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How to Learn Arithmetic Reasoning

Arithmetic Reasoning is a part of mathematics which deals with the number sequence, mathematical operators, ratio and proportion, percentage, power and roots, sets and probability. The application of arithmetic reasoning is found in our day to day life activity like calculating the total amount of expenditure, a percentage of monthly income, to find the area of a land,etc...The common arithmetic operators we use in problem are +,-,*,/, =. 

Arithmetic Reasoning Formula:

        SI = $\frac{(PTR)}{100}$

Speed = $\frac{(distance\ traveled)}{(time\ taken)}$  

The ratio a to b is written as a: b

average of numbers or mean = $\frac{(sum\ of\ all\ the\ numbers)}{( number\ of\ terms)}$  

Arithmetic Reasoning Problems:

1. Solve for a
          2a + 10 = 20

Solution: To find the value of a  first add -10 to both side of the equation.
                2a + 10 - 10 = 20 - 10
2a = 10
Divide both side of the equation by 2

$\frac{2a}{2}$ = $\frac{10}{2}$

       a = 5

2. Convert the decimal number 0.575 to fraction

Solution: Multiply and divide the decimal number by 1000 to get the number in fraction form.

0.575 x $\frac{1000}{1000}$ = $\frac{585}{1000}$  

Reduce the fraction to it’s simplest form.

           $\frac{575}{1000}$ = $\frac{23}{40}$

3. Simplify 5 + [3 - {2 * 4} + 8 - 2]

Solution: The given expression is solved by making use of BODMAS rule.
               The terms inside the brackets are simplified first.
                    = 5 + [3 - {8} + 8 - 2]
               Do the addition and subtraction from left to right.
                    = 5 + [3 - 8 + 8 - 2]
                    = 5 + [3 - 2]
                    = 5 + [1] 
                    = 6

4. Subtract the given Integer Number:
          $\frac{3}{5}$ - $\frac{4}{7}$

Solution: A rational number can be subtracted from the other number by adding its negative to the given first rotational number.

                        = $\frac{3}{5}$ + (- $\frac{4}{7}$)

               To add the terms first convert the number with equivalent rational numbers with a common denominator and then do the                     addition.
       The L.C.M of the denominator is 5 and 7 is 35.
       covert the rational number to its equivalent number by making use of the L.C.M

               $\frac{3}{5}$ = $\frac{3*7}{5*7}$ = $\frac{21}{35}$

               $\frac{4}{7}$ = $\frac{4*5}{7*5}$ = $\frac{20}{35}$

               Multiply of a negative term by a positive term results in negative

               = $\frac{21}{35}$ - $\frac{20}{35}$

               = $\frac{1}{35}$

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From Wikipedia

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.


  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.

From Yahoo Answers

Question:I am searching the WWW but I can't find what I need. I need a site that breaks down how to slove this arithmetic reasoning problems found on the Officer Aptitude Rating exam given by the navy to Qualify for OCS. I'm math eliterate!!!!!

Answers:Well, I was unable to find a site as well. I do have a suggestion.... In doing research, I came across the description of the math portion of the exam.... "The math skills assessed by the ASTB subtests include arithmetic and algebra, with some geometry. The assessments include both equations and word problems. Some items require solving for variables, others are time and distance problems, and some require the estimation of simple probabilities. Skills assessed include basic arithmetic operations, solving for variables, fractions, roots, exponents, and the calculation of angles, areas, and perimeters of geometric shapes." Given each of these topics, maybe http://www.math.com will be a good place to start, looking under algebra and geometry primarily. From browsing the site, it looks like it provides enough information necessary to help you learn the steps needed to work most problems on the exam. Hope this was of some help to you. Best wishes on your exam.

Question:I've been trying to convince my parents to let me do online high school and they just wont give in! I have solid reasons as to why I want to study at home and I'm wondering why they're so stubborn about it. If you were a parent, would you think that these are good reasons to let your child do online schooling? - I'm originally from California but my family moved to Switzerland. The school system is so different here and they focus on shoving French down my throat before any other subject. I've studied at the public school here for a year and counting, and they teach me things that I already know. I don't feel like I'm up-to-par with the things the kids my age are studying back in the US. For example: Before I left, I was in the 8th grade and in Algebra. I actually felt challenged in all of my classes. When I got here, the teacher was teaching our class how to add fractions. - I also don't want to end up like my brother, who was told that he can't go to "college" (the high school here) if he doesn't perfect his French by June. My brother is supposed to be in his senior year of high school in the US, yet they won't even let him start the first year of "college" here because he can't speak French. I feel like I'm cracking under the pressure to learn this language and not have to repeat grades. - I'm harassed at school on a daily basis to the point where I can't even go through the same hallways anymore. I've changed my routes to all my classes just to avoid being bullied. I feel THAT threatened at school. Because of the bullying, I constantly feel stressed and scared. I can't defend myself either because I still have a ton of French to learn and I'll end up looking like even more of an idiot. I'm being reasonable, right? It's not like I want to do online schooling just to sit at home and rot. I feel like it's the best thing to do if I want to stay sane. The only problem is that my mom is extremely old-fashioned and thinks that anything out of the ordinary should be shunned by society! She doesn't realize how bad of an influence public school has on me because I'm so good at controlling myself at home.

Answers:You can easily compare info about these schools in this site - schools.iblogger.org

Question:how does it benefit us to know other peoples learning styles?

Answers:Numero Uno: If I know your preferred way of learning, then I can adjust my training/teaching in such a way to make it easier for you to learn

Question:Scientists can now determine the complete DNA sequences of organisms, including humans. Now that this milestone has been reached, is there a reason to continue to learning about Mendel, alleles, and inheritance patterns.

Answers:Just because you have a few million base pairs of code doesn't mean you have a clue about how the genes are regulated and interact in order to fashion an organism. When you selectively breed and make crosses you can study the interactions of combinations of alleles. Basic introduction to mendelian genetics shows you a maximum of three or four gene interactions with no linkage but an organism is the cascading series of interactions of thousands of genes. Looking at restricted breeding experiments can give insight in how the the allelic combinations respond. This is done to link a variation in phenotype with actual genotypes. This often how specific desirable alleles that influence predisposition to disease resistance are found. http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.genet.40.110405.090511?journalCode=genet

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