Working with Fractions


The standardized tests – the likes of GRE, SAT, GMAT, TOEFFL etc. – exist in various countries all over the world. Over he years, the have proved themselves as one of the better tools to assess one’s ability to successfully complete/continue his or her education towards the first and after that – higher degrees.

Despite many disadvantages, constantly repeated by the antagonists, multiple choice time limited tests have a specific target. Those exams try to estimate our ability to cope with fairly large amount of new information in sort period of time. In short – examine our ability to analyze data quickly. That’s why the questions present in those exams are many times “inconvential” – poses in a way that makes hard to utilize our knowledge and techniques assessed throughout the process of formal school education. Thus, although almost everyone owns the knowledge required to solve the exams questions, most of examinees find it difficult to apply it.

Even the simplest tasks all of a sudden become difficult to cope with – add the time factor to this, as the time limit is, in fact, pretty tense – and you get the far from perfect marks for many students.

Fractions and percents have never been a favorite subject for most of us in math classes. Solid numbers are so much easier to deal with… And in the GMAT you can see calculations like this:

0.5 + 1/5 + 1/10 + 0.175 + 0.5*5 – 3/5 = ?

The answers are not really relevant here (although from a quick look, one should expect the result to be slightly below 3) – as we are talking about general approach. There are fractions and there are decimals – how should we cope with that? Well, it is pretty obvious that we should decide either on working with “regular” fractions or with decimal ones. My proposal is when most of the calculation is addition and subtraction – work with decimals. When multiplication and division prevail – switch to regular fractions.
In the question above we have only one multiplication (which is easily done and gives 0.5*5 = 2.5) and many additions with one subtraction. Let’s convert the regulars to decimals:

1/5 = 2/10 = 0.2; 1/10 = 0.1; 3/5 = 6/10 = 0.6. No we have:

0.5 + 0.2 + 0.1 + 0.175 + 2.5 – 0.6 = ?
0.5 + 0.1 – 0.6 = 0, thus we get 0.2 + 0.175 + 2.5 = 2.875. If you want an answer in “regular” form, you should know that 0.875 = 7/8, thus the result is 2 7/8.

Here us another example

(1/3 * 6/5) + (1.5 * 2 2/3) – (15/3/(2/0.5)) – (-4/3 * 1.25) + 1/3 = ?

Although there are many additions, I would recommend turning everything into regular fractions as the fist stage:

1.5 = 3/2; 2 2/3 = 8/3; 0.5= ½; 1.25 = 1 ¼ = 5/4.

Now the multiplications and divisions are relatively easy:

1/3 * 6/5 = 6/(5*3) = 2/5
3/2 * 8/3 = 8/2 = 4
15/3/(2/1/2) = 15/3/4 = 5/4
-4/3 * 5/4 = -5/3

2/5 + 4 – 5/4 – (-5/3) + 1/3 = 4 + 2/5-5/4 + 6/3 = 6 + 2/5 – 5/4 = 5 + 2/5 – 1/4.
Here you can either convert into decimals (5+ 0.4 – 0.25 = 5.15) or calculate 2/5 – ¼, which is 8/20-5/20=3/20; giving 5 3/20 as a result.

Here are some decimal/regular conversions that are advisable to remember when studying and training for the standardized test:

½ = 0.5
¼ = 0.25
¾ = 0.75
1/8 = 0.125
3/8 = 0.375
5/8 = 0.625
7/8 = 0.875
1/5 = 0.2
2/5 = 0.4
3/5 = 0.6
4/5 = 0.8
1/20 = 0.05
3/20 = 0.15
7/20 = 0.35
9/20 = 0.45
11/20 = 0.55
13/20 = 0.65
17/20 = 0.85
19/20 = 0.95

There is an important thing that should be mentioned regarding the conversion into decimals. ONLY fractions that have their denominator “constructed” form 2 and 5 (and their multiplications) can be converted into decimal fraction of finite length. For example, 7/50 can be presented as decimal, as 50 is 2*5*5. And 7/60 cannot be converted to finite decimal, as 60 is 2*2*5*3. Remember that when deciding on “which way” to go – decimal fractions or regular fractions.


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