youtube.com/watch?v=5C2VvcGEr4Y
Title: This High School Math Problem Will Test Your Skills
Presenter: The Phantom of the Math
This high-school trigonometry problem is presented in the figure below.
Angle $\alpha = 60^\circ$. The Presenter got for L, $15\sqrt{7}$.
I got that and a possible
alternate $30\sqrt{7}$. I ran out of time trying
to reject my larger value for $L$.
Perhaps the reader can settle the issue.
Problem 984 [Not Unipodal: Complex Numbers]
Source: https://www.youtube.com/watch?v=HVLVwJtMnF0
Title: Complex Trig Puzzle Explained! | P596
Presenter: aplusbi
Given \begin{equation} \tan \theta - \sec \theta = i\,, \end{equation} solve for real values of $\theta$, and be general about it.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 985 [Not Unipodal: Integration]
Source: https://www.youtube.com/watch?v=zMOfo9bV2II
Title: A Deceptively Hard Integral
Presenter: Mental Math
Find the integral $I$: \begin{equation} I = \int_0^1 \frac{x^5}{1+x^2} dx\,. \end{equation}
Problem 986 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=0aMU0q3w-mo
Title: A Very Nice Exponential Equation Can You Solve It?
Presenter: Mental Math
Given the relation \begin{equation} \left(\frac{x}{x+11}\right)^{x+11}= \frac{1}{2048}\,, \end{equation} solve for real values of $x$.
Problem 987 [Not Unipodal: Algebra: Logarithms]
Source: https://www.youtube.com/watch?v=cgUtbvIQC3Q
Title: Can You Solve This Exponential Equation in 30 Seconds
Presenter: Mental Math
Given the relation \begin{equation} 2^x + 2^{1-x}=3\,, \end{equation} solve for real values of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 988 [Not Unipodal: Algebra: Logarithms]
Source: https://www.youtube.com/watch?v=39eNat-v4og
Title: Only 1% of Students Solve This Logarithm Problem Correctly!
Presenter: NonsoMath
Given the relation \begin{equation} \log_x 10 + 2\log_{10x} 10= 3\log_{100x}10\,, \end{equation} solve for the values of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 989 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 286358: A Mixed-Rate Problem
Presenter: Patrick
An urn is filled with coins and beads. All the coins are either silver
or gold. Twenty percent of the objects in the urn are beads. Forty
percent of the coins in the urn are silver. What percent of the objects
in the urn are gold coins?
Problem 990 [Not Unipodal: Complex Numbers]
Source: https://www.youtube.com/watch?v=Y8Xp1nPEycE
Title: The Real Deal | P589
Presenter: aplusbi
Given the relation \begin{equation} |z|^2 - 2 \Re(iz)=3\,, \end{equation} solve for the locus of points $\{(a,b)\}$ satisfying the relation.
Problem 991 [Not Unipodal: Trigonometry]
Source: https://www.youtube.com/watch?v=HQxPzY3C6dg
Title: Two Equations, Infinite Curiosity
Presenter: SyberMath
Given the relations \begin{align} 3 \sin A + 4 \cos B &= 6\,,\\ 4 \sin B + 3 \cos A &= 1\,, \end{align} where $A$ and $B$ are interior angles of triangle ABC, solve for the values of $A,B$.
Problem 992 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 255570: A Mixed-Rate Problem
Presenter: Patrick
It is required to make 12 grams of certain chemical compound called Z.
This is made from compounds W, X, and Y in the ratio of 2:1:3. The
compound Y is itself made from W and X. To make 6 grams of Y requires
4 grams of W and 2 grams of X. How much W and X is required to make
the required amount of Z.
Problem 993 [Unipodal]
Source: https://www.youtube.com/watch?v=j1zdTAsfbbI
Title: This Radical Equation is EASIER Than it Looks
Presenter: NonsoMaths
Given the relation \begin{equation} \sqrt{2x^2-7x+1} - \sqrt{2x^2-9x+4} = 1\,, \end{equation} solve for real values of $x$.
Problem 994 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 154928: A Mixed-Rate Problem
Presenter: Patrick
I have 51 handle bars and 116 wheels. Using all the parts, how
many tricycles and bicycles can I assemble?
Problem 995 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 311277: A Mixed-Rate Problem
Presenter: Patrick
Albert buys and sells books, and always purchase it at the same price.
He then sells the books for \$5 more than what he paid for. Two months
before, he broke even after buying 56 books and selling 49. What is his
buying price and selling price?
Problem 996 [Not Unipodal: Logarithms: Algebra]
Source: https://www.youtube.com/watch?v=kl8uoJt2iDw
Title: This Log Equation in Different Bases Stumps Students
Presenter: NonsoMaths
Given the relation \begin{equation} \log_8(9x)= \log_9(8x)\,, \end{equation} solve for real values of $x$.
Problem 997 [Not Unipodal: Logarithms: Algebra]
Source: https://www.youtube.com/watch?v=8rexi2JUk6M
Title: A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} 2^{x^2} = \frac{10}{5^x}\,, \end{equation} solve for real values of $x$.
Problem 998 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=CRfYk9LUBng
Title: Math Algebra Solution
Presenter: Sifat Math Hospital
Given the relation \begin{equation} x^{x^{73}}=73 \,, \end{equation} solve for real values of $x$.
Problem 999 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: http://gmatclub.com/forum/two-kinds-of-vodka-are
-mixed-in-the-ratio-1-2-and-2-1-and-113897.html
Title: A Mixed-Rate Problem
Presenter: Patrick
Two kinds of Vodka are mixed in the ratio 1:2 and 2:1 and they are
sold fetching the profit 10% and 20% respectively. If the vodkas
are mixed in equal ratio and the individual profit percent on them
are increased by 4/3 and 5/3 times respectively, then the mixture
will fetch the profit of
A. 18%
B. 20%
C. 21%
D. 23%
E. Cannot be determined
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