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# Great Answers to Learn From

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### Great Answers to Learn From

\$\$Introduction:\$\$

Hello! You have reached the "Great answers to learn from" page. This page is meant to be a collection of great answers that people could look back upon and learn from. Hopefully if you have a certain topic you want to learn about, you can some posts on that topic here!

For the most part, this thread is organized into "grade levels" of mathematics to make it easier to find what you want to learn about.

Finally, if you have anything you want to add to this post, go ahead and reply to it with what you want added, or you can message me and I'll add it to the proper section.

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This post was organised and up kept by NinjaDevo.  I am very grateful to Ninja for his great efforts.

NinjaDevo is now too busy to upkeep the thread so I have transerred it into my name so that I can edit it as required.

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Roots and exponents:

• Simplifying.

http://web2.0calc.com/questions/can-anyone-do-square-roots

• Finding a root of a negative number

http://web2.0calc.com/questions/can-i-calculate-a-negative-number-to-the-power-of-a-fraction-ie-1-42-1-7

• Rationalising a denominator:

http://web2.0calc.com/questions/in-question-21-nbsp

• Exponets with negative bases:

http://web2.0calc.com/questions/if-a-negative-number-is-in-parenthesis-and-has-a-positive-exponent-how-does-it-get-a-positive-number-outcome-while-a-negative-number-withou

Fractions:

• Dividing two fractions:

http://web2.0calc.com/questions/3-4-5-8

• Find the Least Common Denominator (LCD):

http://web2.0calc.com/questions/what-is-the-lcd-for-5-9-and-2-3

When can fraction be written as a terminating decimal.  Great answer by CPhill.   (26/11/14)

http://web2.0calc.com/questions/chandler-tells-aubrey-that-the-decimal-value-of-1-7-is-not-a-repeating-decimal-should-aubrey-believe-him

Dealing with time in math, on a handheld calculator!

http://web2.0calc.com/questions/what-is-21hours-and-58-minutes-subtracted-by-4-36-p-m-equal

Directed numbers and plotting on a number line

Is zero even or odd?

Understanding different bases

Converting base 10 to base 2

Ratio

http://web2.0calc.com/questions/there-are-two-glasses-on-the-table-one-containing-water-and-the-other-one-milk-they-both-contain-exactly-the-same-amount-by-volume-if-you

Speed and rates:

• Migrating elephants!

http://web2.0calc.com/questions/a-migrating-elephant-herd-started-moving-at-a-rate-of-6-miles-per-hour-one-elephant-stood-still-and-was-left-behind-then-this-stray-elepha

• Speed of a current

http://web2.0calc.com/questions/a-boat-on-a-river-travels-20-miles-downstream-in-only-2-hours-it-takes-the-same-boat-6-hours-to-travel-12-miles-upstream-what-are-the-spee

* Speed distance and trains (time for trains to pass each other)

http://web2.0calc.com/questions/i-came-across-this-question-and-didn-t-know-how-to-solve-it

* Speed of an escalator

http://web2.0calc.com/questions/i-am-in-the-habit-of-walking-up-the-subway-escalator-while-it-is-running-i-climb-20-steps-at-my-normal-pace-and-it-takes-me-60-seconds-to-r

* Speed of taps filling a pool

http://web2.0calc.com/questions/2-taps-a-and-b-can-fill-a-swimming-pool-in-3-hours-if-turned-on-alone-it-takes-tap-a-5-hours-less-than-tap-b-to-fill-the-same-pool-how-ma

Comparing numbers; the apparent brightness of stars!

Percentages:

• Find a total when given a part

http://web2.0calc.com/questions/435-is-15-of-what-number

Variant of Average:

http://web2.0calc.com/questions/the-average-of-six-numbers-is-6-if-3-is-subtracted-from-each-of-four-of-the-numbers-what-is-the-new-average#r1

Adding up all the numbers between two specific numbers:

Forever repeating decimal!

http://web2.0calc.com/questions/0-99999

Order of operations - multiplication before addition:

http://web2.0calc.com/questions/3x60-5x9-2x2

The basics of algebra:

• What does "x" mean?

http://web2.0calc.com/questions/5x-21-2x-what-does-x-mean

• Solving for x by "balancing" out two sides of an equation:

http://web2.0calc.com/questions/solve-for-x-7

Solving a simple equation

http://web2.0calc.com/questions/solving-a-simple-equation

• Why would I use x2?

http://web2.0calc.com/questions/what-does-x-squared-equal

Algebra word problem examples:

• Simultaneous equations:

http://web2.0calc.com/questions/a-shopkeeper-paid-9-380-for-some-shirts-and-jeans-he-bought-4-times-as-many-shirts-as-jeans-each-pair-of-jeans-cost-40-more-than-a-shirt-h

http://web2.0calc.com/questions/how-old-they-are

• Set it up algebraicly, and solve!

http://web2.0calc.com/questions/you-want-to-place-a-towel-bar-that-is-8-1-4-inches-long-in-the-center-of-a-door-that-is-26-1-3-inches-long-how-far-should-you-place-the-bar

Fractions and solving for x:

http://web2.0calc.com/questions/if-then

http://web2.0calc.com/questions/solve-for-x-3-over-4-x-5-over-8-4x

Three different techniques to this answer dealing with polynomials:

http://web2.0calc.com/questions/5a-5-5a-5

difficult indices (quadratic equation) question.       (20/11/14)

http://web2.0calc.com/questions/can-anyone-tell-me-what-are-the-steps-for-this

Simplifying indices - they can be messy

http://web2.0calc.com/questions/can-you-solve-this

Simplifying indices - A pretty cool question   (29/12/14)

Making x the subject of a parabola

http://web2.0calc.com/questions/factor_5      (15/12/14)

Simplifying root questions

http://web2.0calc.com/questions/sqrt-a-2sqrt-a-4-5-7-sqrt-a-5

Finding root using secant method.   Thanks Alan    (29/12/14)

http://web2.0calc.com/questions/find-the-roots-of-the-equation-x2-4x-5-0-the-secant-method

Floor and ceiling functions

http://web2.0calc.com/questions/need-help-on-this-a-lot-thanks

http://web2.0calc.com/questions/find-all-values-of-nbsp-nbsp-such-that-nbsp-if-you-find-more-than-one-value-then-list-the-values-you-find-in-increasing-order-separated

http://web2.0calc.com/questions/let-nbsp-evaluate-this-sum-has-1000-terms-one-for-the-result-when-we-input-each-integer-from-1-to-1000-into-nbsp

http://web2.0calc.com/questions/the-function-nbsp-nbsp-is-defined-as-find-nbsp

http://web2.0calc.com/questions/what-is-the-coefficient-of-nbsp-nbsp-in-nbsp_2

SEQUENCES AND SERIES

Sum of a Sequence - Really difficult

POLYNOMIALS

(There is a whole collection in here from 5/12/14)

http://web2.0calc.com/questions/polynomial-questions#rr0

GRAPHING and FUNCTIONS

Similar functions - I really liked this question        (7/11/14)

http://web2.0calc.com/questions/i-can-t-access-one-of-the-questions-i-put-up-it-is-the-one-i-will-put-below

This one really made me think - Melody  (7/11/14)

This is similar to the above one but a bit easier.

http://web2.0calc.com/questions/the-function-nbsp-nbsp-satisfiesfor-all-nbsp-nbsp-nbsp-find-nbsp_1

Find the range of this function

This function question is not difficult - just unusual     (7/11/14)

Conic Sections:

The original question is a degenerate conic BUT I have included info on other conics

This result is really weird graph.  It has degenerated into two cross lines.

Imaginary numbers:

• Introduction to imaginary numbers

• Finding Complex Roots with imaginary numbers (Difficult)

http://web2.0calc.com/questions/how-do-i-solve-the-sqrt4-16-complex-numbers-4-different-roots

• Complex number proof that ii is a real number:

http://web2.0calc.com/questions/puzzle-question-i-i

Log of complex number

http://web2.0calc.com/questions/ln-10-75-6-98i

Difficult inequality solution

http://web2.0calc.com/questions/how-can-i-solve-the-first-question-nbsp

Solveing a nonlinear inequality. This one looks like a parabola!

http://web2.0calc.com/questions/solve-the-nonlinear-inequality

Well explained by heureka:

http://web2.0calc.com/questions/5x-8-3x-3-whats-the-x

Heureka just being awesome:

http://web2.0calc.com/questions/solve-the-equation_1

An example of scientific notation:

http://web2.0calc.com/questions/if-i-have-1-5-trillion-us-dollars-and-wanted-to-divide-it-up-between-7-billion-people-the-world-s-population-how-much-would-each-person-g

A more complicated algebra question answered by Alan

http://web2.0calc.com/questions/x-2-xy-y-2-28-y-2-x-how-do-you-get-x-and-y

http://web2.0calc.com/questions/horizontal-asymptote-of-y-x-2-6x-3-x-3

• Complicated one:

• Solving with the quadratic formula:

• A little different than the usual quadratic questions:

http://web2.0calc.com/questions/x-7sqrt-x-10-0

Finding the equation of a polynomial when you are given the roots:

Linear Representation (graphing the equation of a line) of a worded problem:

Completing the squares - This is a really interesting application of it.

http://web2.0calc.com/questions/the-temperature-of-a-point-nbsp-nbsp-in-the-plane-is-given-by-the-expression-nbsp-what-is-the-temperature-of-the-coldest-point-in-the-pla_3

What is a logarithm? (thanks to anonymous)

http://www.mathsisfun.com/algebra/logarithms.html

http://web2.0calc.com/questions/i-m-not-actually-sure-how-to-type-this-out-so-i-took-a-screenshot-does-anyone-know-how-to-solve-this-http-prntscr-com-5a6cz7

Half life problem:

http://web2.0calc.com/questions/the-half-life-of-carbon-14-is-around-5700-years-find-the-of-the-original-carbon-14-that-remains-in-a-sample-after-1835-years-have-passed

Solving for x when x is raised to a power:

http://web2.0calc.com/questions/can-t-seem-to-get-this-equation-solved

Finding why an equation with logs equals zero - two different approaches:

http://web2.0calc.com/questions/why-does-log-2ab-log-5-a-log10b-0

This certainly looks like a more difficult logarithmic equation to me!

http://web2.0calc.com/questions/difficult-logarithmic-equation-without-a-calculator

Harder logarithm problem. Three people got the same answer, so it should be right!

http://web2.0calc.com/questions/log-3-6-log-9-6-log-4-6-log-18-6

Triangles! Great picture and explanation by CPhill

http://web2.0calc.com/questions/so-i-have-two-triangles-here-but-are-they-overlapping-i-can-t-draw-i-diagram-that-fits-the-discription-p-nbsp-does-anyone-have-a-suges

Finding the circumference from the diameter:

http://web2.0calc.com/questions/circumference-of-pipe-diameter-3090mm

Height of a triangle, the Pythagorean theorem!

http://web2.0calc.com/questions/how-do-i-figure-out-the-right-of-a-triangle-when-ab-equals-13-and-cb-equals-12

Heron's formula to find the area of a triangle:

http://web2.0calc.com/questions/triangle-s-sides-are-a-5-b-12-and-c-9-what-is-the-area-of-the-triangle

Volume of a sphere:

http://web2.0calc.com/questions/a-spherical-ball-hss-a-diameter-of-45cm-workout-the-volume-of-the-ball

Word problems:

Hay bales (cylinder):

http://web2.0calc.com/questions/i-have-a-bale-of-hay-that-is-5-in-diameter-and-4-wide-if-i-store-the-bale-outside-the-outer-8-inches-gets-damaged-by-sun-and-rain-what

• Wagon wheels (circumference of a circle):

http://web2.0calc.com/questions/neeeed-help-on-this

Finding the 4th side of a quadrilateral that is circumscribed by a circle.

An interesting trapezium area question.  Thanks Chris

http://web2.0calc.com/questions/a-trapezoid-6m-in-altitude-have-bases-of-12-m-and-18m-respectively-if-the-it-is-divided-into-two-parts-by-a-line-parallel-to-the-bases-suc

Unusual area question

An interesting co-ordinate geometry question.

http://web2.0calc.com/questions/equation-to-circles

Inscribed equilateral triangle in a circle

http://web2.0calc.com/questions/length-of-an-equilateral-traingle-within-a-circle

Finding the area of a segment of a circle.  (19/5/15)

Difficult triangle question       (Thanks Heureka   25/6/15)

http://web2.0calc.com/questions/in-triangle-abc-point-d-is-on-side-bc-bd-2-dc-cosdac-cosc-ac-bc

### Statistics and Probability

Z scores:

http://web2.0calc.com/questions/how-do-i-find-the-area-under-the-standard-normal-curve-over-the-interval-specified-below-between-z-44-and-z-44

Permutations where some objects are not unique:

http://web2.0calc.com/questions/permutations_2

http://web2.0calc.com/questions/probability-melody

### Set Theory

Introductory set theory:

http://web2.0calc.com/questions/represent-the-following-number-as-being-members-of-set-k-2-4-2-0-6-0-10-8

Here is one for middle level trig students to have a go at - I have not actually finished it.

http://web2.0calc.com/questions/3-sin-2x-3-1-2-sin-2x

Verifying the identity

http://web2.0calc.com/questions/verify-the-identity-cot-theta-sec-theta-csc-theta

Why is Sinθ approximately equal to θ for very small values of θ

http://web2.0calc.com/questions/why-is-sin-of-theta-appromately-equal-to-theta

How do you choose the correct trigonometric ratio (sin, cos or tan) to use?

Finding an angle on a right sided triangle:

http://web2.0calc.com/questions/i-need-help-finding-the-angle-of-a-right-sided-triangle#rr2

Unusual trig. equation (solve for θ)

http://web2.0calc.com/questions/sin-3-cos-26-solve-for

Bearings and vectors:

Bearings speed and vectors Great Question!

http://web2.0calc.com/questions/a-coast-guard-cutter-detects-an-unidentified-ship-at-20-0km-in-the-direction-15-0-degrees-east-of-north-the-ship-is-traveling-at-26-0km-h-o

Velocities - Looks like a great answer from Alan:

Light rays!

http://web2.0calc.com/questions/if-a-light-ray-is-incident-of-the-glass-surface-with-index-of-refraction-n-1-50-at-0-degree-anglee-what-would-be-the-angle-of-refraction-if#r2

Some more trig. questions:

http://web2.0calc.com/questions/8-sec2-x-4-tan2-x-12-0

http://web2.0calc.com/questions/3cosx-4sinx-1

Area of a triangle - not straight forward and Heron's formula:

http://web2.0calc.com/questions/2-neigbouring-triangles-with-no-right-angles-with-very-little-information-find-the-area-of-them-combined_1

High level Trig:

http://web2.0calc.com/questions/how-can-i-show-that-the-bound-between-sin-v-2-cos-v-2-1-is-true-for-all-degrees-in-the-intervall-0-v-180

University level Trigonometry:

http://web2.0calc.com/questions/how-to-know-when-the-function-is-continuous

Difficult sine and cosine rule problem:

http://web2.0calc.com/questions/i-get-no-diea

Tricky trig limit

http://web2.0calc.com/questions/how-to-evaluate-the-lim-x-pi-4-1-tan-x-sec-2x

Equation with inverse trigonometry   (Gret answer from Heureka)

http://web2.0calc.com/questions/what-is-x-if-the-equation-is-2-tan-1-x-160-tan-1-x-60

#### Really neat highest common divisor proof by Bertie

http://web2.0calc.com/questions/let-the-term-s_n-be-the-sum-of-the-first-n-powers-of-2-for-instance-s_3-2-0-2-1-2-2-7-find-the-largest-possible-value-of

Continuity at a point.

Limits

Limiting sum of a series - uses a differential         (14/5/15)

How far will the bullet travel?

Limits using L'Hopital's Rule

Limits comparing Newton's method and Linear approximation method.

Solving differential equations:
Street light shadows! Great explanations by both Melody and CPhill:

Unusual Calculus/Cordinate geometry question:

http://web2.0calc.com/questions/find-the-most-left-and-right-tangent

http://web2.0calc.com/questions/how-do-you-find-a-total-mass-using-spherical-shells-for-example-the-total-mass-of-a-spherical-galaxy-with-a-mass-density-at-a-distance-s

http://web2.0calc.com/questions/water-is-leaking-out-of-an-inverted-conical-tank-at-a-rate-of-11-000-cm3-min-at-the-same-time-that-water-is-being-pumped-into-the-tank-at-a

http://web2.0calc.com/questions/integrate-2cos3x-3sinx-sin-3x-dx-and-tan-5x-csc-5x-dx  (Melody and Heureka)   I used the reduction formula here :)

Integration with substitutions - answer is inverse trig  (CPhill and Heureka 6/2/15)

Trig integration (best with substitution)         20/6/15

http://web2.0calc.com/questions/integral_4

Newton's method of approximating roots.

http://web2.0calc.com/questions/use-the-newton-s-method-to-approximate-the-solution-of-the-equation-1-x-tan-x-0-starting-at-x_-0-0-5

Partial Fractions - a technique needed for some integration problems

Integrating log functions    (19/11/14    Heureka)

http://web2.0calc.com/questions/how-to-find-integral-of-logarithmic-functions

Integration by parts  (19/10/14)    Melody

http://web2.0calc.com/questions/integrate-the-inverse-sine-of-x---by-parts

http://web2.0calc.com/questions/integral-of-xe-square-root-of-x-dx

http://web2.0calc.com/questions/integral-problem           (Melody and Alan 8/12/14)

### Commercial  Mathematics

Difficult:

• Compare Renting to Buying - Annuities

Finding the present value of a bond.

http://web2.0calc.com/questions/valuation-corporate-bond_1

Forces explained by Alan

http://web2.0calc.com/questions/finding-friction-coefficient-from-kinetic-energy

http://web2.0calc.com/questions/2d-forces-equation#r0

http://web2.0calc.com/questions/an-extreme-skier-starting-from-rest-coasts-down-a-mountain-that-makes-an-angle-of-30-9-with-the-horizontal-the-coefficient-of-kinetic-f

Electrical Resistance     (24/4/15)

http://web2.0calc.com/questions/resistant#r15

Vectors:

• Vector forces:

http://web2.0calc.com/questions/collinear-resolving-angles-of-a-parallelogram

• Converting to vector units:

http://web2.0calc.com/questions/how-to-turn-into-vector-units

• Magnitude of vector forces:

http://web2.0calc.com/questions/u-6i-j-and-v-10i-j-find-the-manitude-of-u-v

• Another vector problem:

http://web2.0calc.com/questions/the-acute-angle-in-degrees-that-makes-a-with-c-when-a-4i-0j-k-c-i-j-3k

Distance/displacement:

http://web2.0calc.com/questions/total-distance

Misc. physics problems:

http://web2.0calc.com/questions/nbsp_3

Complex numbers

Graphical solutions to the 12 root of 1

### Puzzles

Some thinking puzzle problems:

http://web2.0calc.com/questions/prove-that-in-a-gathering-of-n-persons-there-is-always-2-persones-of-them-that-have-same-amount-of-friends-among-these-persones

http://web2.0calc.com/questions/a-drinking-problem

How slow is the clock?   I really liked this problem!       (4/5/15)

Plain Interesting Facts

http://web2.0calc.com/questions/a-girl-was-eight-years-old-on-her-first-birthday-how-could-that-be

Plain interesting Questions

Optimization - Soap bubbles - problem

Melody  Oct 18, 2014

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MathsGod1  Apr 12, 2015
Sort:

#1
+4663
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MathsGod1  Apr 12, 2015
#2
+90192
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Thanks MathsGod1,

It is a good thread, it has not been added to for a long time because NinjaDeva can no longer do it and I don't have time.  It is a pity but i do include a little more threads in the "reference Material" to make up a little.

Ninja used to help me a lot - I continue to be very appreciative of all the effort he put in :)

Melody  Apr 13, 2015

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