Author Archives: vinaire

I am originally from India. I am settled in United States since 1969. I love mathematics, philosophy and clarity in thinking.

PSAT / SAT Prep

Filling in bubble test
Reference: Maths Self Learning

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WORD PROBLEMS REVIEW

300: 01 Simple Equation Problems

310: 02 Percent Problems

320: 03 Advanced Level Age Problems

330: 04 Mixing Problems

340: 05 Measurement (Ratio) Problems

350: 06 Rate Problems

360: 07 Statistics & Probability

370: 08 Geometry Problems

380: 09 Review of Equations

390: 10 Answers to Practice Problems

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PSAT & SAT REVIEW

400: 01 PSAT Diagnostics 1

405: 02 PSAT Diagnostics 2

410: 03 PSAT Algebra

420: 04 PSAT Equation

430: 05 PSAT Inequalities

440: 06 PSAT Geometry

450: 07 PSAT Fractions

460: 08 PSAT Percentage

470: 09 PSAT Averages

480: 10 PSAT Rates

490: 11 PSAT Ratio

500: 12 PSAT Work

510: 13 PSAT Data Interpretation

520: 14 PSAT Quantitative Comparison

530: 15 PSAT Typical Test A

540: 16 PSAT Typical Test B

550: 17  PSAT Typical Test C

560: 18 PSAT Typical Test D

570: 19 PSAT Typical Test E

580: 20 PSAT Typical Test F

590: 21 PSAT Typical Test G

600: 22 PSAT Typical Test H

610: 23 PSAT Typical Test I

620: 24 PSAT Typical Test J

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PRACTICE PROBLEMS

There are hundreds of sets of excellent practice problems in 465. Some of the sets are listed from 651.

651: SET1001 / Solutions

652: SET1002 / Solutions

653: SET1003 / Solutions

654: SET1004 / Solutions

655: SET1005 / Solutions

656: SET1006 / Solutions

657: SET1007 / Solutions

658: SET1008 / Solutions

659: SET1009 / Solutions

660: SET1010 / Solutions

661: SET1011 / Solutions

662: SET1012 / Solutions

663: SET1013 / Solutions

664: SET1014 / Solutions

665: MATHDEN (User ID: guest; Password: guest)

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Pre-Algebra Review

Pre-Algebra
Reference: Maths Self Learning

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PRE-ALGEBRA – BASICS

PA 100: Natural and Whole Numbers

PA 101: Math Homonyms

PA 102: Concepts in Arithmetic (1)

PA 103: Concepts in Arithmetic (2)

PA 104: Arithmetic Overview

PA 105: Math Overview

PA 106: Equations & Transposition

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PRE-ALGEBRA – REVIEW

PA 100: Diagnostics

PA 110: UNIT  01 Real Numbers / Answers to Unit 01

PA 120: UNIT  02 Variable Expressions / Answers to Unit 02

PA 130: UNIT  03 Solving Equations / Answers to Unit 03

PA 140: UNIT  04 Solving Equations: Applications / Answers to Unit 04

PA 150: UNIT  05 Polynomials / Answers to Unit 05

PA 160: UNIT  06 Factoring / Answers to Unit 06

PA 170: UNIT  07 Algebraic Functions / Answers to Unit 7

PA 180: UNIT  08 Graphs & Linear Equations / Answers to Unit 08

PA 190: UNIT  09 Systems of Linear Equations / Answers to Unit 09

PA 200: UNIT  10 Inequalities / Answers to Unit 10

PA 210: UNIT  11 Radical Expressions / Answers to Unit 11

PA 220: UNIT  12 Quadratic Equations / Answers to Unit 12

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Observe the Incomprehensible

M4
Reference: The 12 Aspects of Mindfulness

Mindfulness provides the discipline for looking and contemplation.

You may do this exercise while sipping coffee in a café, or strolling along a river. You may even find a place where you can sit comfortably for a while without being disturbed. Then patiently observe the world go by.

If something does not make sense, then recognize that it does not make sense. Do not try to justify it. Justification simply puts the blame somewhere without resolving the inconsistency. When you are faced with an inconsistency, and you feel an impulse to explain it away, then be alert to what you might be taking for granted. At times it may take some out-of-the-box thinking to realize what is going on.

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MINDFULNESS 4: If something does not make sense then do not explain it away.

  1. Observe as usual. Notice the environment and the people in it.

  2. Start applying the aspects of mindfulness that you have learned so far.

  3. As you observe, see if there is something that does not make sense.

Example: Some people literally view God as a person who has created this universe. However, a person has a form that occupies space. That form and space of a person are also things that are created as part of the universe. So, God cannot be a person and also the creator of the universe. Reference: A Lovely Discussion on God

  1. Recognize it as an inconsistency; do not pretend that you understand it.

  2. If there is an impulse to explain it away then recognize that too.

  3. Look more closely and isolate exactly what it is that you do not understand.

  4. If it is an idea or concept then look it up in a dictionary, encyclopedia, or on Internet.

  5. If it is some contradiction or inconsistency, then check your assumptions. Be alert to what you might be taking for granted. Verify any assumptions.

  6. If it is a failure in life, then check if there is blame being placed somewhere. Blame pretends to be an answer, but it does not provide solution. Ignore all attempts at blame.

  7. If it is an unwanted condition, then check for explanations that are being used. Ignore all explanations that have not led to resolution in the past.

  8. Simply observe around what does not makes sense without explaining it away, until it starts to resolve.

  9. Expand your span of attention and let the perceptions pour in.

  10. Let the realizations present themselves to you without you making any effort.

  11. Use your own judgment as to when to end a session.

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Develop this exercise to a point where you may execute it even while interacting socially with others. Keep observing patiently without explaining things away and things will start to fall in place. Many things may come up naturally to be scrutinized. Simply observe and become aware of them without effort.

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Observe what is Missing

Missing
Reference: The 12 Aspects of Mindfulness

Mindfulness provides the discipline for looking and contemplation.

You may do this exercise while sipping coffee in a café, or strolling along a river. You may even find a place where you can sit comfortably for a while without being disturbed. Then patiently observe the world go by.

If something is missing then recognize that it is missing. Do not imagine something in its place. If someone asks you a question and no answer comes up in your mind, then do not feel obliged to make up an answer. Continue to look around that area where something seems to be missing. Maybe something interesting will come up, maybe not.

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MINDFULNESS 3: If something is missing do not imagine something else in its place. 

  1. Observe as usual. Notice the environment and the people in it.

  2. Start applying the aspects of mindfulness that you have learned so far.

  3. Notice something that is puzzling, and about which full understanding is missing.

    Note: If nothing puzzling comes up then make a list of people you know and locations you are familiar with. Look at the responses that come up in the mind to items on this list. Something puzzling may come to notice.

  4. Do not feel obliged to accept the explanations given; instead focus on what is puzzling.

  5. Carefully consider the broad context of the scene, and the purpose of the activity.

  6. Notice all those things that do not seem to be consistent and coherent.

  7. Closely examine each one of them, and also your viewpoint with respect to them.

  8. Do not imagine reasons to fill the uncomfortable gap. Instead continue to observe in and around the area. Maybe something interesting will come up to make it clear, or maybe not.

  9. Notice questions you have for which “answers” are missing, or which do not make sense.

  10. Notice the impulse to come up with an answer. If there is no answer then acknowledge the fact. Do not make up an answer.

  11. Look at the question in a broader context and continue looking at the areas around it. Maybe the answer will present itself, maybe not.

  12. Expand your span of attention and let the perceptions pour in.

  13. Let the realizations present themselves to you without you making any effort.

  14. Use your own judgment as to when to end a session.

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Develop this exercise to a point where you are doing it naturally even while interacting socially with others. Keep observing patiently following up on what is inconsistent and incoherent until things start to fall in place. Many things may come up naturally to be scrutinized. Simply observe and become aware of them without effort.

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Observe without Assuming

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Reference: The 12 Aspects of Mindfulness

Mindfulness provides the discipline for looking and contemplation.

You may do this exercise while sipping coffee in a café, or strolling along a river. You may even find a place where you can sit comfortably for a while without being disturbed. Then patiently observe the world go by.

As you look do not assume anything. It is easy to assume what one normally expects to be there. For example, if you are looking at the profile of a person, you see only one ear, but you may take it for granted that the person has two ears. Separate the actual perception from the ideas of what should be there.

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MINDFULNESS 2: Observe things as they are, without assuming anything.

  1. Observe as usual. Notice the environment and other people.

  2. Start applying the aspects of mindfulness that you have learned so far.

  3. As you observe, see if there are any assumptions that are hanging around. Spot them as they come up.

  4. Don’t suppress them. Simply become aware of them, and then move on.

  5. As you observe, see if the actual perceptions are different from the ideas in the mind. Notice such differences one by one.

    Example:  You are looking at the profile of a person. You see only one ear. The mind tells you that he is supposed to have two ears. Notice that you cannot verify what the mind is telling you. Taking for granted that the person has two ears, would be an assumption.

  6. Don’t suppress them. Simply become aware of them, and then move on.

  7. As you observe, see if you are being judgmental about some situation. Notice any preconceived ideas present one by one.

    Example: You may look at a person of certain sex, color, profession or cultural background with certain preconceived ideas. Separate the actual perception from the ideas contained in the mind.

  8. Don’t suppress them. Simply become aware of them, and then move on.

  9. As you observe, see if there is something that does not make sense. Notice if your own ideas are contributing to that inconsistency.

  10. Don’t suppress them. Simply become aware of them, and then move on.

  11. Expand your span of attention and let the perceptions pour in.

  12. Let the realizations present themselves to you without you making any effort.

  13. Use your own judgment as to when to end a session.

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Develop this exercise to a point where you are doing it naturally even while interacting socially with others. Keep observing patiently without assuming anything. Keep actual perception separate from the ideas contained in your mind. Many things may come up naturally to be scrutinized. Simply observe them and become aware of them without effort.

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