Mastering Fractions & Decimals for GRE test

The fractions and decimals of GRE quantitative section is to be formulated to adept by means of strategic reasoning and understanding to command over the stipulated problems with concise precision having intuitive sense to save the time with a view to scoring a smart score.

Fractions: Notes to Remember

• A fraction denotes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. There is an infinite number of fractions between any two integers. There's an infinity of fractions between 1 and 2 and so on.
• Suppose we have the fraction X over Y, The top part, the upstairs of a fraction is the numerator. This fraction has a numerator of X. It also denotes that X/Y as X is divided by Y. Y is the denominator. Let us consider X=4, Y= 5. So, it means 4/5 as a fraction denotes 4 is divided by 5 or four-sevenths of a certain reference. If the hypothetical pie chart is cut into five equal pieces, then four seventh means four of those five pieces. This perception gives a vivacious visualization. It aids to do math in brain quickly and to excel the level of understanding simultaneously.
• The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

 Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents, e.g: as in 0.01, 1%, and 10⁻²respectively, all of which are equivalent to 1/100.Other uses for fractions are to represent ratios and to represent division.

• When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger. One way to compare fractions with different numerators and denominators is to find a common denominator.
• Two fractions are equivalent if they have the same numerical value even though they have different numerators and different denominators. So, two-thirds equals ten fifteenths. Those fractions are equal. In other words, the numerator and the denominator of the original fraction, both got multiplied by the same factor. 
• The GRE multiple choice gives the answer choices usually in simplest form. So, it is necessary to simplify the fraction to the simplest form in order to match the choices. On the numerical entry, one can enter a non-simplified fraction, as long as it fits in the box. 

 

Fractions: Values to Remember

The stipulated fractions with values in decimals and percentage are recommended to remember to provide a fast and quick excel in dealing with time of GRE quantitative reasoning.

Fraction	Decimal	Percent
1/100	0.01	1%
1/50	0.02	2%
1/25	0.04	4%
1/20	0.05	5%
1/10	0.1	10%
1/9	0.1111111…	11.11%
1/8	0.125	12.5%
9/8 (1+1/8)	1.125	1.125x100%
1/6	0.1666666….	16.67%
1/5	0.2	20%
1/4	0.25	25%
1/3	0.3333333…. = 3.3̅3̅3	33.33%
2/3	0.6666666….	66.67%
2/5	0.4	40%
1/2	0.5	50%
3/5	0.6	60%
3/4	0.75	75%
4/5	0.8	80%

For example, if one asked to detect the equivalent of \(\frac{(0.6666)\times(1.125)}{(0.3333)\times(0.75)} \) then it needs time to set each digit in calculator or doing it on scratch paper rather we can do it in this way using the table above:\(\frac{(0.6666)\times(1.125)}{(0.3333)\times(0.75) }= \frac{2/3}{1/3}\times \frac{9/8}{3/4}=3\) here 1.125= 1+0.125= 1+ 1/8 = 9/8

 

Forms of Fractions

There are various forms of fractions to appear in the test:

• Simple Fractions

A simple fraction is a rational number can be written as a/b where a and b are both integers. As with other fractions, the denominator (b) cannot be zero. Simple fractions can be positive or negative, proper, or improper. Compound fractions, complex fractions, mixed numerals, and decimals are not simple fractions, though, unless irrational, they can be evaluated to a simple fraction.

• Proper and Improper Fractions
  • Simple fractions can be classified as either proper or improper. In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is if the fraction is greater than −1 and less than 1.
  • It is said to be an improper fraction, or sometimes top-heavy fraction if the absolute value of the fraction is greater than or equal to 1.
  • Examples of proper fractions are 2/5, –3/4, and 16/36; examples of improper fractions are 9/7, –5/3, and 2/2.

 

• Mixed Fractions

A mixed fraction is the sum of a non-zero integer and a proper fraction. This sum is implied without the use of any visible operator such as "+". For example, in referring to two entire cakes and three-quarters of another cake, the whole and fractional parts of the number are written next to each other as \(2+\frac{3}{4}=2\frac{3}{4}\) 

• Complex Fractions

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example \(\frac{12\frac{3}{4}}{26}, \frac{\frac{3}{4}}{\frac{1}{8}}\)

• Compound Fractions

A compound fraction is a fraction of a fraction, or any number of fractions connected with the corresponding multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication. For example, 3/4 of 7/9 is a compound fraction, corresponding to (3/4)*(7/9) = 21/36 

• Comparing the Fractions

In comparing the fractions they may be greater than or less than or equal with respect to each other in GRE quantitative section. We can have this in our knowledge to compare them:

• If we increase numerator then the value of fractions increase and vice versa while the denominator is unchanged.
• If we increase denominator then the value decrease and vice versa when the numerator is unchanged.
• We can simplify or modify the fraction given at out ease by simply performing addition or multiplication with the same positive number on both numerator and denominator. It is more helpful to make the denominator same. If 3/4 & 7/9 are to be compared then we can do (3*9)/(4*9) = 27/36 & (7*4)/(9*4)  = 28/36and we know the fraction is greater which numerator is higher.
• Cross multiplying for positive numbers cross multiplying does not change the numbers of the inequality, i.e. if  3/4  & 7/9   are to be compared then if we perform cross multiplying then we get 27 & 28 in the same position or column of quantitative comparison which shows 27<28 and so  3/4 < 7/9 

We don't know, so we're just writing double question marks. Now simply cross multiply. 

So we get seven over eight on one side, five, five times 11 on the other side, seven times eight is 56, five times 11 is 55. Of course, 56 is larger so of course. The inequality has to point in the same direction before we cross multiply. , and therefore this means that 7 over 11 has to be bigger than 5 over 8. 

Decimals

• The numbers that are represented by decimal numeral are the decimal fractions that is, the rational numbers that may be expressed as a fraction, the denominator of which is a power of ten, e.g.: 0.8= 8/10, 15.87 = 1587/100. When decimals are rational because they are ratios. We write a fraction, a rational number, as a decimal. One of two things happens, it either terminates or repeats as shown in the table above.
• There are also decimal-irrational numbers e.g.: \(\pi\) which cannot be expressed as a fraction with exact values where the digits after the decimal point is not fixed.
• Digits can be placed to the left or right of a decimal point, to show values greater than one or less than one. The decimal point is the most important part of a Decimal Number. The numbers that may be represented in the decimal system are the decimal fractions, that is the fractions of the form a/10ⁿ, where a is an integer, and n is a non-negative integer.
•  Decimal is the term most often used to indicate that a number is not a whole number. For example, in the decimal number 5.32, the decimal component is ‘.32’, while the whole number component is ‘5’. All numbers can be written as decimals: the number 4 is actually 4.0, and can also be written as 4.00, 4.000, 4.000000, etc.

Fraction to Decimal

If one needs to convert a fraction to a decimal, he/she needs to simply divide the denominator of the fraction into the numerator. If both the numerator and denominator are even, then we'll be able to convert the fraction into a non-repeating number. Otherwise, we cannot fully divide the denominator into the numerator, and one has to realize that the resulting decimal number is a repeating decimal. For example, the decimal value of four divided by five, 4/5 = 0.8.

• Fraction to Decimal

In order to convert a decimal into a fraction, the numbers to the right of the decimal place make up the numerator of the resulting fraction. The number of decimal places is to be counted to the right of the decimal place. If there are two decimal places to the right of the decimal place, then create the fraction by placing the number without the decimal point over 100. If there are three decimal places to the right of the decimal place, create the fraction by placing the number without a decimal point over 1000, e.g.: 15.768 = 15678/1000.

• Addition & Subtraction of Decimals

In order to add or subtract decimals, line up the decimal points which needed to add or subtract and then required operation is to be performed. It may be needed to write zeros in the places where such a 0 is implied and not really written. For example, 4.61+8.650=13.260or 13.26.

	4	.	610
+	8	.	650
	13	.	260

• Multiplication of Decimals

In order to multiply decimals, the two numbers are to be multiplied as usual and then to be written the decimal point in the answer based on the total number of decimal places in the original decimal numbers. For example, if we need to multiply 4.3 and 8.2, the multiplication of 43 times 82, which gives us 3526, and the total number of decimal places is two, one from 4.3 and one from 8.2, so adding two decimal places to the answer we get 35.26.

• Division of Decimals

In order to divide two decimals, one needs to set up a division problem as usual to count the number of decimal places in the divisor. Then the decimal points of both numbers is to be moved to the right with a view to getting the answer.   

For example, we can consider the following example:

9.895 ÷ 5.67

The divisor (5.67) has two decimal places. Move the decimal point of both numbers two places to the right:

9.895 ÷ 5.67
989.5 ÷ 567.0
= 1.745