Fractions
What are Fractions?
A fraction represents a numerical value, which defines the parts of a whole.
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The basics of fractions explain the top and bottom numbers of a fraction. The top number represents the number of selected or shaded parts of a whole whereas the bottom number represents the total number of parts.
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Parts of Fractions
The fractions include two parts, numerator and denominator.
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Numerator: It is the upper part of the fraction, that represents the sections of the fraction
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Denominator: It is the lower or bottom part that represents the total parts in which the fraction is divided.
Example: If 5/8 is a fraction, then 5 is the numerator and 8 is the denominator.
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Properties of Fractions
Similar to real numbers and whole numbers, a fractional number also holds some of the important properties. They are:
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Commutative and associative properties hold true for fractional addition and multiplication.
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The identity element of fractional addition is 0, and fractional multiplication is 1.
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The multiplicative inverse of a/b is b/a, where a and b should be non-zero elements.
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Fractional numbers obey the distributive property of multiplication over addition.
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Types of Fractions
Based on the properties of numerator and denominator, fractions are sub-divided into different types. They are:
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Proper fractions
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Improper fractions
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Mixed fractions
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Like fractions
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Unlike fractions
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Equivalent fractions
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How to Convert Fractions to Decimals?
As we already learned enough about fractions, which are part of a whole. The decimals are the numbers expressed in a decimal form which represents fractions, after division.
For example, Fraction 1/2 can be written in decimal form as 0.5.
The best part of decimals are they can be easily used for any arithmetic operations such as addition, subtraction, etc. Whereas it is difficult sometimes to perform operations on fractions. Let us take an example to understand;
Example:
Add 1/6 and 1/4.
solution:
1/6 = 0.17 and 1/4 = 0.25
Hence,
on adding 0.17 and 0.25,
we get,
0.17 + 0.25 = 0.42
How to Simplify Fractions?
To simplify the fractions easily, first, write the factors of both numerator and denominator. Then find the largest factor that is common for both numerator and denominator. Then divide both the numerator and the denominator by the greatest common factor (GCF) to get the reduced fraction, which is the simplest form of the given fraction. Now, let us consider an example to find the simplest fraction for the given fraction.
For example, take the fraction, 16/48
So, the factors of 16 are 1, 2, 4, 8, 16.
Similarly, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Thus, the greatest common factor for 16 and 48 is 16.
i.e., GCF (16, 48) = 16.
Now,
divide both the numerator and denominator of the given fraction by 16, we get
16/48 = (16/16) / (48/16) = 1/3.
Hence, the simplest form of the fraction 16/48 is 1/3.
Decimal Fractions:
Fractions in which denominators are powers of 10 are known as decimal fractions.
Thus ,1/10=1 tenth=.1.
1/100=1 hundredth =.01.
99/100=99 hundredths=.99;
7/1000= 7 thousandths =.007,etc
Conversion of a Decimal into Vulgar Fraction:
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
Thus, 0.25=25/100=1/4.
2.008=2008/1000=251/125.
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Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
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If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign. Thus, 1.84/2.99 = 184/299 = 8/13; 0.365/0.584 = 365/584=5.
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Operations on Decimal Fractions:
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Addition and Subtraction of Decimal Fractions: The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
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Multiplication of a Decimal Fraction by a Power of 10: Shift the decimal point to the right by as many places as is the power of 10. Thus, 5.9632 x 100 = 596,32; 0.073 x 10000 = 0.0730 x 10000 = 730.
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Multiplication of Decimal Fractions: Multiply the given numbers considering them without the decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers. Suppose we have to find the product (.2 x .02 x .002). Now, 2x2x2 = 8. Sum of decimal places = (1 + 2 + 3) = 6. .2 x .02 x .002 = .000008.
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Dividing a Decimal Fraction by a Counting Number: Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend. Suppose we have to find the quotient (0.0204 + 17). Now, 204 ^ 17 = 12. Dividend contains 4 places of decimal. So, 0.0204 + 17 = 0.0012.
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Dividing a Decimal Fraction by a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number. Now, proceed as above. Thus, 0.00066/0.11 = (0.00066*100)/(0.11*100) = (0.066/11) = 0.006V
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Comparison of Fractions:
Suppose some fractions are to be arranged in ascending or descending order of magnitude. Then, convert each one of the given fractions in the decimal form, and arrange them accordingly.
Suppose we have to arrange the fractions 3/5, 6/7 and 7/9 in descending order.
now, 3/5=0.6,6/7 = 0.857,7/9 = 0.777....
since 0.857>0.777...>0.6, so 6/7>7/9>3/5
Recurring Decimal:
If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal. In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set______
Thus 1/3 = 0.3333…. = 0.3; 22 /7 = 3.142857142857...... = 3.142857
Pure Recurring Decimal:
A decimal fraction in which all the figures after the decimal point are repeated, is called a pure recurring decimal.
Converting a Pure Recurring Decimal into Vulgar Fraction:
Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.
Thus ,0.5 = 5/9; 0.53 = 53/59 ;0.067 = 67/999; etc...
Mixed Recurring Decimal:
A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
e.g., 0.17333 = 0.173.
Converting a Mixed Recurring Decimal into Vulgar Fraction:
In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated, In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.
Thus 0.16 = (16-1) / 90 = 15/19 = 1/6; ____
0.2273 = (2273 – 22)/9900 = 2251/9900
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